text:
\section*{Proof for Theorem~\ref{thm:lower_bound}}
\begin{proof}
The singular values of $O(p_{t_l},u_{t_l})$ can be found as the square-root of the eigenvalues of the \emph{symmetric observability matrix},  $\mathbb{O}(p_{t_l},u_{t_l})$, given as~\cite{Strang1976Linear},
\begin{eqnarray}
\mathbb{O}(p_{t_l},u_{t_l}) &=&O^{T}(p_{t_l},u_{t_l})O(p_{t_l},u_{t_l})\nonumber\\
&=&O^{T}(p_{t_l})O(p_{t_l})+O^{T}(u_{t_l})O(u_{t_l}).
\label{sy_ob_matrix}
\end{eqnarray}
\begin{equation}
\left\{
                \begin{array}{ll}
                  \sqrt{{\lambda_{\min}(\mathbb{O}(p_{t_l},u_{t_l}))}}={\sigma_{\min}(O(p_{t_l},u_{t_l}))},\\
                 \sqrt{{\lambda_{\max}(\mathbb{O}(p_{t_l},u_{t_l}))}}={\sigma_{\max}(O(p_{t_l},u_{t_l}))}.
                \end{array}
              \right.
\label{eqn:min/max_singualr_observability_metrix}
\end{equation}

We can use \emph{Weyl and dual Weyl} inequalities to bound the singular values. For Hermitian matrices $X$ and $Y$ with $r$ eigenvalues written in increasing order $\lambda_1(X)\leq \lambda_2(X)\leq\ldots\leq\lambda_r(X)$ and $\lambda_1(Y)\leq \lambda_2(Y)\leq\ldots\leq\lambda_r(Y)$, respectively, the \emph{Weyl inequalities}~\cite{franklin2012matrix} is given by,
\begin{equation}
\lambda_{i+j-1}(X+Y)\geq \lambda_{i}(X)+\lambda_{j}(Y)
\label{eqn:Weyl_inequalities}
\end{equation}
where $i,j\geq 1$ and $i+j-1\leq r$. Similarly, the \emph{dual Weyl inequalities} is given by
\begin{equation}
\lambda_{i+j-r}(X+Y)\leq \lambda_{i}(X)+\lambda_{j}(Y)
\label{eqn:Weyl_inequalities}
\end{equation}
where $i,j\geq 1$ and $i+j-r\leq r$.

Since $\mathbb{O}(p_{t_l},u_{t_l})\in \mathbb{R}^{2\times 2}$, $O^{T}(p_{t_l})O(p_{t_l})\in \mathbb{R}^{2\times 2}$ and $O^{T}(u_{t_l})O(u_{t_l})\in \mathbb{R}^{2\times 2}$ are symmetric matrices, they are Hermitian with the eigenvalues (in ascending order) as $\lambda_1(\mathbb{O}(p_{t_l},u_{t_l}))\leq \lambda_2(\mathbb{O}(p_{t_l},u_{t_l}))$, $\lambda_1(O^{T}(p_{t_l})O(p_{t_l}))\leq \lambda_2(O^{T}(p_{t_l})O(p_{t_l}))$ and $\lambda_1(O^{T}(u_{t_l})O(u_{t_l}))\leq \lambda_2(O^{T}(u_{t_l})O(u_{t_l}))$. Following the \emph{Weyl and dual Weyl} inequalities, we get
\begin{align}
&\lambda_1(\mathbb{O}(p_{t_l},u_{t_l})) \geq \lambda_1(O^{T}(p_{t_l})O(p_{t_l})) + \lambda_1(O^{T}(u_{t_l})O(u_{t_l})),\nonumber\\
&\lambda_2(\mathbb{O}(p_{t_l},u_{t_l})) \leq
\lambda_2(O^{T}(p_{t_l})O(p_{t_l})) + \lambda_2(O^{T}(u_{t_l})O(u_{t_l})).
\label{eqn:Weyl_eigen_four}
\end{align}
Thus,
\begin{align}
&\frac{\lambda_{1}(\mathbb{O}(p_{t_l},u_{t_l}))}{\lambda_{2}(\mathbb{O}(p_{t_l},u_{t_l}))}\geq
\nonumber\\
&\frac{\lambda_1(O^{T}(p_{t_l})O(p_{t_l})) + \lambda_1(O^{T}(u_{t_l})O(u_{t_l}))}{\lambda_2(O^{T}(p_{t_l})O(p_{t_l})) + \lambda_2(O^{T}(u_{t_l})O(u_{t_l}))}.
\label{eqn:lower_bound_inequality}
\end{align}
Then, from Equation \ref{eqn:min/max_singualr_observability_metrix} and Equation \ref{eqn:lower_bound_inequality},  the inverse of the condition number of the local nonlinear observability matrix,
\begin{align}
\label{eqn:lower_bound_inversecond}
&C^{-1}(O(p_{t_l},u_{t_l}))=\sqrt{\frac{\lambda_{1}(\mathbb{O}(p_{t_l},u_{t_l}))}{\lambda_{2}(\mathbb{O}(p_{t_l},u_{t_l}))}}\nonumber\\
&\geq \sqrt{\frac{\lambda_1(O^{T}(p_{t_l})O(p_{t_l})) + \lambda_1(O^{T}(u_{t_l})O(u_{t_l}))}{\lambda_2(O^{T}(p_{t_l})O(p_{t_l})) + \lambda_2(O^{T}(u_{t_l})O(u_{t_l}))}}.
\end{align}
\revo{Thus, we have the lower of the inverse of the condition number, \begin{equation}
    \sqrt{\frac{\lambda_1(O^{T}(p_{t_l})O(p_{t_l})) + \lambda_1(O^{T}(u_{t_l})O(u_{t_l}))}{\lambda_2(O^{T}(p_{t_l})O(p_{t_l})) + \lambda_2(O^{T}(u_{t_l})O(u_{t_l}))}}.
    \label{eqn:lowerbound_inverse}
\end{equation}}
By calculating the eigenvalues of symmetric matrix of target's control contribution,
$$O^{T}(u_{t_l})O(u_{t_l})=\begin{bmatrix}
     u_{lx}^{2} & u_{lx}u_{ly}\\
    u_{ly}u_{lx} & u_{ly}^{2}
\end{bmatrix},$$
we have,
\begin{eqnarray}
&&\lambda_{1}(O^{T}(u_{t_l})O(u_{t_l}))=0\nonumber\\
&&\lambda_{2}(O^{T}(u_{t_l})O(u_{t_l}))=u_{lx}^{2}+u_{ly}^{2}=u_{t_l}^{2}.
\label{eqn:max_min_eigenu}
\end{eqnarray}
\noindent Then the lower bound of $C^{-1}(O(o,u_o))$ \revo{(Equation~\ref{eqn:lowerbound_inverse})} is calculated as  \begin{eqnarray}
\underline{C}^{-1}(O(p_{t_l},u_{t_l})) &=&  \sqrt{\frac{\lambda_1(O^{T}(p_{t_l})O(p_{t_l}))}{\lambda_2(O^{T}(p_{t_l})O(p_{t_l})) + u_{t_l}^{2}}}\nonumber\\
&=& \frac{\sigma_{\min}(O(p_{t_l}))}{\sqrt{\sigma^{2}_{\max}(O(p_{t_l}))+ u_{t_l}^{2}}}
\label{eqn:inverse_lower_bound_simply}
\end{eqnarray}
Equation~\ref{eqn:inverse_lower_bound_simply} gives the main lower bound. Note that $C^{-1}(O(p_{t_l},u_{t_l}))$ cannot be determined since target's control input, $u_{t_l}$, is unknown. However, we know that $||u_{t_l}||_2 \leq u_{t_l,\max}$. Therefore,
\begin{equation}
\underline{C}^{-1}(O(p_{t_l},u_{t_l})) \geq \frac{\sigma_{\min}(O(p_{t_l}))}{\sqrt{\sigma^{2}_{\max}(O(p_{t_l}))+ u_{t_l,\max}^{2}}}
\label{eqn:inverse_lower_bound_withumax}
\end{equation}
This yields our main lower bound result.
\label{proof:theorem1}
\end{proof}
\section*{Proof for \revo{Corollary~\ref{cor:sensor_n_1}}}
\begin{proof}
The local observability matrix for one-sensor-target, $s_i-t_l$ system can be derived from Equation \ref{eqn:new_multi_sensor_target} as,
\begin{equation}
O_{i}(p_{t_l},u_{t_l})=\begin{bmatrix}
    x_{t_l}-x_{s_i} & y_{t_l}-y_{s_i} \\
     u_{lx} & u_{ly}
\end{bmatrix}.
\label{eqn:observability_i_o}
\end{equation}
The sensor-target relative state contribution is
\begin{equation}
  O_{i}(p_{t_l})=\begin{bmatrix}
    x_{t_l}-x_{s_i} & y_{t_l}-y_{s_i}
\end{bmatrix}.
\label{eqn:sensor_tar_realtive}
\end{equation}
The $s_i-t_l$ system is weakly locally observable if $O_{i}(p_{t_l},u_{t_l})$ has full column rank, i.e., $(x_{t_l}-x_{s_i})u_{ly} \neq (y_{t_l}-y_{s_i})u_{lx}$. However, the sensor does not know the target's control input, $u_{t_l}$.

Given the symmetric matrix by sensor-target relative state contribution of $s_i-t_l$ system,
\begin{equation}
    O_{i}^{T}(p_{t_l})O_{i}(p_{t_l})=\begin{bmatrix}
    (x_{t_l}-x_{s_i})^{2},(x_{t_l}-x_{s_i})(y_{t_l}-y_{s_i}) \\
     (x_{t_l}-x_{s_i})(y_{t_l}-y_{s_i}), (y_{t_l}-y_{s_i})^{2}
\end{bmatrix},
\end{equation}
we have,
\begin{equation}
\left\{
                \begin{array}{ll}
                  {\sigma_{\min}(O_{i}(p_{t_l}))} =\sqrt{{\lambda_{\min}(O_{i}^{T}(p_{t_l})O_{i}(p_{t_l}))}}=0,\\
                 {\sigma_{\max}(O_{i}(p_{t_l}))} =\sqrt{{\lambda_{\max}(O_{i}^{T}(p_{t_l})O_{i}(p_{t_l}))}}\\
~~~~~~~~~~~~~~~=\sqrt{(x_{t_l}-x_{s_i})^{2}+(y_{t_l}-y_{s_i})^{2}}.
                \end{array}
              \right.
\label{eqn:min/max_singualr_observability_metrix_relative_state}
\end{equation}
Thus, from Equation \ref{eqn:inverse_lower_bound_simply}, the lower bound for $C^{-1}(O_{i}(p_{t_l},u_{t_l}))$ is $\frac{\sigma_{\min}(O_{i}(p_{t_l}))}{\sqrt{\sigma^{2}_{\max}(O_{i}(p_{t_l}))+ u_{t_l}^{2}}}=0$. Consequently, the lower bound cannot be controlled by the sensor.
\end{proof}
\section*{Proof for \revo{Corollary~\ref{cor:sensor_n_gre2}}}
\begin{proof} According to Equation~\ref{eqn:min/max_singualr_observability_metrix}, we have,
\begin{equation}
\frac{\sigma_{\min}(O(p_{t_l},u_{t_l}))}{\sigma_{\max}(O(p_{t_l},u_{t_l}))}=\frac{\sqrt{\lambda_{\min}(\mathbb{O}(p_{t_l},u_{t_l}))}}{\sqrt{\lambda_{\max}(\mathbb{O}(p_{t_l},u_{t_l}))}}.
\label{eqn:eigensquare_observability_metrix}
\end{equation}
Following Equations~\ref{eqn:lower_bound_inequality} and \ref{eqn:max_min_eigenu}, we have the lower bound for $\frac{\lambda_{\min}(\mathbb{O}(p_{t_l},u_{t_l}))}{\lambda_{\max}(\mathbb{O}(p_{t_l},u_{t_l}))}$, described as
\begin{eqnarray}
\frac{\lambda_{\min}(\mathbb{O}(p_{t_l},u_{t_l}))}{\lambda_{\max}(\mathbb{O}(p_{t_l},u_{t_l}))}\geq \frac{\lambda_{\min}(O^{T}(p_{t_l})O(p_{t_l}))}{\lambda_{\max}(O^{T}(p_{t_l})O(p_{t_l})) + u_{t_l}^{2}},
\label{eqn:upper_lower_bounds}
\end{eqnarray}
Now, we equivalently rewrite the statement of the theorem (using eigenvalues instead of singular values) as:
\emph{if}
\begin{equation}
\left\{
                \begin{array}{ll}
                 \frac{\lambda_{\min}(\revo{O^{'}}^T(p_{t_l})\revo{O^{'}}(p_{t_l}))}{\lambda_{\max}(\revo{O^{'}}^{T}(p_{t_l})\revo{O^{'}}(p_{t_l}))}\geq\frac{\lambda_{\min}(O^{T}(p_{t_l})O(p_{t_l}))}{\lambda_{\max}(O^{T}(p_{t_l})O(p_{t_l}))},\\
                \lambda_{\min}(\revo{O^{'}}^{T}(p_{t_l})\revo{O^{'}}(p_{t_l}))\geq \lambda_{\revo{\min}}(O^{T}(p_{t_l})O(p_{t_l}))
                \end{array}
              \right.
\label{eqn:assumption_singular_condition}
\end{equation}
\emph{then}
\begin{eqnarray}
&&\frac{\lambda_{\min}(\revo{O^{'}}^{T}(p_{t_l})\revo{O^{'}}(p_{t_l}))}{\lambda_{\max}(\revo{O^{'}}^{T}(p_{t_l})\revo{O^{'}}(p_{t_l})) + u_{t_l}^{2}} \nonumber\\
&&\geq \frac{\lambda_{\min}(O^{T}(p_{t_l})O(p_{t_l}))}{\lambda_{\max}(O^{T}(p_{t_l})O(p_{t_l})) + u_{t_l}^{2}}
\label{eqn:conslusion_singular_condition}
\end{eqnarray}
where the $\lambda$ and $\lambda'$ denotes the eigenvalues before and after the sensors increase $C^{-1}(O(p_{t_l}))$ and $\sigma_{\min}(O(p_{t_l})$, respectively.

We start with the left-hand side of Equation~\ref{eqn:conslusion_singular_condition} to get,
\begin{eqnarray}
&&\frac{\lambda_{\min,l}^{'}}{\lambda_{\max,l}^{'} + u_{t_l}^{2}}-\frac{\lambda_{\min,l}}{\lambda_{\max,l} + u_{t_l}^{2}}\nonumber\\
&=&\frac{\lambda_{\min,l}^{'}\lambda_{\max,l}-\lambda_{\max,l}^{'}\lambda_{\min,l}}{(\lambda_{\max,l}^{'} + u_{t_l}^{2})(\lambda_{\max,l} + u_{t_l}^{2})}\nonumber\\
&+&\frac{u_{t_l}^{2}(\lambda_{\min,l}^{'}-\lambda_{\min,l})}{(\lambda_{\max,l}^{'} + u_{t_l}^{2})(\lambda_{\max,l} + u_{t_l}^{2})}
\label{eqn:conclude_proof}
\end{eqnarray}
where $\lambda_{\min,l}^{'}$, $\lambda_{\max,l}^{'}$, $\lambda_{\min,l}$ and $\lambda_{\max,l}$ denote the simplified forms of $\lambda_{\min}(\revo{O^{'}}^{T}(p_{t_l})\revo{O^{'}}(p_{t_l}))$, $\lambda_{\max}(\revo{O^{'}}^{T}(p_{t_l})\revo{O^{'}}(p_{t_l}))$, $\lambda_{\min}(O^{T}(p_{t_l})O(p_{t_l}))$ and $\lambda_{\max}(O^{T}(p_{t_l})O(p_{t_l}))$, respectively. \revo{Note that Equation~\ref{eqn:assumption_singular_condition} can be reordered to match the numerator in the
second line of Equation~\ref{eqn:conclude_proof}. Then we have,}
\begin{equation}
\frac{\lambda_{\min,l}^{'}}{\lambda_{\max,l}^{'} + u_{t_l}^{2}}-\frac{\lambda_{\min,l}}{\lambda_{\max,l} + u_{t_l}^{2}} \geq 0.
\label{eqn:conclusion_proof}
\end{equation}
Hence the claim is proved.

\end{proof}
\begin{rem}
Equation~\ref{eqn:assumption_singular_condition} is sufficient, but not necessary condition to guarantee Equation~\ref{eqn:conslusion_singular_condition}. This is because Equation~\ref{eqn:conslusion_singular_condition} can be established with a weaker condition,
\begin{equation*}
\lambda_{\min,l}^{'}\lambda_{\max,l}-\lambda_{\max,l}^{'}\lambda_{\min,l}+u_{t_l}^{2}(\lambda_{\min,l}^{'}-\lambda_{\min,l})\geq 0.
\label{eqn:weaker_assumption*}
\end{equation*}
We choose the stricter condition (Equation~\ref{eqn:assumption_singular_condition}) because it is conceptually easy to separate and eliminate the influence on the degree of observability from target's control input, $u_{t_l}$, which is unknown and uncontrolled.
\label{remark:rem2}
\end{rem}
\rev{
\section*{Proof for Theorem~\ref{thm:submodular_observability_measure}}
We first give the proofs of the properties of the \emph{symmetric observability matrix by sensor-target relative state contribution}, $\mathbb{O}(p_{t_l})$. Then we obtain the similar result for the \emph{symmetric observability matrix}, $\mathbb{O}(p_{t_l}, u_{t_l})$ by a minor extension. For the target $t_l$, denote its sensor-target relative state vector as $r_{il} = [x_{tl}-x_{si}, y_{tl}-y_{si}], i\in\{1,2,\cdots, N\}$. Denote its ground sensor-target relative state set as $\mathcal{V}=\{r_{1l}, r_{2l}, \cdots, r_{Nl}\}$. For a given $\mathcal{W} \subseteq \mathcal{V}$, we form $R_{\mathcal{W}}=[R_{0l}, r_{wl}]^{T}$ with a (possibly empty) existing matrix $R_{0l}$ and the associated sensor-target relative vector $r_{wl}\in \mathcal{W}$. We calculate the associated symmetric observability matrix by sensor-target relative state contribution as $\mathbb{O}(p_{t_l})_{\mathcal{W}} = R_{wl}^{T}R_{wl}$. To \revo{simplify} notation, we write $\mathbb{O}(p_{t_l})_{\mathcal{W}}$ as $\mathbb{O}_{\mathcal{W}}$.
\begin{lem}
The set function mapping subsets $\mathcal{W} \subseteq V$ to the trace of the associated symmetric observability matrix by sensor-target relative state contribution, $f(\mathcal{W}) = \text{trace}(\mathbb{O}_{\mathcal{W}})$ is modular (submodular) and monotone increasing.
\label{lem_trace_modular}
\end{lem}
\begin{proof}
The symmetric observability matrix by sensor-target relative state contribution associated with $\mathcal{W}$,  $\mathbb{O}_{\mathcal{W}}$ can be calculated as
\begin{equation}
\mathbb{O}_{\mathcal{W}} = R_{wl}^{T}R_{wl} = \sum_{r_{wl}\in \mathcal{W}} r_{wl}^{T}r_{wl}.
\label{eqn:additivity_symmetric}
\end{equation}
Thus, for any $\mathcal{W} \subseteq V$, $\mathbb{O}_{\mathcal{W}}$ is simply a sum of the symmetric observability matrix by sensor-target relative state contribution associated with the individual rows of $R_{\mathcal{W}}$. Given the trace is a linear matrix function, we have
\begin{eqnarray}
f(\mathcal{W}) &=& \text{trace}(\mathbb{O}_{\mathcal{W}}) \nonumber\\
&=& \text{trace}{\sum_{r_{wl}\in \mathcal{W}} r_{wl}^{T}r_{wl}}\nonumber\\
&=&\sum_{r_{wl}\in \mathcal{W}} \text{trace}(r_{wl}^{T}r_{wl}).
\label{eqn:trace_fun}
\end{eqnarray}
If no sensors are assigned to the target $t_l$, define $trace(\emptyset) =0$. Then we have $f(\mathcal{W}) = \text{trace} (\mathbb{O}_{\mathcal{W}})$ is a modular (submodular) and monotone increasing set function.
\label{proof_trace_modular}
\end{proof}

\begin{lem}
The set function mapping subsets $\mathcal{W} \subseteq V$ to the rank of the associated symmetric observability matrix by sensor-target relative state contribution, $f(\mathcal{W}) = \text{rank}(\mathbb{O}_{\mathcal{W}})$ is submodular and monotone increasing.
\label{lem_rank_modular}
\end{lem}
\begin{proof}
Given two linear transformations $Q_1, Q_2 \in \mathbb{R}^{n\times n}$, we have
\begin{align}
&\text{rank}(Q_1+Q_2)\nonumber\\
&=\text{rank}(Q_1)+ \text{rank}(Q_2)\nonumber\\
&-\text{dim}(\text{range}(Q_1) \cap \text{range}(Q_2)).
\end{align}
\label{proof_rank_modular}
From~\cite{lovasz1983submodular}, we know that a set function $f: 2^{\mathcal{V}} \to \mathbb{R}$ is submodular if and only if the derived set functions $f_r : 2^{\mathcal{V}-\{r\}} \to \mathbb{R}$
$$f_r(\mathcal{W}) = f(\mathcal{W}\cup \{r\})-f(\mathcal{W}),$$ are monotone decreasing for all $r\in \mathcal{V}$. We can form the marginal gain functions for rank measure as
\begin{align}
&f_r(\mathcal{W})=\text{rank}(\mathbb{O}_{\mathcal{W}\cup{r}})-\text{rank}(\mathbb{O}_{\mathcal{W}})\nonumber\\
&=\text{rank}(\mathbb{O}_{r})-\text{dim}(\text{range}(\mathbb{O}_{\mathcal{W}}) \cap \text{range}(\mathbb{O}_{r})).
\end{align}
Note that, $\text{rank}(\mathbb{O}_{r})$ is a constant and $\text{dim}(\text{range}(\mathbb{O}_{\mathcal{W}}) )$ only increases with $\mathcal{W}$. Thus, \revo{$f_r$} is monotone decreasing which means $f(\mathcal{W}) = \text{rank}(\mathbb{Q}_{\mathcal{W}})$ is a submodular function. From the additivity property of the $\mathbb{O}_{\mathcal{W}}$ (Equation~\ref{eqn:additivity_symmetric}), it is clear that $f(\mathcal{W}) = \text{rank}(\mathbb{Q}_{\mathcal{W}})$ is monotone increasing.
\end{proof}

\begin{lem}
The set function mapping subsets $\mathcal{W} \subseteq V$ to the log det of the associated symmetric observability matrix by sensor-target relative state contribution, $f(\mathcal{W}) = \log\det(\mathbb{O}_{\mathcal{W}})$ is submodular and monotone increasing if $\mathbb{O}_{\mathcal{W}}$ is non-singular.
\label{lem_logdet_submodular}
\end{lem}
\begin{proof}
We use the similar idea to show the submodularity of the log det measure, namely, showing that the derived set functions, $f_r : 2^{\mathcal{V}-\{r\}} \to \mathbb{R}$
\begin{align*}
&f_r(\mathcal{W})=\log\det(\mathbb{O}_{\mathcal{W}\cup{r}})-\log \det(\mathbb{O}_{\mathcal{W}})\\
&=\log\det(\mathbb{O}_{\mathcal{W}} + \mathbb{O}_{r}) - \log\det(\mathbb{O}_{\mathcal{W}}),
\label{eqn:marginal_logdet}
\end{align*}
are monotone decreasing for any $r\in \mathcal{V}$. Take any $\mathcal{W}_1 \subseteq \mathcal{W}_2 \subseteq \mathcal{V}-\{r\}$. By the additivity property of the $\mathbb{O}_{\mathcal{W}}$ (Equation~\ref{eqn:additivity_symmetric}), it is clear that $\mathcal{W}_1 \subseteq \mathcal{W}_2 \Rightarrow \mathbb{O}_{\mathcal{W}_1} \preceq \mathbb{O}_{\mathcal{W}_2}$. Define $\mathbb{O}(\gamma) = \mathbb{O}_{\mathcal{W}_1} + \gamma(\mathbb{O}_{\mathcal{W}_2} - \mathbb{O}_{\mathcal{W}_1})$ for $\gamma \in [0,1]$. Clearly, $\mathbb{O}(0) = \mathbb{O}_{\mathcal{W}_1}$ and $\mathbb{O}(1) = \mathbb{O}_{\mathcal{W}_2}$. Now define
\begin{equation}
\hat{f}_r(\mathbb{O}(\gamma)) = \log\det (\mathbb{O}(\gamma) + \mathbb{O}_{r}) -\log\det(\mathbb{O}(\gamma)).
\label{eqn:auxiliary}
\end{equation}
Note that $\hat{f}_r(\mathbb{O}(0)) = \hat{f}_r(\mathbb{O}_{\mathcal{W}_1})$ and $\hat{f}_r(\mathbb{O}(1)) = \hat{f}_r(\mathbb{O}_{\mathcal{W}_2})$. We have
\begin{align}
&\dv{}{\gamma} \hat{f}_r(\mathbb{O}(\gamma))\nonumber\\
& = \dv{}{\gamma}[\log\det (\mathbb{O}(\gamma) + \mathbb{O}_{r}) -\log\det(\mathbb{O}(\gamma))]\nonumber\\
& = \text{trace}[(\mathbb{O}(\gamma) + \mathbb{O}_{r})^{-1}(\mathbb{O}_{\mathcal{W}_2}-\mathbb{O}_{\mathcal{W}_1})]\nonumber\\
&~~-\text{trace}[(\mathbb{O}(\gamma))^{-1}(\mathbb{O}_{\mathcal{W}_2}-\mathbb{O}_{\mathcal{W}_1})]\nonumber\\
& = \text{trace}[((\mathbb{O}(\gamma) + \mathbb{O}_{r})^{-1} - (\mathbb{O}(\gamma))^{-1}) (\mathbb{O}_{\mathcal{W}_2}-\mathbb{O}_{\mathcal{W}_1})]\nonumber\\
& \leq 0.
\end{align}
The second equality follows by the matrix derivative formula $\dv{}{t}\log\det X(\gamma) = \text{trace}[X(\gamma)^{-1}\dv{}{\gamma}X(\gamma)]$~\cite{petersen2008matrix}. \revo{Notably, since $\mathbb{O}(\gamma)\succeq 0$ and $\mathbb{O}_{r} \succeq 0$, we have $\mathbb{O}(\gamma) +  \mathbb{O}_{r} \succeq \mathbb{O}(\gamma)$ and thus $(\mathbb{O}(\gamma) + \mathbb{O}_{r})^{-1} - (\mathbb{O}(\gamma))^{-1} \preceq 0$.  Given} $(\mathbb{O}(\gamma) + \mathbb{O}_{r})^{-1} - (\mathbb{O}(\gamma))^{-1} \preceq 0$, and $\mathbb{O}_{\mathcal{W}_2}-\mathbb{O}_{\mathcal{W}_1}\succeq 0$, the last inequality holds. Since
$$\hat{f}_r(\mathbb{O}(1)) = \hat{f}_r(\mathbb{O}(0)) + \int_{0}^{1} \dv{}{\gamma} \hat{f}_r(\mathbb{O}(\gamma)) d\gamma,$$ it follows that $\hat{f}_r(\mathbb{O}(1)) = \hat{f}_r(\mathbb{O}_{\mathcal{W}_2}) \leq \hat{f}_r(\mathbb{O}(0)) = \hat{f}_r(\mathbb{O}_{\mathcal{W}_1})$. Thus, we have $f_r$ is monotone decreasing, and $f$ is submodular. Similarly, the additivity property of the $\mathbb{O}_{\mathcal{W}}$ (Equation~\ref{eqn:additivity_symmetric}) shows that $f$ is monotone increasing.

However, the proof of the monotonicity and submodularity for the log det is based on the condition that $\mathbb{O}(p_{t_l})$ is non-singular. Otherwise, the conclusion does not hold. For example, if a single sensor is assigned to target $t_l$, $\mathbb{O}(p_{t_l})$ is always singular (see the proof of Corollary~\ref{cor:sensor_n_1}), and thus $\log\det \mathbb{O}(p_{t_l}) = -\infty$. If no sensors are  assigned, we define the empty set case as $\log\det(\emptyset) = 0$. Then, the function is not monotone increasing.

The proof of the monotonicity and submodularity for the trace, rank and log det of the \emph{symmetric observability matrix}, $\mathbb{O}(p_{t_l},u_{t_l}):=O^{T}(p_{t_l},u_{t_l})O(p_{t_l},u_{t_l})$ is similar to the proof for that of the \emph{symmetric observability matrix by the sensor-target relative state contribution}, $\mathbb{O}(p_{t_l})$ as provided above. Because, from Equation~\ref{sy_ob_matrix}, we know,
\begin{equation}
  \mathbb{O}(p_{t_l},u_{t_l}) =
\mathbb{O}(p_{t_l})+O^{T}(u_{t_l})O(u_{t_l}).
\end{equation}
The \emph{unknown control input contribution} $O^{T}(u_{t_l})O(u_{t_l}) \succeq 0$ does not affect the properties of the measures for the \emph{symmetric observability matrix}, $\mathbb{O}(p_{t_l},u_{t_l})$, since assigning sensors only influences the \emph{sensor-target relative state contribution part}, $\mathbb{O}(p_{t_l})$ . But for the log det of \emph{symmetric observability matrix}, we need to guarantee  $\mathbb{O}(p_{t_l},u_{t_l})$ is non-singular.
\label{proof_log_det}
\end{proof}
}

\section{Introduction}
Assigning sensors to better track targets is a well-studied \rev{problem~\cite{grocholsky2002information,jiang2003optimal,spletzer2003dynamic,williams2007information,kamath2007triangulation,zavlanos2008dynamic,tekdas2010sensor,nam2015your,tzoumas2016sensor,tzoumas2016near,tzoumas2017scheduling,yang2017algorithm,chopra2017distributed}.} The typical setup is to formulate a one-to-one assignment problem which can be solved using bipartite graph matching algorithms~\cite{kuhn1955hungarian}. Unlike these works, we focus on scenarios where multiple sensors can be assigned to each target. Furthermore, the utility of assigning multiple sensors may not even be a simple addition of individual utilities, and can have diminishing returns. We study two versions of these many-to-one assignment problems and give constant-factor approximation algorithms for each.

We study these assignment problems in settings where assigning the right set of sensors can improve the observability in tracking potentially mobile targets using noisy sensors. The performance of the target state estimators can be improved by exploiting the \emph{observability} of the underlying system~\cite{gadre2004toward,papadopoulos2010cooperative,arrichiello2013observability,williams2015observability}. Our formulation is motivated by scenarios where sensing multiple targets \revo{by the same sensor} can be time-consuming \revo{as is the case in the following applications}. For example, Tokekar et al.~\cite{tokekar2011active} presented a system to track radio-tagged fish in which each radio tag is assigned a unique frequency. In order to get a measurement for one frequency, the sensor spends two seconds listening to the periodic emissions. Therefore, tracking multiple targets by the same sensor can be time-consuming. Motivated by these scenarios, we seek to assign sensors to track targets with the constraint that each sensor is assigned at most one target. However, multiple sensors can be assigned to track the same target. In fact, more sensors can often improve the tracking performance.

We first study a general assignment problem where there is no restriction on the number of sensors assigned to a target. We let the algorithm decide the optimal configuration of sensor teams assigned to each target. If the value function is submodular and monotone,\footnote{We use ``monotone" and ``monotone increasing" interchangeably.} a greedy algorithm gives a $1/2$--approximation~\cite{fisher1978analysis}. \revo{In general, this bound is tighter, i.e., $1-1/e$ by using the randomized continuous greedy algorithm~\cite{vondrak2008optimal}. However, for a deterministic algorithm, $1/2$--approximation is the best-known result~\cite{vondrak2008optimal}.} We show that observability measures such as the trace and rank of the symmetric observability matrix are submodular and monotone and the log determinant is submodular and monotone when the symmetric observability matrix is non-singular. However, the inverse of the condition number, a commonly-used measure, is neither monotone nor submodular\revo{, which is shown by a counterexample.}

We then study a \emph{restricted} version of the problem where the value function can be arbitrary (not necessarily monotone and submodular) but exactly two sensors must be assigned to a target. \revo{In fact, we show in \revo{ Corollaries \ref{cor:sensor_n_1} and \ref{cor:sensor_n_gre2} that at least two range sensors must be assigned for the inverse condition number to be greater than zero.} Even for the case of bearing sensors, it is known that two sensors are necessary; there exists work on assignment and placement of pairs of bearing sensors for improving target tracking~\cite{kamath2007triangulation,tekdas2010sensor}. We study the analogous problem for range sensors.} We prove that the greedy algorithm achieves a $1/3$--approximation of the optimal solution \revo{for assigning pairs of range sensors}.

Observability is a basic concept in control theory and has been widely applied in robotics. Observability for range-only beacon sensors, in particular, has been closely studied for underwater navigation. Gadre and Stilwell~\cite{gadre2004toward} analyzed the local and global observability~\cite{hermann1977nonlinear} for the localization of an autonomous underwater vehicle by an acoustic beacon. The problems of single vehicle localization and multi-vehicle relative localization are studied in \cite{arrichiello2013observability} using an observability criterion introduced in \cite{krener2009measures}. In these works, it is the sensors that are moving. Consequently, the sensors know their control vectors and can thus, compute the observability matrix and its measures. \revo{When tracking a target, however, the control inputs for the targets may be unknown.} In recent work, Williams and Sukhatme~\cite{williams2015observability} studied a multi-sensor localization and target tracking problem where they showed how to leverage graph rigidity to improve the observability for sensor team localization and robust target tracking.

Unlike these works, we consider scenarios where the control inputs for the targets are not known to the sensors. Consequently, we cannot compute the observability matrix exactly. Instead, we present a novel lower bound on the observability for the case of unknown target motion tracked by range-only sensors. Specifically, we show how to lower bound the \emph{condition number}~\cite{krener2009measures} of the partially known observability matrix using only the known part (Section~\ref{sec:lowerbound}). \revo{The result is specific to the problem at hand where the inputs
appear, linearly, on a single line of the observability matrix.}

In addition to theoretical results, we conduct simulations to evaluate the empirical performance of the algorithms. We find that sensors are assigned to targets almost uniformly using the greedy algorithm for the first problem (Section~\ref{sec:simulation}-A). The greedy algorithm for the second problem performs much better than $1/3$ in practice (Section~\ref{sec:simulation}-B).

\section{Overview of the Problem and Results} \label{sec:pro_formulation}
\revo{In this section, we first formally define the problems that are studied and then summarize the main contributions of this work.}

\subsection{Problem Formulation}
We consider a scenario where there are $N$ sensors and $L$ targets in the environment.  We use  $\sigma(t_l)$ to represent the set of sensors assigned to target $t_l$. $\sigma^{-1}(s_i)$ gives the set of targets assigned to sensor $s_i$. We use $\sigma_i(t_l)$ to give the $i^{th}$ sensor assigned to target $t_l$. We order the assigned sensors by using their IDs such that $\sigma_1(t_l) < \sigma_2(t_l) < \sigma_3(t_l) < \ldots$.  Let $\omega(s_i, s_j, t_l)$, and $\omega(\mathcal{S}_{i}, t_l)$ be some measure of the observability of tracking $t_l$ with $s_i$ and $s_j$, and with a set of sensor(s) $\mathcal{S}_{i}$, respectively. \revo{We calculate the observability measure by taking account of the relative positions of sensors and targets (which will appear in the observability matrix) at every time step}.

We start with the problem of assigning a set of sensors to each target. The sensors form teams of varying sizes to track individual targets. Sensors within a team can share measurements so as to better track the targets. We constrain each sensor to be assigned to only one target. This is motivated by scenarios where sensing multiple targets can be time consuming (as is the case with radio sensors~\cite{tokekar2011active,alvarez2018acoustic} or communicating multiple measurements with the team can be time and energy consuming.
\begin{problem}[General Assignment] Given a set of sensors, $\mathcal{S} := \{s_0,\ldots,s_N\}$ and a set of targets, $\mathcal{T} := \{t_0,\ldots,t_L\}$, find an assignment of sets of sensors to targets to
\begin{equation}
\text{maximize} \sum_{l=1}^L \omega(\sigma(t_l), t_l)
\end{equation}
with the added constraint that each sensor is assigned to at most one target.
\label{prob:general}
\end{problem}

 Problem~\ref{prob:general} is the general assignment problem which is difficult to solve for arbitrary $\omega(\cdot)$.
In order to solve the assignment problem for arbitrary value functions, we consider a restricted case where each target is tracked by exactly two sensors.

\begin{problem}[\revo{Non-overlapping} Pair Assignment] Given a set of sensors, $\mathcal{S} := \{s_0,\ldots,s_N\}$ and a set of targets, $\mathcal{T} := \{t_0,\ldots,t_L\}$, find an assignment of \revo{non-overlapping} pairs of sensors to targets:
\begin{equation}
\text{maximize} \sum_{l=1}^L \omega(\sigma_1(t_l),\sigma_2(t_l), t_l)
\end{equation}
with the added constraint each sensor is assigned to at most one target. That is, for all $i = 1,\ldots, N$ we have $|\sigma^{-1}(s_i)|\leq 1$, assuming $N \geq 2 L$.
\label{prob:unique}
\end{problem}

\subsection{Contributions}

\revo{The main contributions of this paper are summarized as follows.
\begin{enumerate}
    \item We derive a lower bound on the inverse of the condition number of the observability matrix which is useful when the control input for the target is not known.
    \item We show that the condition number of the observability matrix is neither submodular nor monotone increasing. We show that the trace and the rank of the symmetric observability matrix are submodular and monotone increasing. We also show the log determinant of the symmetric observability matrix is submodular and monotone when the symmetric observability matrix is non-singular.
    \item We present a greedy assignment algorithm that yields a $1/2$--approximation for Problem~\ref{prob:general} and $1/3$--approximation for Problem \ref{prob:unique}.
    \item We verify the performance of our proposed greedy algorithm with various observability measures (that not necessary submodular) through extensive simulations.
\end{enumerate}
}

We start by showing how to bound the inverse of condition number (Section~\ref{sec:lowerbound}) and then present the assignment algorithms (Section~\ref{sec:algo})

\section{Bounding the Observability} \label{sec:lowerbound}
Consider a mobile target $t_l$ whose position is denoted by $p_{t_l}$. Suppose there are $N$ stationary sensors that can measure the distance\footnote{We use the square of the distance/range for mathematical convenience.} to the target. We have:
\begin{equation}
\left\{
                \begin{array}{ll}
                  \dot{p}_{t_l}=u_{t_l},\\
                  z_{s_i}=h_i(p_{t_l})=\frac{1}{2}\|p_{s_i}-p_{t_l}\|_{2}^{2}, ~i=1,...,N
                \end{array}
              \right.
\label{eqn:multi_sensor_target_system}
\end{equation}
where $p_{t_l}:=[x_{t_l},y_{t_l}]^{T}$ gives the 2D position of the target, and $u_{t_l}:=[u_{lx},u_{ly}]$ defines its control input, which is unknown to the sensor. Let $u_{t_l,\max}\triangleq\max\|u_{t_l}\|_{2}$. $z_{s_i}$ defines the range-only measurement from each sensor $s_i$ whose position is given by $p_{s_i}=[x_{s_i},y_{s_i}]^{T}$. For simplicity, we also assume that the target does not collide with any sensor, i.e., $\|p_{s_i}-p_{t_l}\|_{2} \neq 0$ and no two sensors are deployed at the same position.

We analyze the weak local observability matrix, $O(p_{t_l},u_{t_l})$, of this multi-sensor target tracking system. We show how to lower bound the inverse of the condition number of $O(p_{t_l},u_{t_l})$, given by $C^{-1}(O(p_{t_l},u_{t_l}))$, independent of $u_{t_l}$. We also show that the lower bound, $\underline{C}^{-1}(O(p_{t_l},u_{t_l}))$, is tight.

We compute the local nonlinear observability matrix \rev{by using Lie derivatives}~\cite{williams2015observability,hermann1977nonlinear} for this system (Equation \ref{eqn:multi_sensor_target_system}) as,
\begin{equation}
O(p_{t_l},u_{t_l})=\begin{bmatrix}
    \nabla L_{0}^{h_1} \\
     \nabla L_{1}^{h_1}\\
     \vdots\\
     \nabla L_{0}^{h_2} \\
     \nabla L_{1}^{h_2}\\
     \vdots\\
     \\
    \vdots\\
    \\
    \nabla L_{0}^{h_N} \\
     \nabla L_{1}^{h_N}\\
     \vdots

\end{bmatrix}=\begin{bmatrix}
    x_{t_l}-x_{s_1},y_{t_l}-y_{s_1} \\
     u_{lx},u_{ly}\\
     0,0\\
     \vdots\\
     x_{t_l}-x_{s_2},y_{t_l}-y_{s_2} \\
     u_{lx},u_{ly}\\
     0,0\\
     \vdots\\
     \\
    \vdots\\
    \\
    x_{t_l}-x_{s_N},y_{t_l}-y_{s_N} \\
     u_{lx},u_{ly}\\
     0,0\\
     \vdots

\end{bmatrix}.
\label{observability_multi_sensor_target}
\end{equation}
This equation can be rewritten as,
\begin{eqnarray}
O(p_{t_l},u_{t_l})&=&\begin{bmatrix}
    x_{t_l}-x_{s_1},y_{t_l}-y_{s_1} \\
     x_{t_l}-x_{s_2},y_{t_l}-y_{s_2} \\
    \vdots\\
    x_{t_l}-x_{s_N},y_{t_l}-y_{s_N} \\
     u_{lx},u_{ly}
\end{bmatrix}_{(N+1)\times 2}.
\label{eqn:new_multi_sensor_target}
\end{eqnarray}
The state of the target $t_l$ is \emph{weakly locally observable} if the local nonlinear observability matrix has full column rank~\cite{hermann1977nonlinear}. However, the rank test for the observability of the system does not tell the degree of the observability or how \emph{good} the observability is.  The \emph{condition number}~\cite{krener2009measures}, defined as the ratio of the largest singular value to the smallest, can be used to measure this degree of unobservability. We use the \emph{inverse of condition number} given as,
\begin{equation}
C^{-1}(O(p_{t_l},u_{t_l}))=\frac{\sigma_{\min}(O(p_{t_l},u_{t_l}))}{\sigma_{\max}(O(p_{t_l},u_{t_l}))}
\label{eqn:inverse_condition_number}.
\end{equation}
Note that, $C^{-1} \in [0,1]$. $C^{-1}=0$ means  $O(p_{t_l},u_{t_l})$ is singular and $C^{-1}=1$ means $O(p_{t_l},u_{t_l})$ is \emph{well conditioned}. A larger $C^{-1}$ means better observability (see more details in~\cite{arrichiello2013observability}).

In the local nonlinear observability matrix $O(p_{t_l},u_{t_l})$, $u_{t_l}$ is unknown and not controllable by the sensor. On the other hand, $p_{t_l}-p_{s_i}$, depends on the relative state between each sensor $s_i$ and target $t_l$ and is known to the sensor (assuming an estimate of the target's position is known). Thus, we first partition $O(p_{t_l},u_{t_l})$ into the known and unknown parts as,
\begin{eqnarray}
O(p_{t_l},u_{t_l})=\begin{bmatrix}
    O(p_{t_l}) \\
     O(u_{t_l})
\end{bmatrix},
\label{eqn:new_multi_sensor_target_partition}
\end{eqnarray}
where
\begin{equation}
O(p_{t_l}):=\begin{bmatrix}
    x_{t_l}-x_{s_1},y_{t_l}-y_{s_1} \\
     x_{t_l}-x_{s_2},y_{t_l}-y_{s_2} \\
    \vdots\\
    x_{t_l}-x_{s_N},y_{t_l}-y_{s_N}
\end{bmatrix},
\label{eqn: state_contribution_observability}
\end{equation}
and
\begin{equation}
O(u_{t_l}):=\begin{bmatrix}
     u_{lx},u_{ly}
\end{bmatrix},
\label{eqn: control_contribution_observability}
\end{equation}
indicate the contributions from the sensor-target relative state and target's control input, respectively. Since $O(u_{t_l})$ is unknown, we cannot compute $C^{-1}(O(p_{t_l},u_{t_l}))$ exactly. Instead, we compute its lower bound as below.
\begin{thm}
For the multi-sensor-target system (Equation \ref{eqn:multi_sensor_target_system}) with the number of sensors, $N\geq 2$, the inverse of the condition number is lower bounded by $\dfrac{\sigma_{\min}(O(p_{t_l}))}{\sqrt{\sigma^{2}_{\max}(O(p_{t_l}))+ u_{t_l,\max}^{2}}}$.
\label{thm:lower_bound}
\end{thm}
We present the full proof for this and other results in the appendix.

We wish to improve the worst case, i.e., the lower bound of $C^{-1}(O(p_{t_l},u_{t_l}))$, by optimizing the sensor-target relative state by assigning a different subset of sensors to the target. In the following, we will show that at least two sensors are required to improve the lower bound.
\begin{cor}
The lower bound of the observability metric in one-sensor-target system, $\underline{C}^{-1}(O_{i}(p_{t_l},u_{t_l}))$, \revo{ is identically zero. It} does not depend on the position of the sensor and therefore cannot be controlled by the sensors.
\label{cor:sensor_n_1}
\end{cor}

When the number of sensors, $N\geq 2$, we have a positive result that shows that the sensors can improve the lower bound on the inverse of the condition number of optimizing their positions, \revo{even though the contribution to the observability matrix from the target's input, $O(u_{t_l})$, is unknown and cannot be controlled.}
\begin{cor}
Suppose that the number of sensors assigned to $t_l$, $N \geq 2$. If the sensors increase $C^{-1}(O(p_{t_l}))$ and $\sigma_{\min}(O(p_{t_l})\revo{)}$ (the inverse of condition number and the smallest singular number of the relative state contribution $O(p_{t_l})$), then the lower bound of $C^{-1}(O(p_{t_l},u_{t_l}))$ also increases.
\label{cor:sensor_n_gre2}
\end{cor}
\begin{rem}
The lower bound $\underline{C}^{-1}(O(p_{t_l},u_{t_l}))$ is tight when the target is known to be stationary. If $u_o\in\{0\}$, $O(p_{t_l},u_{t_l})=O(p_{t_l})$ by Equation~\ref{eqn:new_multi_sensor_target} and Equation~\ref{eqn: state_contribution_observability}. Thus, from Equation~\ref{eqn:inverse_condition_number} and Theorem~\ref{thm:lower_bound}, the lower bound  $\underline{C}^{-1}(O(p_{t_l},u_{t_l}))={C}^{-1}(O(p_{t_l},u_{t_l}))$, which implies that the lower bound is tight.
\end{rem}

Next, we use these results for assigning sensors to targets.
\section{Assignment Algorithms} \label{sec:algo}
So far, we have assumed that we know the true position, $p_{t_l}(k)$, of the target at time $k$. In practice, we only have an estimate, $\hat{p}_{t_l}(k)$, for $t_l$ along with its covariance $\Sigma_{t_l}(k)$. The estimate is obtained by fusing measurements using, for example, an Extended Kalman Filter (EKF). \revo{We use the estimated position of the target for the assignment algorithms in the simulation (Section~\ref{sec:simulation}).}

\subsection{General assignment as Submodular Welfare Optimization}\label{sec:general_assign}
 We first study the General Assignment (Problem~\ref{prob:general}) where each target $t_l$ is tracked by a subset of sensors $\sigma(t_l)\subset \mathcal{S}, ~l\in\{1,2,...,L\}$ whose cardinality is not necessarily two.
This is known as \emph{submodular welfare problem} in the literature~\cite{vondrak2008optimal} where the objective is to maximize $\sum_{i=1}^{n}w_i(\mathcal{S}_i)$ for independent sets $\{\mathcal{S}_i| \mathcal{S}_i\subseteq \mathcal{S}, ~i=\{1,2,...n\}\}$ by using monotone and
submodular utility functions $w_i$. A greedy algorithm~\cite{fisher1978analysis} yields a $1/2$--approximation for this problem.
We first show that the lower bound of the inverse of the condition number is neither monotone nor submodular.
\rev{This is not surprising since it has been shown that similar measures for analogous versions of the controllability matrix are also not submodular or monotone increasing~\cite{summers2016submodularity}.}

\begin{thm}
The lower bound of the inverse of condition number function $\omega(\cdot)$ is neither monotone increasing nor submodular.
\label{Them:invcond_not_sub}
\end{thm}
\begin{proof}
We prove the claim by giving two counter-examples.

\noindent\textbf{Case 1}: Given the sensors $s_1(0,0)$, $s_2(2\sqrt{3},-9)$, $s_3(\sqrt{3},3)$ and target $t_1(\sqrt{3},1)$ with $u_{t_{1},max}=1$ in 2-D plane,
$\omega(\{s_1,s_3\})=0.5345>\omega(\{s_1,s_2, s_3\})=0.1823$, which shows $\omega(\cdot)$ is not monotone increasing.

\noindent\textbf{Case 2}: Given the sensors $s_1(0,0)$, $s_2(2\sqrt{3},0)$, $s_3(\sqrt{3},0.1)$, $s_4(\sqrt{3},3)$ and target $t_1(\sqrt{3},1)$ with $u_{t_{1},max}=1$ in 2-D plane,
 $\omega(\{s_1,s_2,s_3\})-\omega(\{s_1,s_2\})=0.3310-0.5345=-0.2035<\omega(\{s_1,s_2,s_4,s_3\})-\omega(\{s_1,s_2, s_4\})=0.8765-0.9258=-0.0493$, which shows $\omega(\cdot)$ is not submodular.
\end{proof}

Therefore, we focus on other measures of observability and summarize the results in Theorem~\ref{thm:submodular_observability_measure}.

\begin{thm}
\rev{The trace and rank of the symmetric observability matrix, $\mathbb{O}(p_{t_l},u_{t_l}):=O^{T}(p_{t_l},u_{t_l})O(p_{t_l},u_{t_l})$, and of the sensor-target relative state contribution matrix, $\mathbb{O}(p_{t_l}):=O^{T}(p_{t_l})O(p_{t_l})$, are submodular \revo{(modular)} and monotone increasing. The log determinant of the two matrices is submodular and monotone increasing if the corresponding matrix is non-singular.} \label{thm:submodular_observability_measure}
\end{thm}

\rev{The reason that we also study the measures of the symmetric observability matrix by sensor-target relative state contribution is that the control input of the target is unknown, and thus the symmetric observability matrix is not available.}
The proof \rev{provided in the appendix} is similar to proving that the trace of the Gramian, the log determinant, and the rank of the Gramian are monotone submodular~\cite{summers2016submodularity}. \rev{Note that, if $\mathbb{O}(p_{t_l})$ is singular, its log determinant is $-\infty$. In our sensor assignment case, if a single sensor is assigned to target $t_l$, $\mathbb{O}(p_{t_l})$ is always singular, i.e., $\det(\mathbb{O}(p_{t_l}))=0$ (see the proof of \revo{Corollary~\ref{cor:sensor_n_1}}). If no sensors are assigned to a target, then the matrix is not defined. Thus, at least two sensors need to be assigned to a target to ensure non-singularity of $\mathbb{O}(p_{t_l})$.}

\subsection{A $1/3$--approximation algorithm for Problem~\ref{prob:unique}} \label{sec:selfishmatching}
Next, we study a more specific assignment (Problem~\ref{prob:unique}) where each target $t_l$ is tracked by a pair of sensors but allow the value function to be arbitrary.
We propose a greedy algorithm to solve this problem. In each round, we calculate the observability metric, $\omega(\sigma_1(t_l),\sigma_2(t_l), t_l)$ for
all triples $(\sigma_1(t_l),\sigma_2(t_l), t_l), ~\sigma_1(t_l),\sigma_2(t_l)\in \mathcal{S},~t_l\in\mathcal{T}$, and select the triple which has the maximum $\omega(\sigma_1(t_l),\sigma_2(t_l), t_l)$, then remove
$\{\sigma_1(t_l),\sigma_2(t_l)\}$ from sensor set $\mathcal{S}$ and remove $t_l$ from target set $\mathcal{T}$, respectively. We present the greedy approach in Algorithm \ref{algorithm:unique_pair_assignment} where $\omega(\textrm{GREEDY})$ denotes total value obtained by the greedy approach. In this case, we can use the inverse of the condition number as $\omega(\cdot)$.
\begin{algorithm}
$k\leftarrow 0, ~\omega(\textrm{GREEDY})\leftarrow 0$\\
\While{true}{
Compute all possible
$\omega(\sigma_1(t_l),\sigma_2(t_l), t_l)$.\\
Pick the triple $(\sigma_1(t_l),\sigma_2(t_l), t_l)$ with maximum $\omega(\sigma_1(t_l),\sigma_2(t_l), t_l)$ defined as $\omega_{\max}$.\\ $\omega(\textrm{GREEDY})\leftarrow \omega(\textrm{GREEDY})+\omega_{\max}$.\\
Remove $\{s_i,s_j\}$ from the sensor set $\mathcal{S}$ and remove $t_l$ from the target set $\mathcal{T}$.\\
$k\leftarrow k + 1$
}
 \caption{Greedy \revo{Non-overlapping} Pair Assignment}
 \label{algorithm:unique_pair_assignment}
\end{algorithm}

\begin{thm}
$\omega({\revo{\emph{GREEDY}}}) \geq \frac{1}{3}\omega(\revo{\emph{OPT}})$ where \revo{\emph{OPT}} is the optimal algorithm for Problem~\ref{prob:unique}. The running time for Algorithm~\ref{algorithm:unique_pair_assignment} is $O(N^2L^2)$.
\label{thm:lem_optimal_fin}
\end{thm}
\begin{proof}
\revo{We first give the proof for the approximation ratio of Algorithm~\ref{algorithm:unique_pair_assignment}.} Recall that $\omega(\textrm{GREEDY})$ and $\omega(\textrm{OPT})$ will be the sum of $\omega(\cdot)$ terms of triples consisting of one target and the two assigned sensors. As a shorthand, we will denote $\omega^g(l)$ and $\omega^*(l)$ to be the values of the triple assigned to $t_l$ by GREEDY and OPT, respectively.

We show that there exists a many-to-one mapping $\mathcal{M}: [1,\cdots, L] \to [1,\cdots, L]$ such that:
\begin{enumerate}
\item $\omega^{*}(k)\leq \omega(\mathcal{M}(k))$; and \item $|\mathcal{M}^{-1}(\revo{y})|\leq 3$ \revo{for all $y\in \mathcal{Y}$ where $\mathcal{Y} \subseteq [1,..., L]$ is the range of $\mathcal{M}$.}
\end{enumerate}
That is, each triple in OPT is mapped to some triple in GREEDY whose value is at least as high and no triple in GREEDY has more than three terms in OPT mapped to it.

We first show that if such a mapping exists, then the main result holds. Then, we prove the existence of such a mapping by constructing a specific $\mathcal{M}$.

If such a mapping $\mathcal{M}$ exists, we prove $\omega(\textrm{GREEDY}) \geq \frac{1}{3}\omega(\textrm{OPT})$ as below.
\begin{align}
\omega(OPT) &= \sum_{k=1}^{L} \omega^{*}(k)\nonumber\\
&\leq \sum_{k=1}^{L} \omega^{g}(\mathcal{M}(k))\nonumber\\
&\revo{= \sum_{y\in \mathcal{Y}} \omega^{g}(y) |\mathcal{M}^{-1}(y)|} \nonumber\\
& \revo{\leq 3 \sum_{y\in \mathcal{Y}} \omega^{g}(y)} \nonumber\\
&\leq 3 \sum_{l=1}^{L} \omega^{g}(l)\nonumber\\
&= 3 \omega(\text{GREEDY}).
\label{eqn:approxiamtion_ratio}
\end{align}
The first inequality is due to $\omega^{*}(k) \leq \omega^{g}(\mathcal{M}(k))$. \revo{The second equality is because $\mathcal{M}$ maps each item in $[1,...,L]$ to the set $\mathcal{Y}$. } The second inequality is due to $|\mathcal{M}^{-1}(\cdot)|\leq 3$. \revo{The third inequality is because $\mathcal{Y}\subseteq [1,...,L]$.}

Next, we show that such a mapping always exists by constructing one. We will define $\mathcal{M}$ in the order in which the triples are picked by GREEDY. Suppose GREEDY picks the triple $(s_i,s_j,t_l)$ in some round. Then $\omega^{g}(l) = \omega(s_i,s_j,t_l)$. We will map at most three triples in OPT to this triple in GREEDY. There are three cases.

\begin{figure}[tbh]
\centering{
\subfigure[Case 1]{
\includegraphics[width=0.25\columnwidth]{figs/opt_case1.eps}}
\subfigure[Case 2]{
\includegraphics[width=0.28\columnwidth]{figs/opt_case2.eps}}
\subfigure[Case 3]{
\includegraphics[width=0.32\columnwidth]{figs/opt_case3.eps}}
}
\rev{\caption{The optimal solution in three cases. \revo{In all cases, the GREEDY chooses the triple $(s_i,s_j,t_l)$. } Case 1: The OPT charges $\omega(s_i,s_j,t_l)$ to the same triple $(s_i,s_j,t_l)$ chosen by the GREEDY.  Case 2: The OPT charges $\omega(s_i,s_j,t_l)$ to at most two triples --- $(s_i,s_j,t_m)$ and  $(s_p,s_q,t_l)$. Case 3: The OPT charges  $\omega(s_i,s_j,t_l)$ to at most three triples --- $(s_i,s_p,t_m)$, $(s_j,s_q,t_n)$ and $(s_u,s_v,t_l)$. Here, $i\neq j \neq p \neq q \neq u \neq v$ and $l \neq m \neq n$. \label{fig:opt_choose}} }
\end{figure}
\begin{enumerate}
\item $(s_i,s_j,t_l)$ is also chosen by OPT (Figure~\ref{fig:opt_choose}-(a)). If $\mathcal{M}(l)$ has not been defined in previous rounds, we define $\mathcal{M}(l) = l$. Note that, here $\omega^g(l) = \omega^*(l)$ and $|\mathcal{M}^{-1}(l)| = 1$. Therefore, the two conditions for a valid mapping are satisfied.

\item  Exactly two of $(s_i,s_j,t_l)$ appear in a triple chosen by OPT (Figure~\ref{fig:opt_choose}-(b)). Consider the case where OPT chooses $(s_i,s_j,t_m)$ and $(s_p,s_q,t_l)$ where $m \neq l$. All other cases are symmetric. If $\mathcal{M}(m)$ has not been defined in a previous round, we define $\mathcal{M}(m) = l$. Note that, if $\mathcal{M}(m)$ was not defined in a previous round, then $\omega^*(m) \leq \omega^g(l)$. Otherwise, GREEDY would pick the triple $(s_i,s_j,s_m)$ in this round. Similarly, if $\mathcal{M}(l)$ is not defined in a previous round, we define $\mathcal{M}(l) = l$. By a similar argument, if $\mathcal{M}(l)$ was not defined in a previous round, $\omega^*(l) \leq \omega^g(l)$. Furthermore, $|\mathcal{M}^{-1}(l)|=2$. Therefore, the two conditions for a valid mapping are satisfied.

\item No two of $(s_i,s_j,t_l)$ appear in the same triple chosen by OPT. Suppose instead they appear in three distinct triples, $(s_u,s_v,t_l)$, $(s_i,s_p,t_m)$, and $(s_j,s_q,t_n)$ as shown in Figure~\ref{fig:opt_choose}-(c). We can use a similar argument as in the previous case. If any of $l,m$ and $n$ were not mapped in a previous round, then we will map them to $l$. Furthermore, since they were not mapped in a previous round, their value in OPT cannot be greater than $\omega^g(l)$. Finally, $|\mathcal{M}^{-1}(l)|\leq 3$. Therefore, the two conditions for a valid mapping are satisfied.
\end{enumerate}
Therefore, given such a mapping $\mathcal{M}$, it follows that $\omega(\textrm{GREEDY}) \geq \frac{1}{3}\omega(\textrm{OPT})$.

\revo{We next prove the running time for Algorithm~\ref{algorithm:unique_pair_assignment}. The ``while" loop runs for $L$ rounds since all the targets must be tracked eventually. Inside the ``while" loop, we compute all possible triples and find the best one. This requires $O(N^2L)$ time. Overall, Algorithm~\ref{algorithm:unique_pair_assignment} runs in $O(N^2L^2)$ time.}
\end{proof}
\revo{

It is possible to generalize this result to the case where exactly $n$ sensors are to be assigned to a target with $n\geq 2$. We can obtain a generalized bound, $\omega(\text{GREEDY}) \geq \frac{1}{n+1}\omega(\text{OPT})$ by using a proof similar to that of Theorem~\ref{thm:lem_optimal_fin}.
}

\section{Simulations}\label{sec:simulation}
We illustrate the performance of the assignment strategies for sensor selection using observability measure as the performance criterion.
A video showing the algorithm in action is submitted as supplementary material.

\subsection{Greedy General Assignment}
\begin{figure*}[tbh]
\centering{
\subfigure[$k=1$ (initial time)]{
\includegraphics[width=0.65\columnwidth]{figs/greedy_general_trace_k1.eps}}
\subfigure[$k=60$]{
\includegraphics[width=0.65\columnwidth]{figs/greedy_general_trace_k60.eps}}
\subfigure[$k=100$]{
\includegraphics[width=0.65\columnwidth]{figs/greedy_general_trace_k100.eps}}
}
\caption{Greedy General Assignment~\cite{fisher1978analysis} in action for tracking three targets with circular motion by using $\text{trace}(\mathbb{O}(p_{t_l}))$. The three colors red, blue and magenta specify three targets, respectively. The pentagram, filled square, solid ellipse (sometimes, it looks like \revo{a} solid circle) and dotted circle indicates the true position, estimate mean position, variance and trajectory for the target, respectively. The black diamond indicates the sensor. The solid line joining the target and sensor indicates that the sensor is assigned to the target.\label{fig:gen_assign_tracking}}
\end{figure*}

We first simulate the greedy assignment~\cite{fisher1978analysis} for Problem~\ref{prob:general} by using \rev{the trace of the \emph{symmetric observability matrix by sensor-target relative state contribution}, $\text{trace}(\mathbb{O}(p_{t_l}))$}
as observability measure which is monotone and submodular \revo{(modular) as shown in Theorem~\ref{thm:submodular_observability_measure}. Notably, for maximizing a modular function, the greedy algorithm yields an optimal solution.} We start with the greedy assignment with 8 stationary sensors and 3 targets moving in a circle and $u_{t_l,\max}=1,~ l\in\{1,2,3\}$ within 100 time steps in a $10\times 10$ environment. At each timestep, we use the greedy approach~\cite{fisher1978analysis} to assign a set of sensors to each target, and use the measurements of the set of sensors to update the estimate of the target by EKF as shown in Figure~\ref{fig:gen_assign_tracking}.

\rev{}
In order to further evaluate the greedy approach for general assignment, we set the number of targets as $L=5$ and number of sensors, $N$, from 20 to 50. For each $N\in\{20,21,...,50\}$, the positions of sensors and targets are randomly generated within $[0,100]\times [0,100]\in\mathbb{R}^{2}$ for 30 trials.
We compare the number of sensors assigned to one specific target, i.e., $t_l, l\in\{1,2,...L\}$ with the $N/L$ as shown in Figure~\ref{fig:trial_submodular_obs.eps}. It shows that the sensors are assigned to each target almost evenly.

\begin{figure}[htb]
\centering{
\includegraphics[width=0.9\columnwidth]{figs/trial_submodular_obs.eps}
}
\caption{Number of sensors assigned to each target.\label{fig:trial_submodular_obs.eps}}
\end{figure}

\subsection{Greedy \revo{Non-overlapping} Pair Assignment}
We then simulate the greedy \revo{non-overlapping} pair assignment (Algorithm~\ref{algorithm:unique_pair_assignment}) for solving Problem~\ref{prob:unique}. With the same environment setting, we start with simulating Algorithm~\ref{algorithm:unique_pair_assignment} with $\text{trace}(\mathbb{O}(p_{t_l}))$  as the observability measure (Figure~\ref{fig:uni_assign_tracking_trace}). \rev{We know that the observability measure for greedy \revo{non-overlapping} pair assignment does not have to be monotone submodular. Therefore, we use the log determinant of the \emph{symmetric observability matrix by sensor-target relative state contribution},  $\log\det(\mathbb{O}(p_{t_l}))$,  and the lower bound on the inverse of the condition number of the observability matrix, $\underline{C}^{-1}(O(p_{t_l},u_{t_l}))$, for the assignment. The assignments for a specific scenario are shown in Figures~\ref{fig:uni_assign_tracking_logdet} and \ref{fig:uni_assign_tracking_invcond}. Note that, even though a pair of sensors is assigned to target $t_l$, $\mathbb{O}(p_{t_l})$ is not always non-singular.}

\begin{figure*}[thb]
\centering{
\subfigure[$k=1$ (initial time)]{
\includegraphics[width=0.65\columnwidth]{figs/greedy_uni_trace_k1.eps}}
\subfigure[$k=60$]{
\includegraphics[width=0.65\columnwidth]{figs/greedy_uni_trace_k60.eps}}
\subfigure[$k=100$]{
\includegraphics[width=0.65\columnwidth]{figs/greedy_uni_trace_k100.eps}}
}
\caption{Greedy \revo{Non-overlapping} Pair Assignment (Algorithm~\ref{algorithm:unique_pair_assignment}) in action for tracking three targets with circular motion by using $\text{trace}(\mathbb{O}(p_{t_l}))$. The three colors red, blue and magenta specify three targets, respectively. The pentagram, filled square, solid ellipse (sometimes, it looks like \revo{a} solid circle) and dotted circle indicates the true position, estimate mean position, variance and trajectory for the target, respectively. The black diamond indicates the sensor. The solid line joining the target and sensor indicates that the sensor is assigned to the target.\label{fig:uni_assign_tracking_trace}}
\end{figure*}

\begin{figure*}[thb]
\centering{
\subfigure[$k=1$ (initial time)]{
\includegraphics[width=0.65\columnwidth]{figs/uni_assign_log_deter_k_1.eps}}
\subfigure[$k=60$]{
\includegraphics[width=0.65\columnwidth]{figs/uni_assign_log_deter_k_60.eps}}
\subfigure[$k=100$]{
\includegraphics[width=0.65\columnwidth]{figs/uni_assign_log_deter_k_100.eps}}
}
\caption{Greedy \revo{Non-overlapping} Pair Assignment (Algorithm~\ref{algorithm:unique_pair_assignment}) in action for tracking three targets with circular motion by using \rev{$\log\det(\mathbb{O}(p_{t_l}))$}. The three colors red, blue and magenta specify three targets, respectively. The pentagram, filled square, solid ellipse (sometimes, it looks like \revo{a} solid circle) and dotted circle indicates the true position, estimate mean position, variance and trajectory for the target, respectively. The black diamond indicates the sensor. The solid line joining the target and sensor indicates that the sensor is assigned to the target.\label{fig:uni_assign_tracking_logdet}}
\end{figure*}

For each target $t_l$, the estimate generated by EKF includes its mean position $\hat{p}_{t_l}$ and variance $\Sigma_{t_l}$, $l\in\{1,2,3\}$. To evaluate the tracking performance, denote the mean error as
\begin{equation}
    \mathrm{err}_{t_l} = \|\hat{p}_{t_l}-p_{t_l}\|_2.
    \label{eqn:mean_error}
\end{equation}
 and the trace of covariance as $\mathrm{tr}(\Sigma_{t_l})$. Figure~\ref{fig:comparsion_gen_uni_log_mean_trace} shows $\mathrm{err}_{t_l}$ and $\mathrm{tr}(\Sigma_{t_l})$ for each target $t_l$ with the greedy algorithms to the two assignment problem and with $\text{trace}(\mathbb{O}(p_{t_l})\revo{)}$ as the measure. \rev{We can observe that the greedy general assignment performs better for some target (e.g., $t_2$), but performs worse for target $t_1$ as compared to the \revo{non-overlapping} pair assignment (Figure~\ref{fig:comparsion_gen_uni_log_mean_trace}-(b))}. \rev{Both algorithms maximize the sum of the observability measures for the three targets. In the \revo{non-overlapping} pair assignment, a pair of sensors is always assigned to each target whereas in the general assignment, there are situations where no sensors are assigned to a target. This can be due to the fact that it is profitable to assign more than two sensors to some target in order to maximize the sum. We can observe the happening in Figure~\ref{fig:gen_assign_tracking} where there exist some time steps (e.g., $k=100$) when less than two sensors are assigned to a target (red), which leads to a bad tracking performance for individual targets in the general assignment case (\revo{Corollary~\ref{cor:sensor_n_1}}). However, in all cases, the greedy general assignment~\cite{fisher1978analysis} ensures that the sum of individual measures is within a factor of 2 of the optimal sum.

Figure~\ref{fig:comparsion_uni_logcondtrace_mean_trace} shows the tracking performance with the inverse condition number and log det is comparable. The performance with trace is worse as compared to the other two.  However, we can only use the trace for the greedy general assignment, since it is submodular and monotone. This exactly reflects the importance of our greedy \revo{non-overlapping} pair assignment that better observability measures, i.e., log det and inverse condition number, which are not necessarily submodular and monotone are also allowed.}

\begin{figure*}[htb]
\centering{
\subfigure[$k=1$ (initial time)]{
\includegraphics[width=0.65\columnwidth]{figs/uni_assign_k_1.eps}}
\subfigure[$k=60$]{
\includegraphics[width=0.65\columnwidth]{figs/uni_assign_k_60.eps}}
\subfigure[$k=100$]{
\includegraphics[width=0.65\columnwidth]{figs/uni_assign_k_100.eps}}
}
\caption{Greedy \revo{Non-overlapping} Pair Assignment (Algorithm~\ref{algorithm:unique_pair_assignment}) in action for tracking three targets with circular motion by using $\underline{C}^{-1}(O(p_{t_l},u_{t_l}))$. The three colors red, blue and magenta specify three targets, respectively. The pentagram, filled square, solid ellipse (sometimes, it looks like \revo{a} solid circle) and dotted circle indicates the true position, estimate mean position, variance and trajectory for the target, respectively. The black diamond indicates the sensor. The solid line joining the target and sensor indicates that the sensor is assigned to the target.\label{fig:uni_assign_tracking_invcond}}
\end{figure*}

\begin{figure}[htb]
\centering{
\subfigure[]{\includegraphics[width=0.9\columnwidth]{figs/com_greedy_general_uni_trace_meanerr.eps}}
\subfigure[]{\includegraphics[width=0.9\columnwidth]{figs/com_greedy_general_uni_trace_coverr.eps}}
}
\caption{Comparison of mean error (a) and trace of covariance (b) for three-target tracking within 100 time steps by Greedy General Assignment~\cite{fisher1978analysis} with $\text{trace}(\mathbb{O}(p_{t_l}))$ and Greedy \revo{Non-overlapping} Pair Assignment (Algorithm~\ref{algorithm:unique_pair_assignment})  with $\text{trace}(\mathbb{O}(p_{t_l}))$. \label{fig:comparsion_gen_uni_log_mean_trace}}
\end{figure}

\begin{figure}[htb]
\centering{
\subfigure[]{\includegraphics[width=0.9\columnwidth]{figs/comp_unipair_three_metric_meanerr.eps}}
\subfigure[]{\includegraphics[width=0.9\columnwidth]{figs/comp_unipair_three_metric_coverr.eps}}
}
\caption{Comparison of mean error (a) and trace of covariance (b) for three-target tracking within 100 time steps by Greedy \revo{Non-overlapping} Pair Assignment (Algorithm~\ref{algorithm:unique_pair_assignment})  with $\underline{C}^{-1}(O(p_{t_l}))$, $\log\det(\mathbb{O}(p_{t_l}))$ and $\text{trace}(\mathbb{O}(p_{t_l}))$. \label{fig:comparsion_uni_logcondtrace_mean_trace}}
\end{figure}

\revo{\subsection{Baseline Comparisons}}
The \revo{Non-overlapping} Pair Assignment (Problem~\ref{prob:unique}) is NP-Complete. Therefore, finding $\omega(\textrm{OPT})$ is infeasible in polynomial time. In order to empirically evaluate the Greedy \revo{Non-overlapping} Pair Assignment (Algorithm~\ref{algorithm:unique_pair_assignment}), \revo{we use two baselines. When the number of sensors and targets is small, we compute the optimal solution using brute-force. When the number of sensors and targets is large, we compute an upper bound on the optimal solution value by solving a relaxed version of Problem~\ref{prob:unique}. In this relaxed version, one sensor is allowed to be assigned to multiple targets (unlike Problem~\ref{prob:unique}). However, we still require a pair of sensors to be assigned to at most one specific target.}

We formulate the new assignment as Relaxed Pair Assignment (Problem \ref{prob:perfect}). It is clear that solving Relaxed Pair Assignment problem optimally gives us an upper bound of the optimality for \revo{Non-overlapping} Pair Assignment problem. We can use this upper bound for the comparison of the greedy approach in \revo{Non-overlapping} Pair Assignment.
\begin{problem}[Relaxed Pair Assignment] Given a set of sensors, $\mathcal{S} := \{s_0,\ldots,s_N\}$ and a set of targets, $\mathcal{T} := \{t_0,\ldots,t_L\}$, find an assignment of \revo{non-repetition} pairs of sensors to targets:
\begin{equation}
\text{maximize} \sum_{l=1}^L \omega(\sigma_1(t_l),\sigma_2(t_l), t_l)
\end{equation}
with the added constraint that all pairs are \revo{non-repetition}, that is, $\forall k, l= 1,\ldots, L$, $k\neq l$, $\sigma_1(t_k) \neq \sigma_1(t_l)$ \revo{or} $\sigma_2(t_k) \neq \sigma_2(t_l)$.
\label{prob:perfect}
\end{problem}

The Relaxed Pair Assignment can be solved optimally by using maximum weight perfect bipartite matching (MWPBM)~\cite{cormen2009introduction}.
Note that a sensor can be matched in multiple distinct pairs and assigned to multiple targets. \revo{This violates the constraint in Problem~\ref{prob:unique} where each sensor can be matched to at most one pair and assigned at most once. } The MWPBM can be solved using the Hungarian algorithm~\cite{kuhn1955hungarian} in polynomial time.

\revo{While the Relaxed Pair Assignment computes an upper bound for $\omega(\textrm{OPT})$, we can compute $\omega(\textrm{OPT})$ exactly using brute-force when $N$ and $L$ are small  by enumerating all the possibilities. There are $\prod_{l=0}^{L-1} \binom{N-2l}{2}$ possible cases. Thus, the brute-force algorithm has an exponential running time.

Figure~\ref{fig:comparsion_perfect_unique_bf} shows the total value of the greedy algorithm, $\omega(\text{GREEDY})$, the brute-force algorithm, $\omega(\text{OPT})$, and the MWPBM, $\omega(\text{MWPBM})$ for log det and inverse condition number. We simulate the following environment: $N=2L$, the positions of sensors and targets are generated randomly within $[0,100]\times [0,100]\in\mathbb{R}^{2}$ for 30 trials for each $L$, and the target's maximum control input is $u_{o,\max}=1$.

We run the comparison code on a MacBook Pro with 2.6 GHz Intel Core i5 and  8 GB Memory. When $L=7$ and $N=14$, MATLAB runs out-of-memory when running the brute-force algorithm (there are $681080400$ possible cases for each trial). When $L=6$ and $N=12$, brute-force could not finish after running for 25 hours. Thus, we only consider the case when $L$ is varied from $1$ to $5$.

From Figure~\ref{fig:comparsion_perfect_unique_bf}, we observe that $\omega(\text{MWPBM})$ is the highest because the MWPBM gives the upper bound of  $\omega(\text{OPT})$. More importantly, $\omega(\text{GREEDY})$ is close to $\omega(\text{OPT})$ and much higher than the theoretical bound of $\frac{1}{3}\omega(\textrm{OPT})$. }

\revo{Next, we compare $\omega(\textrm{GREEDY})$ and $\omega(\textrm{MWPBM})$, without brute-force, for larger values of $L$ and $N$. We vary $L$ from 1 to 20 and set $N=2L$.}  For both observability measures, Figure~\ref{fig:comparsion_perfect_unique} shows that $\omega(\textrm{GREEDY})$ is close to $\omega(\textrm{MWPBM})$ and much higher than $\frac{1}{3}\omega(\textrm{MWPBM})$. Thus, even though we give a theoretical $1/3$--approximation for the greedy algorithm, it performs much better in practice.

\begin{figure}[htb]
\centering{
\subfigure[]{\includegraphics[width=0.8\columnwidth]{figs/compare_relax_uni_logdet.eps}}
\subfigure[]{\includegraphics[width=0.8\columnwidth]{figs/compare_relax_uni_invcond.eps}}
}
\caption{Comparison of \rev{the total value charged} by greedy approach (Algorithm~\ref{algorithm:unique_pair_assignment}) with the brute-force algorithm and the maximum perfect pair matching by using log det $\log(\det(\mathbb{O}(p_{t_l})))$ (a) and inverse condition number $\underline{C}^{-1}(O(p_{t_l}, u_{t_l}))$ (b), respectively.\label{fig:comparsion_perfect_unique_bf}}
\end{figure}

\begin{figure}[htb]
\centering{
\subfigure[]{\includegraphics[width=0.8\columnwidth]{figs/comparsion_perfect_unique_submod.eps}}
\subfigure[]{\includegraphics[width=0.8\columnwidth]{figs/comparsion_perfect_unique.eps}}
}
\caption{Comparison of \rev{the total value charged} by greedy approach (Algorithm~\ref{algorithm:unique_pair_assignment}) with the maximum perfect pair matching by using log det $\log(\det(\mathbb{O}(p_{t_l})))$ (a) and inverse condition number $\underline{C}^{-1}(O(p_{t_l}, u_{t_l}))$ (b), respectively.\label{fig:comparsion_perfect_unique}}
\end{figure}

\section{Conclusion}
\label{sec:Conclusion}
In this paper, we solved sensor assignment problems to improve the observability for target tracking. We derived the lower bound on the inverse of the condition number of the observability matrix for a system with a mobile target and $N$ stationary sensors. The lower bound considers only the known part of the observability matrix --- the sensor-target relative position and an upper bound on the target's speed. We showed how this lower bound can be employed for sensor selection.
We considered two sensor assignment problems for which we presented constant-factor approximation algorithms. While we presented the two algorithms as alternatives to each other, they can also be combined to give better results. \rev{We proved the log det, $\log\det(\mathbb{O}(p_{t_l}))$ is submodular and monotone only when $\mathbb{O}(p_{t_l})$ is non-singular and performs better in terms of tracking than the trace (which is submodular and monotone). We can combine the greedy general assignment with the greedy \revo{non-overlapping} pair assignment to use log det as the observability measure. First, we use the \revo{non-overlapping} assignment to assign a pair of sensors to each target to make sure that $\mathbb{O}(p_{t_l})$ is non-singular. Then, we can improve the assignment strategy by using the greedy general assignment strategy to assign more sensors to each target.}

Our immediate work is focused on assigning sensors to cover an area instead of tracking a group of targets. Another avenue is designing an efficient set covering strategy based on observability measures which are submodular and monotone. \revo{In many scenarios, the sensors are actually robots that are mobile~\cite{spletzer2003dynamic,martinez2006optimal,hollinger2009efficient,chung2006decentralized,huang2015bank,zhou2017active,zhou2018active,khodayi2018distributed}. in such cases, in addition to solving the assignment problem, we can also optimize the trajectories of the sensors. An immediate future work is to devise joint assignment and planning algorithms for better observability. }

Many of the results presented can be easily extended to 3D tracking. We conjecture that at least three sensors will be required for ensuring non-trivial lower bounds of the inverse condition number in 3D. The \revo{non-overlapping} pair assignment can be extended to assigning \revo{non-overlapping} triplets. We conjecture that this algorithm will yield a $1/4$-approximation.

redpajama_set_name: RedPajamaArXiv


In this miniature nature art original watercolor paintingI painted blue columbine purple monkshood and lavender flower mint. This is a lovely botanical garden watercolor with a bunch of detail for a small work of art.
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text:
Book Reviews By For Your Marriage Staff and Associates
Join us each month for a review of a book pertaining to marriage, dating, family life, children, parenting, and all other things For Your Marriage.
The Catholic Guide to Dating After Divorce
by Lisa Duffy
Related Topics: Church Teaching, Dating, Divorce, Getting Serious, Marriage
The Catholic Church is often called a nourishing mother, and those of her children who suffer through a divorce are no less deserving of her guidance and support. For this reason, Lisa Duffy's book The Catholic Guide to Dating After Divorce offers a compassionate lifeline so many Catholics need to help direct their lives after divorcing. Perhaps a more accurate title might be The Catholic Guide to HEALING After Divorce, because this book is so much more than a "how-to" book on finding a perfect mate; rather, this guide offers an entirely holistic approach to help the divorced Catholic find healing and peace. Those reading this book will be affirmed in Christ's love, fortified in His teachings on marriage, and encouraged by Duffy's own personal journey.
Lisa Duffy's friendly and uplifting tone radiates off the page. The book's opening chapter encourages the reader, first and foremost, to draw strength from Christ and not to turn away from His Church during this difficult time. Duffy knows firsthand the isolation that can come after a divorce, and she admits that it was a constant struggle for her to remember that God's love is unconditional. God has a unique plan for each person and Duffy asks the reader to be open to discovering that plan.
To further the deep, personal evaluation necessary for healing and growth, each chapter ends with both a quiz and reflection questions. Duffy also encourages the use of a journal as one reads this guide, because as she advises, "getting these thoughts and ideas out of [the] head and onto paper will not only provide a cathartic experience but also help [one] assess [one's] self more objectively."
The Catholic Guide to Dating After Divorce shares five qualities that free a person to love, and the first crucial quality is availability. Duffy writes honestly about her own mistakes; she began dating before she was truly available both in the eyes of the Church and emotionally. By God's grace and some serious soul searching, she eventually did the very important work necessary to be completely available to date and fall in love. Duffy points to three things that will help a person to discern their availability. First, she advises individuals to consider the possibility of reconciliation with an ex-spouse. Second, she encourages readers to go through the declaration of nullity (or annulment) process. Duffy explains that receiving a declaration of nullity about one's first marriage is not merely necessary for marrying in the Church, but "if approached with a heart that is open to seeking God's will and desires healing," the annulment process itself can help an individual grow in self-knowledge, learn from mistakes, and grow spiritually while gaining complete confidence in the direction of one's life. Third, Duffy emphasizes the importance of healing spiritually and emotionally in order to be available to love another person unconditionally. Spending time in prayer and giving of oneself through volunteer work in the Church or community are aspects of the healing process. The quality of availability discussed in this initial chapter acts as a springboard for the other qualities discussed by Duffy, and thus this chapter is by far the most important of the book, and the most likely to help the reader rebuild after a divorce.
The next three chapters discuss being affectionate, being a communicator, and being faithful. Of these important qualities, the chapter on being a communicator is most valuable. Divorce, as Duffy points out, often involves a breakdown of communication. Learning to avoid harmful patterns and foster healthy communication skills are keys to a successful future relationship. Learning to identify particular areas of struggle and then applying a practical way to overcome those vices with the corresponding virtue is one of the many nuggets of wisdom found in this chapter. Duffy attempts to cover a great deal of material in one chapter, including the four temperaments, the five love languages, as well as communication pitfalls and bad habits. While the scope of the book does not allow for an in-depth look at all these topics, the author provides a very thorough appendix of helpful books that discuss these subjects in further detail.
In the chapter "Being Faithful," Duffy shares her own experiences with God's mercy in her life through Eucharistic adoration and reception of the Church's sacraments. The reflection questions in this chapter are great points of consideration for any Catholic thinking about entering into another relationship, because as Duffy explains, "as faith grows so will confidence, trust in God, peace of mind and happy disposition." A person at peace is a person ready to give and receive love.
Magnanimity or largeness of spirit is the final quality discussed in Dating After Divorce, and Duffy reiterates that a magnanimous person is capable of moving past hurt, even forgiving someone who has wounded them deeply. As Pope Francis says, such a person has a big heart open to God and others. A magnanimous person knows their life has a greater purpose; this is both attractive to others and a recipe for a successful future relationship.
Lisa Duffy concludes her guidebook by sharing her own joy in meeting her husband and experiencing the miracle of motherhood. This book weaves together personal stories, clear Church teaching, and great practical advice in an empathetic and abundantly hopefully manner. The Catholic Guide to Dating After Divorce provides a much-needed support to Catholics that need healing after divorce.
Kathleen O'Beirne is a wife, mother of four, a freelance writer and works as a volunteer in the Marriage Preparation Program for the Arlington Diocese.
Disclaimer: Book reviews do not imply and are not to be used as official endorsement by the USCCB of the work or those associated with the work. Book reviews are solely intended as a resource regarding publications that might be of interest to For Your Marriage visitors.
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Finding the right person can be easier than being the right person.

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Annie Wilmot Lance, was born at Kingsbridge, Devon in 1857, daughter of Walter Lance, a relieving officer & registrar of births and deaths, and his wife Eliza née Savage, who married at Blandford, Devon in 1846. In 1861, a 4 year old, living at Rope Close, Blandford, Devon with her parents, 36 year old Walter, born Dorchester and 38 year old Eliza, born Froyle, Hampshire and five siblings, Walter Herbert 14, Eliza Letitia 12, Emily 9, George 5 and Louisa 2, all born in Dorset. In 1881, a 23 year old 'secretary clerk' at Ipswich High School, Berners Street, Ipswich. A flower painter who exhibited at Ipswich Fine Art Club in 1884, from Hillside, Christchurch Street, Ipswich 'Cottage at Applethwaite', she also exhibited from the same address, at the Royal Academy and the Royal Society of British Artists 1882-1893. She died at 13 Clarendon Avenue, Leamington Spa, Warwickshire on 13 March 1939, aged 82, she was unmarried. Her time in Ipswich is elusive and she may have been staying with Henry Robertson [q.v.] at 102 Christchurch Street, Ipswich during her Ipswich painting period.


The Poseidon, an ocean liner bigger than the Queen Elizabeth and Mary combined, is charting its course on New Year's Eve when just after midnight a 90' tidal wave hits. It is the last thing that everyone onboard sees before drowning — the Poseidon is turned upside down, with only a handful of survivors. The lucky few take a journey through the bowels of the ship in an attempt to survive.
If you are familiar with The Poseidon Adventure, why not let others know?
The other sinking boat movie.
Before James Cameron's Titanic, there was The Poseidon Adventure. A thrilling survival movie in a sinking ship.
there is a new movie directed by John Putch.
Platinum Disc releases a new product.


text:
Daiiibabyyy - Updated Jan 2023
Harrisburg, Pennsylvania, USA
More Facts of Daiiibabyyy
Taurus, Pisces, Scorpio
Instagram Star, TikTok Star
$1 Million - $5 Million
two (sister: Lyric Thomas and brother: Curly Head Monty)
Daiiibabyyy Instagram
Personal Web
Daiiibabyyy Personal Web
View more / View less Facts of Daiiibabyyy
Daiiibabyyy is a popular social media personality. Daiiibabyyy is best known for posting lip-sync videos on TikTok and the photo-sharing app Instagram.
5 Daiiibabyyy – Social Media
6 Daiiibabyyy – Net Worth 2022
Daiiibabyyy came to this world on July 19, 2003. She is from Harrisburg, Pennsylvania, USA. She turns x as of 2020 and her birth sign is Cancer. Moreover, she spent her childhood with her siblings, Lyric Thomas and Curly Head Monty. Ging to her educational background, she was homeschooled. Her father is of Jamaican origin.
Caption: Daiiibabyyy posing for a photo. Source: Instagram
All her siblings are famous on the internet. Her brother, Curly Head Monty is a famous TikTok star and is known for his amazing dancing skills. He is also famous on YouTube as he has been featured in the list of 'Top 10 Best Dance YouTube Videos' on several occasions.
Similarly, her sister Lyric Thomas is a famous rapper and dancer. She appeared in the fourth season of 'Lifetime' channel's reality television series, 'The Rap Game'. In the show, Lyric was seen along with other famous rappers like Street Bud and Jordan Young.
And, following in her siblings' footsteps, Daiiibabyyy began posting video clips on her TikTok account in 2016. Then, within months from creating her account, she started gaining popularity because of her lip-sync videos. At present, she has more than 1.1 million fans and over 7 million emoji loves on TikTok.
And after gaining popularity on TikTok, she went on to create an Instagram page, where she started posting a number of pictures and video clips. She mostly uses Instagram's split-screen features to convey certain stories through pictures. She is also famous for posting some of her dance video clips, in which she is seen showing off her belly dancing skills.
Apart from Instagram and TikTok's popularity, she is also active on other social media platforms, such as Snapchat and Twitter. While she doesn't own a YouTube channel yet, her TikTok videos can be found on various YouTube channels owned by her fans and followers.
Caption: Daiiibabyyy while making a TikTok video. Source: TikTok
Speaking of her personal life, she is currently single. She is not dating anyone at the moment. Daiiibabyyy has got enough time for linking into a relationship in the future. However, she might be at the heart of many boys of her age due to her talent and appearance. It is just a matter of time to seeing her in a relationship.
Similarly, she has never been into any sort of rumors and controversies. She is aware of the unnecessary rumors and scandals and has always stayed away from it without letting them harm her personal or professional life. However, as of now, she is busy focusing on her promising future.
Daiiibabyyy has a fine and sexy body figure. She has a healthy body status. Her body measurements are not available. However, she has a matching height and weight which suits her body figure. Moreover, she has a beautiful pair of dark brown eyes and curly black hair. Daiibabyyy often dyes her hair into different colors like silver, brown, blonde, blue, red, pink, and many more.
Caption: Daiiibabyyy clicking a photo. Source: Instagram
Daiiibabyyy – Social Media
Daiiibabyyy is active on Instagram with more than 487K followers. Her Instagram handle is '@therealdaiibabyy'. She also has a backup Instagram account under the username '@therealdaiibabyy_backup'. There, she has earned over 2K followers.
Likewise, she is also available on Snapchat under the username '@realdaiibabyy'. Daiibabyyy has gathered more than 1.1 million followers and more than 7 million likes on her TikTok account. She is not active on Twitter.
Daiiibabyyy – Net Worth 2022
As a social media personality, she would earn a decent amount of money through his profession. Undoubtedly, her wealth will increase in the upcoming years, assuming that she successfully continues her career. However, she has a net worth of around $1 million – $5 million.
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NEW YORK, Sept. 20, 2017 /PRNewswire/ -- CELLPHANE, Inc. today announced the launch of BROC SPROUT 2, a 100% nutraceutical-grade whole broccoli sprout capsule containing the two essential components necessary for the human body to internally produce Sulforaphane. The dynamic, phytochemical Sulforaphane molecule is widely considered by the scientific community to be a critical trigger of the human cells' defense system against conditions such as breast and prostate cancer, high blood pressure, high cholesterol, diabetes, asthma and obesity to name a few.
BROC SPROUT 2 is being launched on www.jet.com and www.Amazon.com.
The remarkable benefits of Sulforaphane, nature's most potent natural activator of the human cell's Nrf2 defense system, have been scientifically-documented since it was first identified in 1992 at The Johns Hopkins University School of Medicine.
"Sulforaphane is an amazing biomolecule with the capacity to decisively enhance the defense function of practically every cell in the human body," said Franco Lagudi, president of CELLPHANE, Inc. "Its benefits have been well-documented in numerous scientific research articles, describing not only its ability to activate the body's own defense system in combatting disease-causing elements that affect lives on a daily basis but also its ability to enhance our everyday good health by favorably modifying DNA expression." Websites including the U.S. National Library of Medicine, Oregon State University and The Johns Hopkins University School of Medicine provide information on studies supporting the benefits of Sulforaphane.
Young whole broccoli sprouts are the most potent natural dietary source of Sulforaphane. They contain the two crucial components needed for the body to create Sulforaphane: the naturally-occurring Myrosinase enzyme and the Glucoraphanin compound. Only when these two essential components are mixed with saliva or water is Sulforaphane produced. Click here to learn more about this process.
BROC SPROUT 2 capsules are one-of-a-kind, as the superior potency of young whole broccoli sprouts is preserved while making them into capsules, therefore ensuring the delivery of Sulforaphane to the body's cells. It's a 100% natural product made solely from young whole broccoli sprouts without the use of herbicides, pesticides or harmful chemicals of any kind.
There are products currently on the market that claim to produce Sulforaphane, or even claim to contain Sulforaphane, Lagudi points out. Some of these products are branded as 'extracts', others are derived from seeds or contain other ingredients; some of these even brand the glucoraphanin as Sulforaphane Glucosinolate (SGS). Virtually none contain the naturally-occurring Myrosinase enzyme. "It's important for consumers to understand that it doesn't matter how much of the Glucoraphanin compound is present if a product does not contain the naturally-occurring Myrosinase enzyme (as found in young whole broccoli sprouts). Without Myrosinase, no Sulforaphane can be generated by the product," he adds.
BROC SPROUT 2 is a collaboration with leading scientists in the field of Nutrigenomics wherein they have optimized the unique growing and drying systems used by CELLPHANE. "The study of Nutrigenomics centers on how the molecules in food trigger the protective genes in human cells; this is a research area with amazing potential," said Lagudi. Such protective genes are Nature's way of preventing and treating disease.
When an individual swallows BROC SPROUT 2 capsules, the chemical reaction between the Glucoraphanin compound and the Myrosinase enzyme takes place and Sulforaphane is produced within their body.
For more information on BROC SPROUT 2 visit: www.brocsprout2.com.


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Janowo (deutsch Johannisberg) ist ein Wohnplatz in der Woiwodschaft Westpommern in Polen.

Der Wohnplatz liegt in Hinterpommern, etwa 35 Kilometer südöstlich von Stettin und etwa neun Kilometer nordöstlich von Pyrzyce (Pyritz), etwa 600 Meter vom südöstlichen Ortsrand des Dorfes Stary Przylep (Alt Prilipp) entfernt.

Im Jahre 1832 verlegte Johann David Schönfeldt, der Besitzer des Freischulzengutes von Alt Prilipp, seinen Hof aus dem Dorf heraus und erbaute sich südöstlich des Dorfes einen Gutshof, der den Namen Johannisberg erhielt. Schönfeldt hatte sich zu seinem Freischulzengut weiteres Land hinzugekauft.

Im Jahre 1868 gehörten zu Johannisberg auch eine Windmühle und eine Ziegelei. Zu dem Gut Johannisberg gehörten damals 544 Morgen Land. Um 1912 hatte Johannisberg 101 Einwohner.Meyers Orts- und Verkehrslexikon des Deutschen Reiches. Band 1. Leipzig 1912, S. 46. Nachdruck: Baltimore 2000, ISBN 0-8063-1634-9. (Online, unter Alt Prilipp)

Vor 1945 gehörte Johannisberg zur Gemeinde Alt Prilipp und mit dieser zum Kreis Pyritz der preußischen Provinz Pommern. im Informationssystem Pommern.

Nach dem Zweiten Weltkrieg kam Johannisberg, wie ganz Hinterpommern, an Polen. Es erhielt den polnischen Ortsnamen Janowo. Heute gehört es zur Gmina Warnice (Landgemeinde Warnitz) und mit dieser zum Powiat Pyrzycki (Pyritzer Kreis) der polnischen Woiwodschaft Westpommern.

Literatur
 Heinrich Berghaus: Landbuch des Herzogtums Pommern. Teil II. Band 3. Anklam 1868, S. 607–608. (Online, im Artikel über Alt Prilipp)

Fußnoten

Ort in Pommern
Ort der Woiwodschaft Westpommern
Gmina Warnice
Gegründet 1832

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text:
Home » Weekly Torah Portion » Advanced » The Guiding Light
https://www.aish.com/tp/i/gl/566416061.html
Sorcery and Lucky Charms
by Rabbi Yehonasan Gefen
Yosef, in the guise of the Viceroy of Egypt, sends back his brothers and commands his son to hide his special silver goblet in the sack of Binyamin. While the brothers are on their way home, Yosef sends his son to confront them and claim that one of them had stolen his goblet. His son describes the goblet as "the one from which my master drinks, and with which he regularly divines (nachesh yenachesh)."[1] He tells the brothers of the importance of this goblet in that it enables Yosef to tell the future.
When the brothers are returned to Yosef, in his role as viceroy, he emphasizes the importance of this goblet and how it enables him to do nichush (divining). Rabbi Yissachar Frand asks why there is so much emphasis on 'nichush'? Why is it so important for Yosef to convey to his brothers his reliance on this form of sorcery?
Rabbi Shimson Raphael Hirsch helps shed light on this question with his explanation of the deeper meaning of the word 'nichush'. He notes that it is connected to the word, nachash, meaning snake. Unlike all other animals, a snake does not walk in a straight line, rather it slithers, taking a crooked path. Normally in life, in order to achieve anything, a person has to go from one point to another point, going through a process involving hard work and effort. This is the law of cause and effect - one works, he makes money, he is good, he is rewarded.
However, there is a way to 'beat the system', to achieve a goal without having to go through the hard work that is normally required. This is the concept of all forms of sorcery, one of which is nichush. In Yosef's case it was a special goblet that enabled him to 'cheat the system' by seeing the future. This is comparable to the path of a snake; because it is crooked, it avoids the straight path of hard work and takes the easy way out. Yosef wanted to convince his brothers that he was an Egyptian. A Jew who follows the Torah knows that the only way to achieve goodness in this world is to be good and to act appropriately. Reward will come to those deserving of it.
Rabbi Frand continues:
"Egyptians don't think like that. They want to receive the goodness without having to work for it. They want to achieve undeserved goodness. When superstition controls one's life, he is liberated from having to work in order to achieve. If it is "in the cards" or "in the stars" - according to this philosophy - one can get things even though he does not deserve them. This allows one to act however he wants."
Yosef, in trying to disguise himself from his brothers, is communicating to them that he swears by this 'non-Jewish' approach to life. Indeed, when he tells them that a man of his position needs his goblet in order to practice nichush, he implicitly insinuates that his great success and rise to his position of power was because of his power of nichush.
Such an idea is foreign to Yaakov's children and should be foreign to anyone raised in a home of Jewish values. We believe that one earns reward by deserving reward. There are no superstitious shortcuts to expect or to rely upon.
However, there can be a tendency even for a Jew to want to rely on 'segulos' (charms) to solve their problems or ease their suffering. Rabbi Yitzchak Berkovits points out that this is somewhat missing the point. For example, God does not send us difficulties merely so that we can perform some kind of segula even if it is effective in ending the pain. Rather, God wants us to grow from the challenge into a greater people. This does not necessarily mean that all segulos are negative, but one should not forget the purpose of challenges - that God is telling us to grow.
In this vein, when some Torah sages are asked about segulos to help people, they often answer with 'segulos' that are related to increased Torah observance and Torah learning. A young, newly married man once asked a Sage for a segula to increase Shalom Bayit (marital harmony) on Erev Shabbat when there tends to be a lot of stress. The Sage answered that he should take a broom and help clean the house! He was communicating to the young man that the best 'segulos' are those that do not take short-cuts to achieve success, but that involve hard work and improving one's character traits.
1. Bereishis, 44:5.
Rabbi Yehonasan Gefen
Rabbi Yehonasan Gefen is a close student of Rabbi Yitzchak Berkovits and a senior member of the Jerusalem Kollel. Rabbi Gefen's has spent over 12 years of intensive Torah and Talmudic study at Aish HaTorah and the Jerusalem Kollel, and has a degree in history and politics from the University of Birmingham, UK. Rabbi Gefen has written numerous articles for Hamodia and other publications. His writing inspires people across the spectrum of Jewish observance to live with the vibrancy and beauty of Torah.
Rabbi Gefen has recently published his fifth book. His new book delves deeply into the characters of great kings such as David, Shlomo, Chizkiyahu, and Yoshiyahu and those of the most maligned of kings, such as Yeravam, Achav, and Menashe, focusing on the most significant aspects unique to each king. Its readers will develop a whole new understanding of the kings of Israel and the overall institution of the kingship of Israel. Click here to order: https://www.feldheim.com/king-david-and-the-kings-of-israel
Visit Rabbi Gefen's site at http://rabbiygefen.blogspot.com/


text:
Q: JavaScript result happening before callbacks complete I'm totally new to JS having jumped in a few days ago to try make a chrome extension, so sorry if this is a simple problem, but I can't seem to figure it out.
My original function was to simply download an image and increment the stored count by 1 and add on the file size. However on a page of images it hit the write limits of chrome so I decided to count the values and write them at the end.
Initially the return value happened much later than when the function was executed (so it returned nothing), so I looked up how to fix it and got it working with a callback. However, although it waits for the callbacks, the code just continues past the callbacks and the part afterwards is executed before anything else, meaning the final count will always be 0.
    // Loop through all urls
    var approx_filesize = 0;
    for(var i = 1; i < all_urls.length; i++){
        var image_url = all_urls[i];
        _download_image(image_url, folder_name, function(item){
            approx_filesize += parseInt(item);
        });
    }

    // This happens before any _download_image functions complete
    alert('end' + approx_filesize);

    // Write to storage
    chrome.storage.sync.get({
        download_count: 0,
        download_size: 0
    }, function(items) {
        chrome.storage.sync.set({
            download_count: parseInt(items.download_count) + all_images_data.length - 1,
            download_size: parseInt(items.download_size) + approx_filesize
        }, function() {
        });
    });

I just tried moving the loop into its own callback function and still had no luck, the alert runs before the first function completes.
function image_url_loop_callback(callback, folder_name, all_urls){

    var approx_filesize = 0;
    for(var i = 1; i < all_urls.length; i++){
        var image_url = all_urls[i];

        _download_image(image_url, folder_name, function(filesize){
            approx_filesize += parseInt(filesize);
        });
    }
    callback(approx_filesize);
}

image_url_loop_callback(function(approx_filesize){
    alert(approx_filesize);
}, folder_name, all_urls);

How do I make it so that the loop completes before anything else is done?
Edit: Got it working with promise, here's the adjusted code:
new Promise( function(resolve, reject) {
    var count = 1;
    var num_items = all_urls.length;
    var approx_filesize = 0;
    for(var i = 1; i < num_items; i++){
        var image_url = all_urls[i];

        _download_image(image_url, folder_name, function(item){
            approx_filesize += parseInt(item);
            count ++;
            if(count == num_items){
                resolve([num_items, approx_filesize]);
            }
        });
    }
}).then( function(stuff) {
    var num_items = stuff[0];
    var approx_filesize = stuff[1];
    chrome.storage.sync.get({
        download_count: 0,
        download_size: 0
    }, function(items) {
        chrome.storage.sync.set({
            download_count: parseInt(items.download_count) + num_items,
            download_size: parseInt(items.download_size) + approx_filesize
        }, function() {
        });
    });
});

A: Basically, you have to handle the asynchronous aspect of JavaScript.
To do so, you have to use a Promise.
This works this way:
new Promise( () => {

    // My asynchronous code

}).then( () => {

    // My code which need to wait for the promise resolution.

});

If you are working with only the latest versions of browsers, you can also have a look at async/await keywords which make asynchronous handling much easier than regular promises (but still are promises).
EDIT: As this answer required further explanation and proper code snippets, I edited it to answer a comment.
This example maybe easier to understand:
let myFoo = "Hello";

test = new Promise( (resolve) => {

    console.log(myFoo);

    myFoo = "World!";

    setTimeout(() => {

        resolve();

    }, 4000);

}).then( () => {

    console.log(myFoo);

});

This will print "Hello" immediately, and "World!" 4 seconds after.
This is how you work with promises. You can perfectly edit variables which are defined in a scope outside of the promise. Please don't use var, just stick to let and define a decent scope.

A: Due to javascript's async nature you have to use promises:
https://developers.google.com/web/fundamentals/getting-started/primers/promises

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Seal coating is an important pavement maintenance procedure that can prolong the life of your pavement and improve its appearance. It also provides one of the greatest returns on your maintenance dollar. Asphalt sealcoating is to protect the asphalt on parking lots or drive ways and make looks better seal for the asphalt that protects your investment from wear and tear, rain, and all the harsh chemicals leaking from vehicles. Our professional sealcoating services provide a seal that you can rely on. It also restores the slate black appearance, while reducing future maintenance costs.