We consider an environment with multiple reward functions. One of them represents goal achievement and the others represent instantaneous safety conditions. We consider a scenario where the safety rewards should always be above some thresholds. The thresholds are parameters with values that differ between users. We efficiently compute a family of policies that cover all threshold-based constraints and maximize the goal achievement reward. We introduce a new parameterized threshold-based scalarization method of the reward vector that encodes our objective. We present novel data structures to store the value functions of the Bellman equation that allow their efficient computation using the value iteration algorithm. We present results for both discrete and continuous state spaces.
I generally like the paper. The paper discussed a constrained value iteration setting where the safety contraints must be greater some threshold, and thresholds \delta are parameters. The paper attempts to develop an value iteration algorithm to compute a class of optimal polices with such a parameter. The algorithm is mainly based on a special design of representation/data structure of PWC function, which can be used to store value functions and allows to efficiently compute several relevant operations in bellman equation. A graph-based data structure is developed for continuous state domains and hence value iteration can be extended to such cases. 
In general, the paper presents an interesting direction which can potentially help solve RL problems with the proposed constraint setting. However, the paper spends lots of effort explaining representations, but only a few sentences explaining about how the proposed representations/data structures can help find a somehow generic value iteration solution, which allows to efficiently compute/retrieve a particular solution once a \delta vector is specified. The paper should show in detail (or at least give some intuitive explanations) that using the proposed method can be more efficient than solving a value iteration for each individual constraint given that the constraints are independent. Specifically, the author uses the patient case to motivate the paper, saying that different patients may have different preferred thresholds and it is good to find class of policies so that any one of those policies can be retrieved once a threshold is specified. However, in this case, when dealing with only one patient, the dimension of reward is reduced to 1 (d = 1), while the computation of the algorithm is exponential in d, plus that the retrieval time is not intuitive to be better, so it is unsure whether computing such a class of policies worth.
In terms of novelty, the scalarization method of the vector-valued reward seems intuitive, since breaking a constraint means a infeasible solution. Furthermore, it is also unclear why the representation of PWC in discrete case is novel. A partial order on a high-dimensional space is naturally to be based on dominance relation, as a result, it seems natural to store value function by using right-top coordinates of a (hyper)rectangle.
As for the clarity, though the author made the effort to explain clearly by using examples after almost every definition/notation, some important explanations are missing. I would think the really interesting things are the operations based on those representations. For example, the part of computing summation of two PWC function representation is not justified. Why the summation can be calculated in that way? Though the maximum operation is intuitive, however, the summation should have some justification. I think a better way to explain those things is to redefine a new bellman operator, which can operate on those defined representations of PWC function. 
I think it could be a nice work if the author can improve the motivation and presentation. Experiments on some simple domains can be also helpful.