  For a graph $G= (V,E)$, a double Roman dominating function (DRDF) is a
function $f : V \to \{0,1,2,3\}$ having the property that if $f (v) = 0$, then
vertex $v$ must have at least two neighbors assigned $2$ under $f$ or {at
least} one neighbor $u$ with $f (u) = 3$, and if $f (v) = 1$, then vertex $v$
must have at least one neighbor $u$ with $f (u) \ge 2$. In this paper, we
consider the double Roman domination problem, which is an optimization problem
of finding the DRDF $f$ such that $\sum_{v\in V} f (v)$ is minimum. We propose
{five integer linear programming (ILP) formulations and one mixed integer
linear programming formulation with polynomial number of constraints for this
problem. Some additional valid inequalities and bounds are also proposed for
some of these formulations.} Further, we prove that {the first four models
indeed solve the double Roman domination problem, and the last two models} are
equivalent to the others regardless of the variable relaxation or usage of a
smaller number of constraints and variables. Additionally, we use one ILP
formulation to give an $H(2(\Delta+1))$-approximation algorithm. All proposed
formulations and approximation algorithm are evaluated on randomly generated
graphs to compare the performance.

Combinatorics;Discrete Mathematics;Optimization and Control