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For a graph $G= (V,E)$, a double Roman dominating function (DRDF) is afunction $f : V \to \{0,1,2,3\}$ having the property that if $f (v) = 0$, thenvertex $v$ must have at least two neighbors assigned $2$ under $f$ or {atleast} 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, weconsider the double Roman domination problem, which is an optimization problemof 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 integerlinear programming formulation with polynomial number of constraints for thisproblem. Some additional valid inequalities and bounds are also proposed forsome of these formulations.} Further, we prove that {the first four modelsindeed solve the double Roman domination problem, and the last two models} areequivalent to the others regardless of the variable relaxation or usage of asmaller number of constraints and variables. Additionally, we use one ILPformulation to give an $H(2(\Delta+1))$-approximation algorithm. All proposedformulations and approximation algorithm are evaluated on randomly generatedgraphs to compare the performance.

Integer Linear Programming Formulations for Double Roman Domination Problem