Integer Linear Programming Formulations for Double Roman Domination Problem
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For a graph G= (V,E), a double Roman dominating function (DRDF) is a function f : V →{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 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) > 2. In this paper, we consider the double Roman domination problem (DRDP), which is an optimization problem of finding the DRDF f such that ∑_v∈ V f (v) is minimum. We propose several integer linear programming (ILP) formulations with a polynomial number of constraints for this combinatorial optimization problem, and present computational results for graphs of different types and sizes. Further, we prove that some of our ILP formulations are equivalent to the others regardless of the variables relaxation or usage of less number of constraints and variables. Additionally, we use one ILP formulation to give an H(2(Δ+1))-approximation algorithm and provide an in approximability result for this problem. All proposed ILP formulations and approximation algorithm are evaluated on randomly generated graphs to compare the performance.
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