Solving the signed Roman domination and signed total Roman domination problems with exact and heuristic methods

TL;DR

This paper introduces exact and heuristic methods, including ILP, CP, and VNS, to solve signed Roman domination problems, demonstrating their effectiveness on various instances.

Contribution

It presents two ILP formulations, a CP model, and a VNS heuristic for signed Roman domination, with proofs and a polyhedral study, advancing solution approaches for these problems.

Findings

ILP models outperform other methods on most instances.

All methods find optimal solutions for small and medium instances.

VNS effectively guides search using a penalty function.

Abstract

In this paper we deal with the signed Roman domination and signed total Roman domination problems. For each problem we propose two integer linear programming (ILP) formulations, the constraint programming (CP) formulation and variable neighborhood search (VNS) method. We present proofs for the correctness of the ILP formulations and a polyhedral study in which we show that the polyhedrons of the two model relaxations are equivalent. VNS uses specifically designed penalty function that allows the appearance of slightly infeasible solutions. The acceptance of these solutions directs the overall search process to the promising areas in the long run. All proposed approaches are tested on the large number of instances. Experimental results indicate that all of them reach optimal solutions for the most of small and middle scale instances. Both ILP models have proven to be more successful than the other two methods.