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

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.

Several Roman domination graph invariants on Kneser graphs

“…More precisely, the model given by Burger et al (2013) is used to obtain the results presented in Remark 1 and Remark 2, Tatjana Zec et. al and the model exposed by Filipović et al (2022) is used to obtain the results presented in Remark 3. The formulations of these ILP models are given in Appendix A.…”

“…We used the ILP model from Filipović et al (2022) to find SRDN for some special cases of Kneser graphs which are provided in Remark 3. The SRDFs which correspond to these solutions and ILP model details are presented in Appendix A.…”

“…V \ V2 ∅ The ILP model from Filipović et al (2022) was implemented in Cplex solver Lima and Seminar (2010) to obtain relation between TRDN and domination number as well as with RDN values of SRDN for small Kneser graphs. It is stated as follows.…”