An improved binary programming formulation for the secure domination problem

TL;DR

This paper introduces a more efficient binary programming formulation for the secure domination problem, reducing computational complexity and solving instances faster than previous models, also extending to a related secure connected domination problem.

Contribution

A new binary programming formulation with fewer constraints and variables for secure domination, and the first formulation for secure connected domination problem.

Findings

Fewer constraints and variables improve solving efficiency.

Significantly reduced runtime compared to previous formulation.

First formulation for secure connected domination problem.

Abstract

The secure domination problem, a variation of the domination problem with some important real-world applications, is considered. Very few algorithmic attempts to solve this problem have been presented in literature, and the most successful to date is a binary programming formulation which is solved using CPLEX. A new binary programming formulation is proposed here which requires fewer constraints and fewer binary variables than the existing formulation. It is implemented in CPLEX, and tested on certain families of graphs that have previously been considered in the context of secure domination. It is shown that the runtime required for the new formulation to solve the instances is significantly less than that of the existing formulation. An extension of our formulation that solves the related, but further constrained, secure connected domination problem is also given; to the best of the authors' knowledge, this is the first such formulation in literature.