Exact and heuristic algorithms for the weighted total domination problem

TL;DR

This paper introduces new exact and heuristic algorithms for the weighted total domination problem in graphs, significantly improving solution speed and quality over previous methods, and providing insights into instance characteristics affecting performance.

Contribution

The paper presents two new Mixed-Integer Programming models, solution frameworks, and a genetic algorithm for the weighted total domination problem, enhancing computational efficiency and solution quality.

Findings

Exact algorithms are up to 500 times faster than previous methods.

Instances with up to 125 vertices are solved to optimality within 1800 seconds.

The genetic algorithm often finds optimal or near-optimal solutions quickly.

Abstract

Dominating set problems are among the most important class of combinatorial problems in graph optimization, from a theoretical as well as from a practical point of view. In this paper, we address the recently introduced (minimum) weighted total domination problem. In this problem, we are given an undirected graph with a vertex weight function and an edge weight function. The goal is to find a total dominating set D in this graph with minimal weight. A total dominating set D is a subset of the vertices such that every vertex in the graph, including vertices in D, is adjacent to a vertex in D. The weight is measured as the sum of all vertex weights of vertices in D, plus the edge weights in the subgraph induced by D, plus for each vertex not in D the minimum weight of an edge from it to a vertex in D. In this paper, we present two new Mixed-Integer Programming models for the problem, and design solution frameworks based on them. These solution frameworks also include valid inequalities, starting heuristics and primal heuristics. In addition, we also develop a genetic algorithm, which is based on a greedy randomized adaptive search procedure version of our starting heuristic. We carry out a computational study to assess the performance of our approaches when compared to the previous work for the same problem. The study reveals that our exact solution algorithms are up to 500 times faster compared to previous exact approaches and instances with up to 125 vertices can be solved to optimality within a timelimit of 1800 seconds. Moreover, the presented genetic algorithm also works well and often finds the optimal or a near-optimal solution within a short runtime. Additionally, we also analyze the influence of instance-characteristics on the performance of our algorithms.