Binary Programming Formulations for the Upper Domination Problem

TL;DR

This paper introduces two binary programming formulations for the Upper Domination problem, enhancing solution efficiency with additional constraints and analyzing their performance on different graph types.

Contribution

The paper presents the first binary programming models for Upper Domination and demonstrates their effectiveness with added constraints across various graph classes.

Findings

Additional constraints improve formulation performance

First formulation generally outperforms the second

Second formulation better for very sparse graphs

Abstract

We consider Upper Domination, the problem of finding the minimal dominating set of maximum cardinality. Very few exact algorithms have been described for solving Upper Domination. In particular, no binary programming formulations for Upper Domination have been described in literature, although such formulations have proved quite successful for other kinds of domination problems. We introduce two such binary programming formulations, and show that both can be improved with the addition of extra constraints which reduce the number of feasible solutions. We compare the performance of the formulations on various kinds of graphs, and demonstrate that (a) the additional constraints improve the performance of both formulations, and (b) the first formulation outperforms the second in most cases, although the second performs better for very sparse graphs. Also included is a short proof that the upper domination number of any generalized Petersen graph P(n,k) is equal to n.