A Game-Theoretic Approach to Solving the Roman Domination Problem

TL;DR

This paper introduces a game-theoretic framework for solving the Roman domination problem, proving the existence of Nash equilibria, designing distributed algorithms, and demonstrating improved solutions through simulations.

Contribution

It proposes a novel Roman domination game, establishes its properties, and develops efficient algorithms for finding optimal solutions in a distributed manner.

Findings

GSA converges to a Nash equilibrium in O(n) rounds.

EGSA finds better solutions in O(n^2) rounds.

Numerical simulations show EGSA outperforms existing algorithms.

Abstract

The Roamn domination problem is one important combinatorial optimization problem that is derived from an old story of defending the Roman Empire and now regains new significance in cyber space security, considering backups in the face of a dynamic network security requirement. In this paper, firstly, we propose a Roman domination game (RDG) and prove that every Nash equilibrium (NE) of the game corresponds to a strong minimal Roman dominating function (S-RDF), as well as a Pareto-optimal solution. Secondly, we show that RDG is an exact potential game, which guarantees the existence of an NE. Thirdly, we design a game-based synchronous algorithm (GSA), which can be implemented distributively and converge to an NE in $O(n)$ rounds, where $n$ is the number of vertices. In GSA, all players make decisions depending on the local information. Furthermore, we enhance GSA to be enhanced GSA (EGSA), which converges to a better NE in $O(n^2)$ rounds. Finally, we present numerical simulations to demonstrate that EGSA can obtain a better approximate solution in promising computation time compared with state-of-the-art algorithms.