Empirical Analysis Of Heuristic and Approximation Algorithms for the The Mutual-Visibility Problem

TL;DR

This paper empirically evaluates heuristic and approximation algorithms for the NP-complete mutual-visibility problem, revealing their performance on various graph sizes and highlighting challenges in larger instances.

Contribution

It provides the first comprehensive empirical analysis of multiple algorithms for the mutual-visibility problem across diverse graph datasets.

Findings

Algorithms perform well on small graphs, matching theoretical bounds.

Solution sizes diverge from bounds on larger graphs, indicating complexity.

Genetic algorithms and heuristics perform best on known optimal graphs.

Abstract

The NP-complete mutual-visibility (MV) problem currently lacks empirical analysis on its practical behaviour despite theoretical studies. This paper addresses this gap by implementing and evaluating three distinct algorithms -- a direct random heuristic, a hypergraph-based approximation, and a genetic algorithm -- on diverse synthetic graph datasets, including those with analytically known $\mu(G)$ values and general graph models. Our results demonstrate that for smaller graphs, the algorithms consistently achieve MV set sizes aligning with theoretical bounds. However, for larger instances, achieved solution sizes notably diverge from theoretical limits; this, combined with the absence of tight bounds, complicates absolute quality assessment. Nevertheless, validation on known optimal graphs showed the Genetic Algorithm and other heuristics empirically performing best among tested methods.