Strength of the Upper Bounds for the Edge-Weighted Maximum Clique Problem

TL;DR

This paper compares the theoretical and computational strength of three main upper bounds for the Edge-Weighted Maximum Clique Problem, revealing their limitations and relative performance on benchmark and random instances.

Contribution

It provides a detailed analysis of the unbounded ratio of upper bounds and their comparative strength, supported by extensive computational experiments.

Findings

None of the upper bounds guarantees a performance ratio.

For each pair of bounds, there are instances where either is tighter.

Computational experiments reveal practical differences among bounds.

Abstract

We theoretically and computationally compare the strength of the three main upper bounds from the literature on the optimal value of the Edge-Weighted Maximum Clique Problem (EWMCP). We provide a set of instances for which the ratio between any of the three upper bounds and the optimal value of the EWMCP is unbounded, showing that none of them can give a performance guarantee. We further analyze the relative strength among the three upper bounds by determining, for every choice of a ratio between any two of them, the largest values it can attain and providing families of instances for which such values can be reached. Our results show that, for each pair of upper bounds, there exist appropriately chosen instances on which either bound is tighter than the other. Our theoretical analysis is complemented by extensive computational experiments on two benchmark datasets: the standard DIMACS instances and randomly generated instances, providing practical insights into the empirical strength of the upper bounds.