# Optimality Clue for Graph Coloring Problem

## TL;DR

This paper introduces a novel method called optimality clue that uses randomized heuristics to estimate the likelihood of a solution being optimal in the Graph Coloring Problem, validated on benchmark instances.

## Contribution

It presents a new approach to verify solution optimality in GCP by estimating the number of colorings using randomized heuristics, enabling practical optimality proofs.

## Key findings

- Effective in confirming optimality on benchmark instances
- Provides a probabilistic upper bound for the number of colorings
- Works with standard heuristics like HEAD for large graphs

## Abstract

In this paper, we present a new approach which qualifies or not a solution found by a heuristic as a potential optimal solution. Our approach is based on the following observation: for a minimization problem, the number of admissible solutions decreases with the value of the objective function. For the Graph Coloring Problem (GCP), we confirm this observation and present a new way to prove optimality. This proof is based on the counting of the number of different k-colorings and the number of independent sets of a given graph G. Exact solutions counting problems are difficult problems (\#P-complete). However, we show that, using only randomized heuristics, it is possible to define an estimation of the upper bound of the number of k-colorings. This estimate has been calibrated on a large benchmark of graph instances for which the exact number of optimal k-colorings is known. Our approach, called optimality clue, build a sample of k-colorings of a given graph by running many times one randomized heuristic on the same graph instance. We use the evolutionary algorithm HEAD [Moalic et Gondran, 2018], which is one of the most efficient heuristic for GCP. Optimality clue matches with the standard definition of optimality on a wide number of instances of DIMACS and RBCII benchmarks where the optimality is known. Then, we show the clue of optimality for another set of graph instances. Optimality Metaheuristics Near-optimal.