More Effective Randomized Search Heuristics for Graph Coloring Through Dynamic Optimization

TL;DR

This paper demonstrates that evolutionary algorithms, especially island models, can efficiently solve bipartite graph coloring problems through dynamic optimization, significantly outperforming static approaches and other EAs.

Contribution

It introduces a dynamic optimization approach for graph coloring, showing that island models guarantee exponential speedups on bipartite graphs, surpassing previous methods.

Findings

RLS efficiently finds proper 2-colorings in dynamic bipartite graphs.

Offspring populations like (1+λ) RLS provide exponential speedup.

Island models achieve optimal runtime Θ(m) on all m-edge bipartite graphs.

Abstract

Dynamic optimization problems have gained significant attention in evolutionary computation as evolutionary algorithms (EAs) can easily adapt to changing environments. We show that EAs can solve the graph coloring problem for bipartite graphs more efficiently by using dynamic optimization. In our approach the graph instance is given incrementally such that the EA can reoptimize its coloring when a new edge introduces a conflict. We show that, when edges are inserted in a way that preserves graph connectivity, Randomized Local Search (RLS) efficiently finds a proper 2-coloring for all bipartite graphs. This includes graphs for which RLS and other EAs need exponential expected time in a static optimization scenario. We investigate different ways of building up the graph by popular graph traversals such as breadth-first-search and depth-first-search and analyse the resulting runtime behavior. We further show that offspring populations (e. g. a (1+$\lambda$) RLS) lead to an exponential speedup in $\lambda$. Finally, an island model using 3 islands succeeds in an optimal time of $\Theta(m)$ on every $m$-edge bipartite graph, outperforming offspring populations. This is the first example where an island model guarantees a speedup that is not bounded in the number of islands.