Time Complexity Analysis of Randomized Search Heuristics for the Dynamic Graph Coloring Problem

TL;DR

This paper analyzes the expected time for randomized heuristics to adapt solutions in dynamic graph coloring problems, revealing conditions where reoptimization is faster or harder, and proposing tailored mutation strategies for efficiency.

Contribution

It provides a theoretical analysis of reoptimization times for various randomized heuristics in dynamic graph coloring, highlighting the impact of graph structure and mutation tailoring.

Findings

Reoptimization often faster than from-scratch optimization.

Certain bipartite graphs pose reoptimization challenges similar to initial optimization.

Tailored mutation operators can significantly reduce reoptimization time.

Abstract

We contribute to the theoretical understanding of randomized search heuristics for dynamic problems. We consider the classical vertex coloring problem on graphs and investigate the dynamic setting where edges are added to the current graph. We then analyze the expected time for randomized search heuristics to recompute high quality solutions. The (1+1)~Evolutionary Algorithm and RLS operate in a setting where the number of colors is bounded and we are minimizing the number of conflicts. Iterated local search algorithms use an unbounded color palette and aim to use the smallest colors and, consequently, the smallest number of colors. We identify classes of bipartite graphs where reoptimization is as hard as or even harder than optimization from scratch, i.e., starting with a random initialization. Even adding a single edge can lead to hard symmetry problems. However, graph classes that are hard for one algorithm turn out to be easy for others. In most cases our bounds show that reoptimization is faster than optimizing from scratch. We further show that tailoring mutation operators to parts of the graph where changes have occurred can significantly reduce the expected reoptimization time. In most settings the expected reoptimization time for such tailored algorithms is linear in the number of added edges. However, tailored algorithms cannot prevent exponential times in settings where the original algorithm is inefficient.