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

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.

On the elusivity of dynamic optimisation problems

“…Rohlfshagen et al [6] theoretically analyse the runtime of a (1+1) evolutionary algorithm on two simple frameworks according to the magnitude and frequency of changes, and illustrate two counter-intuitive assumptions about the dynamic domain where restart is preferred over adaptation for slightly changing problems. Further theoretical works also analyse the runtime analysis of different algorithms for different well-studied dynamic problems [12,13,14,15]. Yu et al [16] demonstrate that relocating the moving optima may be difficult in severely and quickly changing continuous optimisation problems.…”