Solving Clique Covering in Very Large Sparse Random Graphs by a Technique Based on k-Fixed Coloring Tabu Search

Abstract

We propose a technique for solving the k-fixed variant of the clique covering problem (k-CCP), where the aim is to determine, whether a graph can be divided into at most k non-overlapping cliques. The approach is based on labeling of the vertices with k available labels and minimizing the number of non-adjacent pairs of vertices with the same label. This is an inverse strategy to k-fixed graph coloring, similar to a tabu search algorithm TabuCol. Thus, we call our method TabuCol-CCP. The technique allowed us to improve the best known results of specialized heuristics for CCP on very large sparse random graphs. Experiments also show a promise in scalability, since a large dense graph does not have to be stored. In addition, we show that Γ function, which is used to evaluate a solution from the neighborhood in graph coloring in $\mathcal{O}(1)$ time, can be used without modification to do the same in k-CCP. For sparse graphs, direct use of Γ allows a significant decrease in space complexity of TabuCol-CCP to $\mathcal{O}(|E|)$, with recalculation of fitness possible with small overhead in $\mathcal{O}(\log \deg(v))$ time, where deg(v) is the degree of the vertex, which is relabeled.