Effective Data Reduction for the Vertex Clique Cover Problem

Abstract

The vertex clique cover (VCC) problem—the problem of computing a minimum cardinality set of cliques covering all vertices of a graph—is a classic NP-hard problem. Despite recent advances in parameterized algorithms that have been used to solve NP-hard problems in practice, the VCC problem has been almost completely unexplored. In particular, data reduction rules, which transform the input graph to a smaller equivalent instance, are well studied and highly effective at solving other NP-hard problems (e.g., the minimum vertex cover problem) in practice on sparse graphs of millions of vertices. Practical rules for the VCC problem, on the other hand, are nearly nonexistent: instead, the complementary graph coloring problem has received the lion's share of attention, and the available rules for that problem are either theoretical or they do not translate to effective rules for solving the VCC problem on sparse graphs. In this paper, we introduce a large suite of data reduction rules for the VCC problem. These rules enable us to solve large, sparse, real-world graphs significantly faster than the state of the art. Of the 52 graphs tested, without any additional techniques, our reduction rules completely solve 14 graphs with up to 326K vertices in a few milliseconds. Furthermore, applying our rules as a preprocessing step accelerates the state-of-the-art iterated greedy (IG) approach due to Chalupa, enabling us to find higher-quality solutions up to multiple orders of magnitude faster than previously possible. Furthermore, we integrate our data reductions into the branch-and-reduce framework, exactly solving instances on up to millions of vertices. As an added bonus, our data reduction rules partially explain why the clique cover number and independence number have been observed to match for many sparse instances—our data reduction rules apply to both the maximum independent set and VCC problems.