Solving Edge Clique Cover Exactly via Synergistic Data Reduction

TL;DR

This paper introduces a synergistic data reduction approach that significantly improves the exact solution of the NP-hard edge clique cover problem, enabling solutions for large graphs previously unsolvable and facilitating better evaluation of heuristics.

Contribution

The paper presents a novel method combining data reduction for ECC and VCC problems, transforming and lifting reductions to solve large instances exactly, outperforming previous techniques.

Findings

Enables solving large sparse graphs with millions of vertices and edges.

Finds exact minimum ECCs much faster than previous methods.

Provides the first large-instance exact solutions to evaluate heuristics.

Abstract

The edge clique cover (ECC) problem -- where the goal is to find a minimum cardinality set of cliques that cover all the edges of a graph -- is a classic NP-hard problem that has received much attention from both the theoretical and experimental algorithms communities. While small sparse graphs can be solved exactly via the branch-and-reduce algorithm of Gramm et al. [JEA 2009], larger instances can currently only be solved inexactly using heuristics with unknown overall solution quality. We revisit computing minimum ECCs exactly in practice by combining data reduction for both the ECC \emph{and} vertex clique cover (VCC) problems. We do so by modifying the polynomial-time reduction of Kou et al. [Commun. ACM 1978] to transform a reduced ECC instance to a VCC instance; alternatively, we show it is possible to ``lift'' some VCC reductions to the ECC problem. Our experiments show that combining data reduction for both problems (which we call \emph{synergistic data reduction}) enables finding exact minimum ECCs orders of magnitude faster than the technique of Gramm et al., and allows solving large sparse graphs on up to millions of vertices and edges that have never before been solved. With these new exact solutions, we evaluate the quality of recent heuristic algorithms on large instances for the first time. The most recent of these, \textsf{EO-ECC} by Abdullah et al. [ICCS 2022], solves 8 of the 27 instances for which we have exact solutions. It is our hope that our strategy rallies researchers to seek improved algorithms for the ECC problem.