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, which we do 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'' 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 enables solving large sparse graphs on up to millions of vertices and edges that have never before been solved. With these new exact solutions in hand, we objectively 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], is able to exactly solve 8 of the 21 instances for which we have exact solutions. It is our hope that our strategy rallies researchers to seek improved exact and heuristic methods for the ECC problem.

翻译：暂无翻译