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：边团覆盖问题——目标是找到覆盖图中所有边的最小基数团集合——是一个经典的NP难问题，受到理论和实验算法社区的高度关注。虽然小型稀疏图可通过Gramm等[JEA 2009]的分支约简算法精确求解，但更大规模的实例目前只能使用启发式方法进行非精确求解，且解的整体质量未知。我们重新审视实践中精确计算最小边团覆盖的方法，通过结合边团覆盖和顶点团覆盖问题的数据约简技术来实现——具体而言，我们修改了Kou等[Commun. ACM, 1978]的多项式时间约简，将约简后的边团覆盖实例转化为顶点团覆盖实例；或者，我们展示了将顶点团覆盖约简“提升”到边团覆盖问题的可行性。实验表明，结合两类问题的数据约简（我们称之为协同数据约简）使得最小边团覆盖的精确求解速度比Gramm等人的技术快数个数量级，并能求解此前从未解决过的、包含数百万顶点和边的大型稀疏图。凭借这些新的精确解，我们首次客观评估了近期启发式算法在大规模实例上的解质量。其中最新算法——Abdullah等[ICCS 2022]提出的\textsf{EO-ECC}——能在我们拥有精确解的21个实例中精确求解8个。我们希望我们的策略能激励研究者为边团覆盖问题寻求更优的精确与启发式方法。