Data reduction and exact algorithms for clique cover

Abstract

To cover the edges of a graph with a minimum number of cliques is an NP-hard problem with many applications. We develop for this problem efficient and effective polynomial-time data reduction rules that, combined with a search tree algorithm, allow for exact problem solutions in competitive time. This is confirmed by experiments with real-world and synthetic data. Moreover, we prove the fixed-parameter tractability of covering edges by cliques.

Tractable Parameterizations for the Minimum Linear Arrangement Problem

“…That is, reordering vertices of a given type does not affect the total edge lengths of the ordering. Now, it is well known that a graph with edge clique cover number at most k has at most 2 k different types [13]. Thus, our algorithm searches through all O(2 k !)…”

Clique Cover on Sparse Networks

“…As a comparison, we considered Gramm et al's algorithm [15] for the decision problem version of clique cover: given the size of clique cover k as an input parameter, their algorithm works by first applying a set of reduction rules to reduce the problem instance, and then using a search tree algorithm on the reduced instance, in time exponential in k. While their algorithm works well in cases where the solution size is small, it would perform poorly against our test data which contains several hundreds of cliques. This is mainly because their reduction rules did not significantly decrease the size of our input graphs (especially biological networks): the solutions for the reduced instances would still contain a large number of cliques, resulting in inefficient running time for the search tree algorithm.…”

“…Recently, Gramm et al [15] showed that the clique cover problem is FPT when the size of the cover is chosen for the parameter k. Similarly, Mujuni and Rosamond [24] have shown that the clique partition problem is FPT with the output size chosen as the parameter. These algorithms run in polynomial time in the input size but exponential time in the number of cliques in the solution.…”

Clique Cover and Graph Separation

“…From the point of view of parameterized complexity, Edge Clique Cover was studied by Gramm et al [32]. A simple kernelization algorithm is known that reduces the size of the graph to at most 2 k vertices; the best known fixed-parameter algorithm is a brute-force search on this 2 k -vertex kernel.…”

“…A simple kernelization algorithm is known that reduces the size of the graph to at most 2 k vertices; the best known fixed-parameter algorithm is a brute-force search on this 2 k -vertex kernel. The question of a polynomial kernel for Edge Clique Cover, probably first verbalized by Gramm et al [32], was repeatedly asked in the parameterized complexity community, for example on the last Workshop on Kernels (WorKer, Vienna, 2011). We provide an AND-cross-composition from (unparameterized) Edge Clique Cover to Edge Clique Cover parameterized by k, thereby establishing that a polynomial kernelization is unlikely to exist.…”