Known Algorithms for Edge Clique Cover are Probably Optimal

Abstract

In the Edge Clique Cover (ECC) problem, given an undirected graph $G$ and an integer $k$, we ask whether one can choose $k$ cliques in $G$ such that each edge of $G$ is contained in at least one of the chosen cliques. Gramm et al. [ACM J. Exp. Algorithmics, 13 (2008)] have shown a set of simple rules that reduce the number of vertices of $G$ to $2^k$ while preserving the answer to the instance at hand, that is, they have shown a kernel for the problem with at most $2^k$ vertices. No algorithm is known with significantly better running time bound than a brute-force search on this kernel. In this paper, we show that the approach of Gramm et al. is essentially optimal: we present a polynomial-time algorithm that reduces an arbitrary instance of $3$-CNF-SAT with $n$ variables and $m$ clauses to an equivalent ECC instance $(G,k)$ with $k = \mathcal{O}(\log n)$ and $|V(G)| = \mathcal{O}(n + m)$. Consequently, there is no $2^{2^{o(k)}}{\rm poly}(n)$ time algorithm for the ECC problem, unless the Exponential Time Hypothesis fails. Moreover, our reduction also implies that, unless P $ = $ NP, the ECC problem does not admit a subexponential kernel, i.e., a kernel of size $2^{o(k)}$.