Known algorithms for EDGE CLIQUE COVER are probably optimal

TL;DR

This paper proves that the known vertex reduction approach for the EDGE CLIQUE COVER problem is essentially optimal, establishing tight bounds and showing no significantly faster algorithms are likely to exist under standard complexity assumptions.

Contribution

It demonstrates that the existing vertex reduction method is optimal and establishes tight lower bounds for the problem's complexity under the Exponential Time Hypothesis.

Findings

Vertex reduction approach is essentially optimal.

No faster algorithms than the current bounds are likely under ETH.

First results linking natural fixed-parameter problems to tight complexity bounds.

Abstract

In the EDGE CLIQUE COVER (ECC) problem, given a graph G and an integer k, we ask whether the edges of G can be covered with k complete subgraphs of G or, equivalently, whether G admits an intersection model on k-element universe. Gramm et al. [JEA 2008] have shown a set of simple rules that reduce the number of vertices of G to 2^k, and no algorithm is known with significantly better running time bound than a brute-force search on this reduced instance. 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 3-CNF-SAT formula with n variables and m clauses to an equivalent ECC instance (G,k) with k = O(log n) and |V(G)| = O(n + m). Consequently, there is no 2^{2^{o(k)}}poly(n) time algorithm for the ECC problem, unless the Exponential Time Hypothesis fails. To the best of our knowledge, these are the first results for a natural, fixed-parameter tractable problem, and proving that a doubly-exponential dependency on the parameter is essentially necessary.