Known algorithms for EDGE CLIQUE COVER are probably optimal
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 kelement 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 bruteforce 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 3CNFSAT 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, fixedparameter tractable problem, and proving that a doublyexponential dependency on the parameter is essentially necessary.
 Publication:

arXiv eprints
 Pub Date:
 March 2012
 arXiv:
 arXiv:1203.1754
 Bibcode:
 2012arXiv1203.1754C
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Computer Science  Computational Complexity;
 Computer Science  Discrete Mathematics;
 F.2.2
 EPrint:
 To appear in SODA 2013