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 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.

Cliquewidth III: The Odd Case of Graph Coloring Parameterized by Cliquewidth

“…Theorem 1.1 shows that GC behaves qualitatively different from every other problem previously studied on graphs of bounded cliquewidth. Indeed, to the best of our knowledge, GC parameterized by cliquewidth is the first (natural) parameterized problem known to require exponential dependence on the parameter in the exponent of n. Note here that there do exist problems for which the tight upper and lower bounds on the dependence of the running time on the parameter are double exponential, triple exponential, or even nonelementary (see e.g [15,22,34,37]. However these lower bounds are all for the g(k) factor of FPT algorithms, and not for the exponent of the input size n.…”

Edge clique cover of claw‐free graphs

“…. , v 11 , where for 1 ≤ i ≤ 10, v i is adjacent to v i+1 , v i+2 (reading subscripts modulo 10) and v 0 is adjacent to v 1 , v 3 , v 5 , v 7 , v 9 and v 11 is adjacent to v 2 , v 4 , v 6 , v 8 , v 10 (see Figure 1). Let G 1 be obtained from G 0 by deleting v 11 and let G 2 be obtained from G 1 by deleting v 10 .…”

Settling the complexity of computing approximate two-player Nash equilibria

“…Recall that the NP-analogue ETH [47] is widely used (e.g. [49,55,1,25,30]), often in stronger variants such as SETH [46,26] and NSETH [27].…”