Finding Large Set Covers Faster via the Representation Method

The worst-case fastest known algorithm for the Set Cover problem on universes with n elements still essentially is the simple O∗(2n)-time dynamic programming algorithm, and no non-trivial consequences of an O∗(1.01n)-time algorithm are known. Motivated by this chasm, we study the following natural question: Which instances of Set Cover can we solve faster than the simple dynamic programming algorithm? Specifically, we give a Monte Carlo algorithm that determines the existence of a set cover of size σn in O∗(2(1−Ω(σ))n) time. Our approach is also applicable to Set Cover instances with exponentially many sets: By reducing the task of finding the chromatic number χ(G) of a given n-vertex graph G to Set Cover in the natural way, we show there is an O∗(2(1−Ω(σ))n)-time randomized algorithm that given integer s = σn, outputs NO if χ(G) > s and YES with constant probability if χ(G) ≤ s− 1. On a high level, our results are inspired by the ‘representation method’ of Howgrave-Graham and Joux [EUROCRYPT’10] and obtained by only evaluating a randomly sampled subset of the table entries of a dynamic programming algorithm. 1998 ACM Subject Classification G.2.2 [Graph Algorithms] Hypergraphs