On Problems as Hard as CNFSAT

The field of exact exponential time algorithms for NP-hard problems has thrived over the last decade. While exhaustive search remains asymptotically the fastest known algorithm for some basic problems, difficult and non-trivial exponential time algorithms have been found for a myriad of problems, including Graph Coloring , Hamiltonian Path , Dominating Set and 3- CNF-Sat . In some instances, improving these algorithms further seems to be out of reach. The CNF-Sat problem is the canonical example of a problem for which the trivial exhaustive search algorithm runs in time O (2 n ), where n is the number of variables in the input formula. While there exist non-trivial algorithms for CNF-Sat that run in time o (2 n ), no algorithm was able to improve the growth rate 2 to a smaller constant, and hence it is natural to conjecture that 2 is the optimal growth rate. The strong exponential time hypothesis (SETH) by Impagliazzo and Paturi [JCSS 2001] goes a little bit further and asserts that, for every (cid:15) < 1, there is a (large) integer k such that k - CNF-Sat cannot be computed in time 2 (cid:15)n . In this paper, we show that, for every (cid:15) < 1, the problems Hitting Set , Set Splitting , and NAE-Sat cannot be computed in time O (2 (cid:15)n ) unless SETH fails. Here n is the number of elements or variables in the input. For these problems, we actually get an equivalence to SETH in a certain sense. We conjecture that SETH implies a similar statement for Set Cover , and prove that, under this assumption, the fastest known algorithms for Steiner Tree , Connected Vertex Cover , Set Partitioning , and the pseudo-polynomial time algorithm for Subset Sum cannot be significantly improved. Finally, we justify our assumption about the hardness of Set Cover by showing