On Problems as Hard as CNF-SATx

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), 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), 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 < 1, there is a (large) integer k such that that K-CNF-SAT cannot be computed in time 2 . In this paper, we show that, for every < 1, the problems HITTING SET, SET SPLITTING, and NAE-SAT cannot be computed in time O(2 ) 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 STEINTER 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 that the parity of the number of set covers x The full version of this paper can be found on the arXiv [10]. ∗IDSIA, University of Lugano, Switzerland. marek@idsia.ch. Partially supported by National Science Centre grant no. N206 567140, Foundation for Polish Science and ONR Young Investigator award when at the University at Maryland. †University of Wisconsin–Madison, USA. holger@cs.wisc. edu. Research partially supported by the Alexander von Humboldt Foundation and NSF grant 1017597. ‡University of California, USA. dlokshtanov@cs.ucsd. edu. §Computer and Automation Research Institute, Hungarian Academy of Sciences (MTA SZTAKI), Budapest, Hungary. dmarx@cs.bme. hu. Research supported by ERC Starting Grant PARAMTIGHT (280152). ¶Utrecht University, The Netherlands. j.nederlof@uu.nl. Supported by NWO project ”Space and Time Efficient Structural Improvements of Dynamic Programming Algorithms”. ‖Japan Advanced Institute of Science and Technology, Japan. okamotoy@jaist.ac. jp. Partially supported by Grantin-Aid for Scientific Research from Japan Society for the Promotion of Science. ∗∗University of California, USA. paturi@cs.ucsd. edu. This research is supported by NSF grant CCF-0947262 from the Division of Computing and Communication Foundations. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. ††Institute of Mathematical Sciences, India. saket@imsc.res.in. ‡‡Max-Planck-Institut für Informatik, Germany. wahl@mpi-inf.mpg. de. cannot be computed in time O(2 ) for any < 1 unless SETH fails. Keywords-Strong Exponential Time Hypothesis, Exponential Time Algorithms, Sparsification Lemma