On Problems as Hard as CNFSAT
Cygan, M and Dell, H and Lokshtanov, D and Marx, Dániel and Nederlof, J (2016) On Problems as Hard as CNFSAT. ACM TRANSACTIONS ON ALGORITHMS, 12 (3). p. 41. ISSN 15496325 10.1145/2925416

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Abstract
The field of exact exponential time algorithms for nondeterministic polynomialtime hard problems has thrived since the mid2000s. While exhaustive search remains asymptotically the fastest known algorithm for some basic problems, nontrivial exponential time algorithms have been found for a myriad of problems, including GRAPH COLORING, HAMILTONIAN PATH, DOMINATING SET, and 3CNFSAT. In some instances, improving these algorithms further seems to be out of reach. The CNFSAT 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 nontrivial algorithms for CNFSAT 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 epsilon < 1, there is a (large) integer k such that kCNFSAT cannot be computed in time 2(epsilon n). In this article, we show that, for every epsilon < 1, the problems HITTING SET, SET SPLITTING, and NAESAT cannot be computed in time O(2(epsilon 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 pseudopolynomial 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 solutions to SET COVER cannot be computed in time O(2(epsilon n)) for any epsilon < 1 unless SETH fails.
Item Type:  Article 

Subjects:  Q Science > QA Mathematics and Computer Science > QA75 Electronic computers. Computer science / számítástechnika, számítógéptudomány 
Divisions:  Informatics Laboratory 
SWORD Depositor:  MTMT Injector 
Depositing User:  MTMT Injector 
Date Deposited:  29 Jan 2017 20:38 
Last Modified:  21 Jul 2019 14:03 
URI:  https://eprints.sztaki.hu/id/eprint/9047 
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