On Problems as Hard as CNFSAT

Details On Problems as Hard as CNFSAT On Problems as Hard as CNFSAT

2011-12-10

arxiv.org/abs/1112.2275v1

Computer Science

Data Structures and Algorithms

22 pages, 1 figure

Scientific paper

Exact exponential time algorithms for NP-hard problems have thrived over the last decade. Non-trivial exponential time algorithms have been found for a myriad of problems, including Graph Coloring, Hamiltonian Path, Dominating Set and 3-CNF-SAT, that is, satisfiability of 3-CNF formulas. For some basic problems, however, there has been no progress over their trivial solution. For others, non-trivial solutions have been found, but improving these algorithms further seems to be out of reach. The CNF-SAT problem is the canonical example of a problem for which the brute force 2^n * poly(n) time algorithm remains the best known. The assumption that k-CNF-SAT requires 2^n time in the worst case when k grows to infinity is known as the strong exponential time hypothesis (SETH) of Impagliazzo and Paturi. In this paper we reveal connections between well-studied problems, and show that improving over the currently best known algorithms for several of them would violate SETH. Specifically, we show that for every epsilon < 1, an O(2^{epsilon n})-time algorithm for Hitting Set, Set Splitting, or NAE-SAT would violate SETH. Here n is the number of elements (or variables) in the input set system (or formula). We conjecture that an O(2^{epsilon n})-time algorithm for Set Cover would violate SETH as well, and prove under this assumption, that improving the fastest known algorithms for Steiner Tree, Connected Vertex Cover, Set Partition or the pseudo-polynomial time algorithm for Subset Sum would violate SETH. Finally, we justify our assumption about the hardness of Set Cover. In particular we prove that, assuming SETH, there does not exist an O(2^{epsilon n})-time algorithm that counts the number of set covers in a set system modulo two.

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