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Worst-case upper bounds for MAX-2-SAT with an application to MAX-CUT. (English) Zbl 1051.68078
Summary: The maximum 2-satisfiability problem (MAX-2-SAT) is: given a Boolean formula in 2-CNF, find a truth assignment that satisfies the maximum possible number of its clauses. MAX-2-SAT is MAX-SNP-complete. Recently, this problem received much attention in the contexts of (polynomial-time) approximation algorithms and (exponential-time) exact algorithms. In this paper, we present an exact algorithm solving MAX-2-SAT in time poly\((L)\cdot 2^{K/5}\), where \(K\) is the number of clauses and \(L\) is their total length. In fact, the running time is only poly\((L)\cdot 2^{K_2/5}\), where \(K_2\) is the number of clauses containing two literals. This bound implies the bound poly\((L)\cdot 2^{L/10}\). Our results significantly improve previous bounds: poly\((L)\cdot 2^{K/2.88}\) [R. Niedermeier and P. Rossmanith, J. Algorithms 36, 63–88 (2000; Zbl 0959.68049)] and poly\((L)\cdot 2^{K/3.44}\) [implicit in N. Bansal and V. Raman, Lect. Notes Comput. Sci. 1741, 247–258 (1999; Zbl 0971.68069)].
As an application, we derive upper bounds for the (MAX-SNP-complete) maximum cut problem (MAX-CUT), showing that it can be solved in time poly\((M)\cdot 2^{M/3}\), where \(M\) is the number of edges in the graph. This is of special interest for graphs with low vertex degree.

MSC:
68Q25 Analysis of algorithms and problem complexity
Software:
MAX-2-SAT
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