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Exact Algorithms for Maximum Two-Satisfiability

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Encyclopedia of Algorithms

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Years and Authors of Summarized Original Work

  • 2004; Williams

Problem Definition

In the maximum 2-satisfiability problem (abbreviated as Max 2-Sat), one is given a Boolean formula in conjunctive normal form, such that each clause contains at most two literals. The task is to find an assignment to the variables of the formula such that a maximum number of clauses are satisfied.

Max 2-Sat is a classic optimization problem. Its decision version was proved NP-complete by Garey, Johnson, and Stockmeyer [7], in stark contrast with 2-Sat which is solvable in linear time [2]. To get a feeling for the difficulty of the problem, the NP-completeness reduction is sketched here. One can transform any 3-Sat instance F into a Max 2-Sat instance F′, by replacing each clause of F such as

$$\displaystyle\begin{array}{rcl} c_{i} = (\ell_{1} \vee \ell_{2} \vee \ell_{3}),& & {}\\ \end{array}$$

where â„“1, â„“2, and â„“3 are arbitrary literals, with the collection of 2-CNF clauses

$$\displaystyle\begin{array}{rcl}...

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Authors and Affiliations

  1. Department of Computer Science, Stanford University, Stanford, CA, USA

    Ryan Williams

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  1. Ryan Williams
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Correspondence to Ryan Williams .

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Editors and Affiliations

  1. Department of Electrical Engineering and Computer Science, Northwestern University, Evanston, IL, USA

    Ming-Yang Kao

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Williams, R. (2016). Exact Algorithms for Maximum Two-Satisfiability. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2864-4_227

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