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  • Lower and Upper Bounds for Random Mimimum Satisfiability Problem

    Author(s)
    Huang, Ping
    Su, Kaile
    Griffith University Author(s)
    Su, Kaile
    Year published
    2015
    Metadata
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    Abstract
    Given a Boolean formula in conjunctive normal form with n variables and m=rn clauses, if there exists a truth assignment satisfying (1−2−k−q(1−2−k))m clauses, call the formula q -satisfiable. The Minimum Satisfiability Problem (MinSAT) is a special case of q -satisfiable, which asks for an assignment to minimize the number of satisfied clauses. When each clause contains k literals, it is called Min k SAT. If each clause is independently and randomly selected from all possible clauses over the n variables, it is called random MinSAT. In this paper, we give upper and lower bounds of r (the ratio of clauses ...
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    Given a Boolean formula in conjunctive normal form with n variables and m=rn clauses, if there exists a truth assignment satisfying (1−2−k−q(1−2−k))m clauses, call the formula q -satisfiable. The Minimum Satisfiability Problem (MinSAT) is a special case of q -satisfiable, which asks for an assignment to minimize the number of satisfied clauses. When each clause contains k literals, it is called Min k SAT. If each clause is independently and randomly selected from all possible clauses over the n variables, it is called random MinSAT. In this paper, we give upper and lower bounds of r (the ratio of clauses to variables) for random k -CNF formula with q -satisfiable. The upper bound is proved by the first moment argument, while the proof of lower bound is the second moment with weighted scheme. Interestingly, our experimental results about MinSAT demonstrate that the lower and upper bounds are very tight. Moreover, these results give a partial explanation for the excellent performance of MinSatz, the state-of-the-art MinSAT solver, from the perspective of pruning effects. Finally, we give a conjecture about the relationship between the minimum number and the maximum number of satisfied clauses on random SAT instances.
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    Journal Title
    Lecture Notes in Computer Science
    Volume
    9130
    DOI
    https://doi.org/10.1007/978-3-319-19647-3_11
    Subject
    Other information and computing sciences not elsewhere classified
    Publication URI
    http://hdl.handle.net/10072/125362
    Collection
    • Journal articles

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