A New Lower Bound on the Maximum Number of Satisfied clauses in Max-SAT and its algorithmic applications. / Crowston, Robert; Gutin, Gregory; Jones, Mark; Yeo, Anders.

In: Algorithmica, Vol. doi: 10.1007/s00453-011-9550-1, 2012.

Research output: Contribution to journalArticlepeer-review

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In: Algorithmica, Vol. doi: 10.1007/s00453-011-9550-1, 2012.

Research output: Contribution to journalArticlepeer-review

### BibTeX

@article{d3aa64215bd14121bae15b4e5f1a6b4d,
title = "A New Lower Bound on the Maximum Number of Satisfied clauses in Max-SAT and its algorithmic applications.",
abstract = "A pair of unit clauses is called conflicting if it is of the form $(x)$, $(\bar{x})$. A CNF formula is unit-conflict free (UCF) if it contains no pairof conflicting unit clauses. Lieberherr and Specker (J. ACM 28, 1981) showed that for each UCF CNF formula with $m$ clauses we can simultaneously satisfy at least $\pp m$ clauses, where $\pp =(\sqrt{5}-1)/2$. We improve the Lieberherr-Specker bound by showing that for each UCF CNF formula $F$ with $m$ clauses we can find, in polynomial time, a subformula $F'$ with $m'$ clauses such that we can simultaneously satisfy at least $\pp m+(1-\pp)m'+(2-3\pp)n''/2$ clauses (in $F$), where$n''$ is the number of variables in $F$ which are not in $F'$.We consider two parameterized versions of MAX-SAT, where the parameter is the number of satisfied clauses above the bounds $m/2$ and $m(\sqrt{5}-1)/2$. The former bound is tight for general formulas, and the later is tight for UCF formulas. Mahajan and Raman (J. Algorithms 31, 1999) showed that every instance of the first parameterized problem can be transformed, in polynomial time, into an equivalent one with at most $6k+3$ variables and $10k$ clauses. We improve this to $4k$ variables and $(2\sqrt{5}+4)k$ clauses. Mahajan and Raman conjectured that the second parameterized problem is fixed-parameter tractable (FPT). We show that the problem is indeed FPT by describing a polynomial-time algorithm that transforms any problem instance into an equivalent one with at most $(7+3\sqrt{5})k$ variables. Our results are obtained using our improvement of the Lieberherr-Specker bound above.",
author = "Robert Crowston and Gregory Gutin and Mark Jones and Anders Yeo",
year = "2012",
doi = "doi: 10.1007/s00453-011-9550-1",
language = "English",
volume = "doi: 10.1007/s00453-011-9550-1",
journal = "Algorithmica",
issn = "0178-4617",
publisher = "Springer New York",

}

### RIS

TY - JOUR

T1 - A New Lower Bound on the Maximum Number of Satisfied clauses in Max-SAT and its algorithmic applications.

AU - Crowston, Robert

AU - Gutin, Gregory

AU - Jones, Mark

AU - Yeo, Anders

PY - 2012

Y1 - 2012

N2 - A pair of unit clauses is called conflicting if it is of the form $(x)$, $(\bar{x})$. A CNF formula is unit-conflict free (UCF) if it contains no pairof conflicting unit clauses. Lieberherr and Specker (J. ACM 28, 1981) showed that for each UCF CNF formula with $m$ clauses we can simultaneously satisfy at least $\pp m$ clauses, where $\pp =(\sqrt{5}-1)/2$. We improve the Lieberherr-Specker bound by showing that for each UCF CNF formula $F$ with $m$ clauses we can find, in polynomial time, a subformula $F'$ with $m'$ clauses such that we can simultaneously satisfy at least $\pp m+(1-\pp)m'+(2-3\pp)n''/2$ clauses (in $F$), where$n''$ is the number of variables in $F$ which are not in $F'$.We consider two parameterized versions of MAX-SAT, where the parameter is the number of satisfied clauses above the bounds $m/2$ and $m(\sqrt{5}-1)/2$. The former bound is tight for general formulas, and the later is tight for UCF formulas. Mahajan and Raman (J. Algorithms 31, 1999) showed that every instance of the first parameterized problem can be transformed, in polynomial time, into an equivalent one with at most $6k+3$ variables and $10k$ clauses. We improve this to $4k$ variables and $(2\sqrt{5}+4)k$ clauses. Mahajan and Raman conjectured that the second parameterized problem is fixed-parameter tractable (FPT). We show that the problem is indeed FPT by describing a polynomial-time algorithm that transforms any problem instance into an equivalent one with at most $(7+3\sqrt{5})k$ variables. Our results are obtained using our improvement of the Lieberherr-Specker bound above.

AB - A pair of unit clauses is called conflicting if it is of the form $(x)$, $(\bar{x})$. A CNF formula is unit-conflict free (UCF) if it contains no pairof conflicting unit clauses. Lieberherr and Specker (J. ACM 28, 1981) showed that for each UCF CNF formula with $m$ clauses we can simultaneously satisfy at least $\pp m$ clauses, where $\pp =(\sqrt{5}-1)/2$. We improve the Lieberherr-Specker bound by showing that for each UCF CNF formula $F$ with $m$ clauses we can find, in polynomial time, a subformula $F'$ with $m'$ clauses such that we can simultaneously satisfy at least $\pp m+(1-\pp)m'+(2-3\pp)n''/2$ clauses (in $F$), where$n''$ is the number of variables in $F$ which are not in $F'$.We consider two parameterized versions of MAX-SAT, where the parameter is the number of satisfied clauses above the bounds $m/2$ and $m(\sqrt{5}-1)/2$. The former bound is tight for general formulas, and the later is tight for UCF formulas. Mahajan and Raman (J. Algorithms 31, 1999) showed that every instance of the first parameterized problem can be transformed, in polynomial time, into an equivalent one with at most $6k+3$ variables and $10k$ clauses. We improve this to $4k$ variables and $(2\sqrt{5}+4)k$ clauses. Mahajan and Raman conjectured that the second parameterized problem is fixed-parameter tractable (FPT). We show that the problem is indeed FPT by describing a polynomial-time algorithm that transforms any problem instance into an equivalent one with at most $(7+3\sqrt{5})k$ variables. Our results are obtained using our improvement of the Lieberherr-Specker bound above.

U2 - doi: 10.1007/s00453-011-9550-1

DO - doi: 10.1007/s00453-011-9550-1

M3 - Article

VL - doi: 10.1007/s00453-011-9550-1

JO - Algorithmica

JF - Algorithmica

SN - 0178-4617

ER -