A New Lower Bound on the Maximum Number of Satisfied Clauses in Max-SAT and its Algorithmic Application.
R. Crowston, G. Gutin, M. Jones and A. Yeo
In: IPEC 2010, DEc 2010, India.

## Abstract

In a formula $F=(V,C)$ in conjunctive normal form (CNF), $V$ is the set of variables and $C$ is the multiset of clauses. We denote by ${\rm sat}(F)$ the maximum number of clauses of $F$ that can be satisfied by a truth assignment. It is well-known that for every CNF formula $F=(V,C)$, ${\rm sat}(F)\ge |C|/2$ and the bound is tight when $F$ consists of conflicting unit clauses $(x)$ and $(\bar{x})$. Since each truth assignment satisfies exactly one clause in each pair of conflicting unit clauses, it is natural to reduce $F$ to the unit-conflict free (UCF) form. If $F=(V,C)$ is UCF, then Lieberherr and Specker (J. ACM 28, 1981) proved that ${\rm sat}(F)\ge \pp |C|$, where $\pp =(\sqrt{5}-1)/2$ and $|C|$ is the number of clauses in $C$ (each clause is counted as many time as it appears in the multiset $C$). A formula $F'=(V',C')$ is called a subformula of a CNF formula $F=(V,C)$ if $C'\subseteq C$ and $V'$ is the set of variables in $C'.$ We prove that for each UCF CNF formula $F=(V,C)$ a subformula $F'=(V',C')$ can be found in polynomial time such that ${\rm sat}(F)\ge \pp |C|+(1-\pp)|C'| + (2-3\pp)|Y|/2$, where $Y$ is the set of variables in $C\setminus C'$. This improves the Lieberherr-Specker lower bound on ${\rm sat}(F)$. We show that our new bound has algorithmic applications by considering the following two parameterized problems introduced by Mahajan and Raman (J. Algorithms 31, 1999). The first problem is as follows: given a CNF formula $F=(V,C)$ decide whether ${\rm sat}(F)\ge |C|/2 + k,$ where $k$ is the parameter. Mahajan and Raman showed that this problem may be transformed 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. The second problem is as follows: given a UCF CNF formula $F=(V,C)$ decide whether ${\rm sat}(F)\ge \pp |C| + k,$ where $k$ is the parameter. Mahajan and Raman conjectured that the problem is fixed-parameter tractable (FPT). We use the new bound to 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.