NEW RESEARCH LINES FOR MAX-SAT

Exploiting the Recent Resolution Rule for Max-SAT

Federico Heras

Barcelona, Spain

Keywords:

Max-SAT, Combinatorial optimization, Inference, Local search, Systematic search, Logic programming.

Abstract:

This paper presents the current state-of-the-art techniques for Max-SAT solving and points out new research

lines in order to exploit the beneﬁts of the novel resolution rule for Max-SAT.

1 INTRODUCTION

Given a propositional formula in conjunctive normal

form (CNF), determining whether there exists a vari-

able assignment such that the formula evaluates to

true is called the Boolean Satisﬁability Problem, com-

monly abbreviated as SAT.

Max-SAT is the optimization variant of SAT:

Given a propositional formula , the unweighted Max-

SAT problem then is to ﬁnd a variable assignment

that maximizes the number of satisﬁed clauses. In

weighted Max-SAT, each clause has an associated

weight and the goal is to maximize the total weight

of the satisﬁed clauses.

In this paper we focus on (weighted) Max-SAT

which is a well-knownNP Hard problem. It is consid-

ered one of the fundamental combinatorial optimiza-

tion problems and many important problems can be

naturally expressed as Max-SAT. They include aca-

demic problems such as Max-Cut or Max-Clique, as

well as real problems in domains like routing, bioin-

formatics, electronic markets, etc.

Search methods for Max-SAT solving are usu-

ally classiﬁed in two categories: systematic and lo-

cal search. For a given Max-SAT instance, a system-

atic search algorithm traverse the whole search space

and can either ﬁnd the optimal solution or proves that

no solution exists. This is why they are considered

complete. On the other hand, a local search algo-

rithm initially selects a point of the search space, and

moves from the current solution to a neighbor candi-

date. Then, it stops when a solution is found or when

a limit is reached such as a time limit or a maximum

number of steps. Typically, local search solvers are

incomplete, because they do not assure to ﬁnd a solu-

tion neither to prove its optimality.

An alternative way to solve problems is inference.

The aim of inference is to make explicit some implicit

information from the problem instance. Algorithms

purely based on inference are complete as systematic

search. However, they are prohibitive because of their

high memory requirements (Rish and Dechter, 2000).

In practice, limited forms of inference are applied in-

side search algorithms in order to reduce the number

of search steps. This small amount of inference is

usually called incomplete (or limited) inference. Re-

cently, a general inference mechanism for Max-SAT

was introduced in (Larrosa et al., 2008) which extends

classical resolution. It is called the resolution rule for

Max-SAT. It is a sound and complete (Larrosa et al.,

2008; Bonet et al., 2007) rule.

Modern Max-SAT solvers based on systematic

search follow the well-know branch and bound

scheme and they apply a limited number of Max-SAT

resolution steps during search in order to simplify the

problem (Larrosa et al., 2008; Heras et al., 2008; Li

et al., 2007). As a consequence, the search process is

boosted several orders of magnitude. The limited res-

olution process affects to mutually inconsistent sub-

sets of clauses that are detected during search by pat-

terns or via unit propagation (Larrosa et al., 2008;

Li et al., 2007; Heras et al., 2008). When we ana-

lyze current works on Max-SAT we observe they are

reﬁnements to improve the pattern detection mecha-

nisms or the effectiveness of unit propagation and to

obtain better results (Lin et al., 2008).

The aim of this paper is to propose new solv-

ing techniques based on the recent resolution rule for

Max-SAT. First, we study the viability of a complete

inference-based algorithm. Second, we propose new

648

Heras F. (2010).

NEW RESEARCH LINES FOR MAX-SAT - Exploiting the Recent Resolution Rule for Max-SAT.

In Proceedings of the 2nd International Conference on Agents and Artiﬁcial Intelligence - Artiﬁcial Intelligence, pages 648-651

DOI: 10.5220/0002761706480651

 SciTePress

methods of limited inference. Finally, we propose to

use inference in order to improve the performance of

local search algorithms.

2 PRELIMINARIES

In the sequel X = {x

, x

, . . . , x

} is the set of boolean

variables. A literal is either a variable x

or its nega-

tion ¯x

. The variable to which literal l refers is noted

var(l). Given a literal l, its negation

l is ¯x

if l is x

and

is x

if l is ¯x

. A clause C is a disjunction of literals. In

the following, possibly subscripted capital letters A,

B, C, D, and E will always represent clauses. The size

of a clause is the number of literals that it has. An

assignment is a set of literals not containing a vari-

able and its negation. Assignments of maximal size

n are called complete, otherwise they are called par-

tial. An assignment satisﬁes a literal iff it belongs to

the assignment, it satisﬁes a clause iff it satisﬁes one

or more of its literals and it falsiﬁes (or unsatisﬁes) a

clause iff it contains the negation of all its literals. In

the latter case we say that the clause is conﬂicting as

it always happens with the empty clause, noted .

A weighted clause is a pair (C, w), where C

is a clause and w is the cost of its falsiﬁcation,

also called its weight. Many real problems contain

clauses that must be satisﬁed. We call such clauses

mandatory or hard and associate with them a special

weight ⊤. Non-mandatory clauses are also called soft

clauses. A weighted formula in conjunctive normal

form (WCNF) F is a multiset of weighted clauses.

A model is a complete assignment that satisﬁes all

mandatory clauses. The cost of an assignment is the

sum of weights of the clauses that it falsiﬁes. Given

a WCNF formula, Weighted Max-SAT is the problem

of ﬁnding a model of minimum cost. Note that if a

formula contains only mandatory clauses, weighted

Max-SAT is equivalent to classical SAT.

Following (Larrosa et al., 2008; Bonet et al.,

2007), the resolution rule can be extended from SAT

to Max-SAT as,

{(x∨ A, u), ( ¯x∨ B, w)} ≡











(A∨ B, m),

(x∨ A, u− m),

( ¯x∨ B, w− m),

(x∨ A∨

B, m),

( ¯x∨

A∨ B, m)











where m = min{u, w}.

The identiﬁcation of mandatory clauses with ⊤ al-

lows to extend some well-known simpliﬁcation rules

from SAT to Max-SAT such as unit propagation. Unit

propagation for Max-SAT (Larrosa et al., 2008) can

be applied as in SAT (Silva and Sakallah, 1996) only

when unit hard clauses exist on the formula. Given a

unit hard clause (l, ⊤), literal l is propagated which

means that it is assigned accordingly to satisfy the

clause. Such an assignment can falsify literals in other

clauses that may become also hard unit clauses. The

procedure is repeated until no more hard unit clauses

are generated or until a conﬂict is detected, that is, a

hard clause is falsiﬁed.

We say that a weighted formula F

′

is a relaxation

F (noted F

′

⊑ F ) if the optimal cost of F

′

is less

than or equal to the optimal cost in F (non-models

are considered to have cost inﬁnity). We say that two

weighted formulas F

′

and F are equivalent (noted

′

≡ F ) if F

′

⊑ F and F ⊑ F

′

Given a SAT or Max-SAT formula, a subset of

(weighted) clauses is inconsistent if all them can-

not be simultaneusly satisﬁed by any variable assign-

ment. The clauses involved in an inconsistent subset

of clauses can be determined with several steps of res-

olution. This process is called refutation. If all the

clauses involved in a refutation are used at most once,

it is called a tree-like refutation, otherwise it is con-

sidered a general refutation.

Given an unsatisﬁable SAT formula (i.e. a Max-

SAT formula that only contains hard clauses), a sub-

set of clauses whose conjunction is still unsatisﬁable

is called an unsatisﬁable core of the original formula.

Unsatisﬁability cores of SAT problem instances are

currently provided with general refutations, that is,

existing procedures do not assure to use each clause

just once at most (Lifﬁton and Sakallah, 2008).

3 NEW RESEARCH LINES

In this Section we propose different research venues

that exploit the novel resolution rule for Max-SAT.

Most recent works for Max-SAT present reﬁnements

to improve the performance of the works in (Chu

Min Li and Planes, 2005; Li et al., 2007; Larrosa

et al., 2008; Heras et al., 2008). In what follows, we

propose new methods based on complete inference,

incomplete inference and local search.

3.1 Complete Inference

It would be interesting to investigate the effectiveness

of a complete inference-based algorithm for Max-

SAT and how it compares with local and systematic

search. It is well-known that this approach has se-

vere memory limitations but it may be effective for

problem instances with a very small induced width.

The induced width is a property associated to each

NEW RESEARCH LINES FOR MAX-SAT - Exploiting the Recent Resolution Rule for Max-SAT

649

problem instance. Its computation implies an NP-

Hard problem but accurate approximationsare known

(Rish and Dechter, 2000). Given the information pro-

vided by the induced width we can decide whether a

problem instance can be solved only with inference or

not.

3.2 Incomplete Inference

To start with, we can limit the complete method intro-

duced in Section 3.1 using the value of the induced

width. Then, we can use it as a pre-processing or

during systematic search when the induced width of

the problem resulting of some variable assignments

is small enough. An initial work in this ﬁeld can

be found Max-SAT in (Argelich et al., 2008). How-

ever, from previous approaches in other frameworks,

it seems that this approach is not very competitive in

general (Rish and Dechter, 2000).

Current approaches for solving the SAT problems

apply some kind of incomplete (or limited) inference.

For example, modern SAT solvers apply unit propaga-

tion in order to detect conﬂicting clauses and then to

learn new clauses in order to boost the search proce-

dure (Silva and Sakallah, 1996). Also, different pre-

processing methods based on limited inference are ap-

plied for local (Cha and Iwama, 1996) and systematic

search (E´en and Biere, 2005).

Recently some techniques from the SAT comunity

have been extended to Max-SAT. First, unit propa-

gation and clause learning can be applied directly to

Max-SAT when the clauses involved are hard (Lar-

rosa et al., 2008; Heras et al., 2008) (i.e, all clauses in-

volved are like (C, ⊤)). But, unit propagation is used

also in Max-SAT to produce as many empty clauses as

possible (i.e., (, m)). As consequence, the problem

is simpliﬁed and search is boosted. In (Chu Min Li

and Planes, 2005) a method for detecting inconsis-

tent subsets of clauses via unit propagation is pre-

sented and then in (Heras et al., 2008) such subset

is transformed to an equivalent but simpler subset by

applying the resolution rule for Max-SAT. Note that

the clauses involved in the inconsistent subset are de-

tected via a tree-like refutation.

Observe that one of the most important differences

between the classical resolution rule for SAT and the

novel resolution rule for Max-SAT is that clauses in

SAT can be used once and again during a resolution

process returning an equivalent formula. Differently,

in the Max-SAT context clauses may disappear after a

resolution step and they should be used only once or

at most as many times as their weight is greater than

0 to assure a resulting equivalent formula.

Based on the same idea of using unit propagation

to produce new weighted empty clauses, we propose

to use techniques of detecting unsatisﬁable cores to

produceempty clauses. The idea is to apply SAT tech-

niques in order to detect unsatisﬁable cores and then

apply Max-SAT resolution if and only if the refutation

is tree-like to assure the correctness of the transfor-

mation. In particular, the Max-SAT problem is solved

assuming all clauses are hard with a SAT solver like

(Silva and Sakallah, 1996) and the information about

the unsatisﬁable core is recorded during search. Once

the SAT solver has proved the unsatisﬁability, a refu-

tation is built using the information of the unsatisﬁ-

able core. Then, Max-SAT resolution is applied as

dictated by the refutation if and only if it is tree-like.

However, we think this is unlikely to happen in gen-

eral and time can be prohibitive (each detection of an

unsatisﬁable core requires solving a SAT instance).

A more speciﬁc recent work on 2-SAT (i.e.,

all clauses have size two) (Buresh-Oppenheim and

Mitchell, 2006) shows how minimum tree-like refuta-

tions can be built for 2-SAT in polynomial time. Ob-

serve that the same process can be applied for Max-

SAT but in order to simplify the formula as much as

possible. This may be specially powerful for prob-

lems with lots of weighted clauses of arity 2 such as

the general Max-2-SAT problem, the Max-CUT prob-

lem or the spin-glass problem.

Finally, we propose to study new forms of lim-

ited inference for problems with a very speciﬁc struc-

ture. In (Heras and Larrosa, 2008) we presented pre-

processing that exploits the synergy of two resolution-

based rules in order to simplify problems by a set of

hard clauses of size 2 and a set of soft clauses of size

1. Observe that prominent optimization problems can

be encoded in this way such as the Maximum Clique

Problem, Minimum Vertex Covering, Maximum Inde-

pendet Set, etc. Results indicated that a systematic

search solver is boosted several orders of magnitude

after the pre-processing for some problem instances.

3.3 Local Search

The ﬁrst successful local search solver for SAT was

GSAT (Selman et al., 1992) that is a simple best-

improvement search algorithm. Performance im-

provements were achieved by the usage of techniques

to avoid falling in local minima, that is, areas where

no improvements can be reached. Major improve-

ments were obtained with the development of the

WALKSAT architecture (Selman et al., 1993). In each

search step, WALKSAT algorithms ﬁrst choose a cur-

rently unsatisﬁed clause and then ﬂip a variable oc-

curring in this clause, that is, the value of the variable

is changed from true to false or viceversa. Current

ICAART 2010 - 2nd International Conference on Agents and Artificial Intelligence

650

research is focused on proposing new schemes for

selecting the variable to be ﬂipped and mechanisms

to avoid local minima. The use of inference in local

search is restricted to the classical resolution rule and

the SAT problem (Cha and Iwama, 1996).

We propose to apply the recent resolution rule to

simplify the problem in local search solvers so that

they can obtain better solutions in less search steps

or time. In (Heras, 2009) we have recently extended

the pre-processing in (Heras and Larrosa, 2008) and

the limited inference of (Heras et al., 2008) has also

been considered as a pre-processing. Preliminary re-

sults indicate that feeding a local search with the pre-

processed instance usually improve its performance

(Heras, 2009). In particular, algorithms based on the

WALKSAT architecture obtain substantially better so-

lutions. Furthermore, the hard instances presented in

(Xu et al., 2007)

were all solved for the ﬁrst time.

4 CONCLUSIONS

In this paper we have presented several ways to ben-

eﬁt from the novel resolution rule for Max-SAT (Lar-

rosa et al., 2008). We have pointed out that com-

plete inference-based algorithms may be effective

on problems with small induced width. Regarding

incomplete inference, we have presented a general

method based on detecting unsatisﬁable cores but re-

stricted to the case in which a tree-like refutation can

be built. We have also proposed incomplete infer-

ence techniques based on building minimum tree-like

refutations restricted to Max-2-SAT as introduced in

(Buresh-Oppenheim and Mitchell, 2006). We have

suggested that speciﬁc limited inference for problems

with a very particular structure can be very powerful

in practice (Heras and Larrosa, 2008). Finally, we

have argued that local search can also beneﬁt from in-

ference (Heras, 2009).

ACKNOWLEDGEMENTS

The author is grateful to Rafael Gimenez for useful

comments.

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