Impact of Extended Clauses on Local Search Solvers for Max-SAT

Federico Heras

Universitat Pompeu Fabra, Barcelona, Spain

Keywords:

Max-SAT, Stochastic Local Search, Optimization, Constraint Satisfaction.

Abstract:

Previous research has demonstrated that several techniques based on the resolution rule for Max-SAT are

effective in improving results and boost the search, either as a preprocessing step or when embedded into

speciﬁc Max-SAT solving algorithms, such as branch-and-bound and Stochastic Local Search (abbreviated

SLS) algorithms. These techniques typically lead to a simpliﬁed and reduced Max-SAT formula, thereby

enabling the algorithms to ﬁnd solutions more efﬁciently. In this paper, we take a different approach by

introducing a preprocessing step that, in contrast to prior methods, increases the size of the Max-SAT formula

based on the Extension Rule. Our objective is to examine how this expansion of the problem instance impacts

the performance of SLS algorithms. The empirical results indicate that for a subset of SLS algorithms, this

approach yields improved solutions. This ﬁnding is signiﬁcant as it challenges the conventional wisdom that

smaller, simpliﬁed formulas are always better for all kind of solvers.

1 INTRODUCTION

(Weighted) Max-SAT is a well-known NP-Hard prob-

lem. The objective of the Max-SAT problem is to

identify an assignment of variables that satisﬁes the

maximum number (or weights) of the provided set of

clauses. It is recognized as one of the fundamental

problems in combinatorial optimization, with many

signiﬁcant problems naturally formulable as Max-

SAT instances. Among others, they include problems

like Maximum One, Maximum Cut (Max-CUT), Max-

imum Clique (Max-Clique) with many practical ap-

plications in bioinformatics, physics and electronic

markets (Bansal and Bafna, 2008; Guerri and Mi-

lano, 2003; Pardalos and Rebennack, 2010; Strick-

land et al., 2005) and other industrial problems.

A sound and complete inference method for Max-

SAT was introduced (Larrosa et al., 2008) called

the resolution rule for Max-SAT. The implementation

of restricted rules based on the resolution for Max-

SAT has been proven advantageous across various al-

gorithmic approaches, whether utilized as a prepro-

cessing step or integrated directly into the algorithm.

These include branch-and-bound algorithms (Heras

et al., 2008; Larrosa et al., 2008; Li et al., 2007a;

Heras and Larrosa, 2008; Heras and Ba

neres, 2013;

Cherif et al., 2020), stochastic local search (SLS) al-

gorithms (Heras and Ba

neres, 2010; Abram

e and Ha-

bet, 2012), and algorithms that rely on iteratively call-

ing to a SAT oracle (Heras and Marques-Silva, 2011;

Py et al., 2022). In those works, the application of 1

step or several steps of the resolution rule resulted in

a simpliﬁed and smaller Max-SAT formula.

Stochastic Local Search (SLS) algorithms employ

heuristic approaches that begin by selecting a point

within the search space and then iteratively transition

from the current solution to a neighboring candidate

solution. Certain methods (Anbulagan et al., 2005;

Heras and Ba

neres, 2010; Abram

e and Habet, 2012)

demonstrate that employing incomplete inference can

enhance the performance of SLS algorithms, partic-

ularly those structured on the WalkSat architecture

(Selman et al., 1994).

In this paper, we take a different approach by ap-

plying a transformation that, contrary to prior meth-

ods, increases the size of the Max-SAT formula. Es-

sentially, we will apply the Extension Rule (ER) (Lar-

rosa and Schiex, 2003; Atserias and Lauria, 2019;

Rollon and Larrosa, 2022) which basically takes one

clause of the original formula and creates two ex-

tended clauses from it, which means replacing the

original clause by two copies of such clause but with

an additional literal. This rule is also referred to as

split rule. Then, we propose a novel preprocessor

based on applying the Extension Rule for all clauses

in the formula.

We assess the effect of such preprocessor with

several experiments including different benchmarks

Heras, F.

Impact of Extended Clauses on Local Search Solvers for Max-SAT.

DOI: 10.5220/0013078200003890

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 17th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2025) - Volume 3, pages 78-85

ISBN: 978-989-758-737-5; ISSN: 2184-433X

and SLS algorithms. The empirical investigation

demonstrates that for a subset of SLS algorithms and

benchmarks, this approach produces better results.

The structure of this paper is as follows. We begin

by introducing some preliminary concepts. Next, we

present the Extension Rule and its connection to the

Neighborhood Rule. We then introduce a new prepro-

cessor based on this rule. Following this, we conduct

empirical experiments to evaluate the preprocessor’s

effectiveness. We then discuss how our work relates

to previous research. Finally, we conclude with sum-

marizing our ﬁndings and future research directions.

2 PRELIMINARY CONCEPTS

In this section, we formally introduce the Max-SAT

problem and related notation.

The Max-SAT Problem. X = {x

, x

, . . . , x

} is a set

of Boolean variables. A literal is either a variable x

its negation ¯x

. The variable to which a literal l refers

is denoted by var(l). Given a literal l, its negation

l is

¯x

if l is x

and it is x

if l is ¯x

A clause C is a disjunction of literals. Capital let-

ters will represent clauses. The size of a clause, noted

|C|, is the number of literals that it has. Clauses of

size one and two are called unit and binary clauses,

respectively. A formula in conjunctive normal form

(CNF) is a set of clauses.

An assignment is a set of literals A =

, l

, . . . , l

} such that for all l

∈ A, its vari-

able var(l

) = x

is assigned to value t (true) or f

( f alse). If variable x

is assigned to t ( f ), literal

( ¯x

) is satisﬁed and literal ¯x

) is falsiﬁed. If

all variables in X are assigned, the assignment is

called complete, otherwise it is called partial. An

assignment satisﬁes a literal iff the literal belongs

to the assignment, the assignment satisﬁes a clause

iff one or more of its literals are satisﬁed and the

assignment falsiﬁes a clause iff the clause contains

the negation of all its literals. The empty clause □

cannot be satisﬁed by deﬁnition.

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

clause and the weight w is the cost of its falsiﬁca-

tion. A weighted formula in conjunctive normal form

(WCNF) F is a set of weighted clauses. Many real

problems contain clauses that must be satisﬁed. We

denote these clauses hard and a special weight ⊤ is

associated to them. Note that, any weight w ≥ ⊤ in-

dicates that the associated clause must be necessarily

satisﬁed. Thus, we can replace w by ⊤ without chang-

ing the problem. Consequently, we can assume all

weights are in the interval [0, ⊤]. Non-hard clauses

are called soft clauses. We use the symbol ≡ (e.g.,

F ≡ F

′

) to denote that two formulas are equivalent.

A model is a complete assignment that satisﬁes all

hard clauses. The cost of an assignment is the sum of

the weights of the falsiﬁed clauses. Given a WCNF

formula, the objective of the Weighted Max-SAT is to

ﬁnd a model with minimum cost.

Example 1. Let be F a weighted for-

mula with 3 clauses F = {( ¯x

, 1), ( ¯x

, 1),

∨ x

, ⊤)}. Clauses ( ¯x

, 1) and ( ¯x

, 1) are

unit soft clauses with weight 1. Clause (x

∨ x

, ⊤) is

a binary hard clause. The assignment A

= { ¯x

, ¯x

}

falsiﬁes the hard clause (x

∨ x

, ⊤) and for this

reason A

is not a model. Assignment A

= {x

, x

}

satisﬁes the hard clause and falsiﬁes both soft unit

clauses. Hence, assignment A

is a model with cost

2. Finally, the assignment A

= {x

, ¯x

} satisﬁes the

hard clause and only falsiﬁes one soft clause. A

an optimal model with cost 1.

Let u and w be two weights. Their sum is deﬁned

as u ⊕ w = min{u + w, ⊤} in order to keep the re-

sult within the interval [0, ⊤]. Assuming u ≥ w, their

subtraction is deﬁned as u ⊖ w = u − w if u ̸= ⊤ and

u⊖w = ⊤ otherwise. The De Morgan rule is unsound

for Max-SAT. Instead, the following rule should be

repeatedly used until the conjunctive normal form is

achieved: (A ∨l ∨C, w) ≡ {(A ∨

C, w), (A ∨

l ∨C, w)}.

The empty clause may appear in a formula. If

its weight is ⊤, i.e. (□, ⊤), the formula does not

have any model. Following (Larrosa et al., 2008),

the resolution rule for Max-SAT is {(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}.

(x ∨ A, u) and ( ¯x ∨ B, w) are called clashing clauses;

(A ∨ B, m) is called the resolvent; and (x ∨ A, u ⊖ m)

and ( ¯x ∨ B, w ⊖ m) are called as posterior clashing

clauses. Finally, (x ∨ A ∨

B, m) and ( ¯x ∨

A ∨ B, m) are

called compensation clauses.

Note that in Max-SAT truth tables are tables with

a cost associated to each truth assignments. A brute-

force solving method consists in computing the truth

table of the input for mula and ﬁnding the minimal

cost model. For instance the cost for formula F =

{(x

∨ x

, 1), (x

, 2), ( ¯x

∨ ¯x

, ⊤)} is as in Table 1:

Table 1: Truth table example.

f f 3

f t 0

t f 2

t t ⊤

First and second columns are the variables and the

respective values they can take, as usual in truth ta-

Impact of Extended Clauses on Local Search Solvers for Max-SAT

bles. The third column shows the sum of weights of

unsatisﬁed clauses in F = {(x

∨ x

, 1), (x

, 2), ( ¯x

∨

¯x

, ⊤)} for each possible assignment. Hence, assign-

ment A

= {x

, ¯x

} is the only optimal model with

cost 0.

The Max-Cut Problem. In what follows, we ex-

plain how to encode the Max-Cut problem as Max-

SAT as we will use some Max-Cut instances in the

experimental section. The Max-Cut problem consists

in ﬁnding a cut of maximum size. It can be easily

modeled as Max-SAT. One variable x

is associated

to each graph vertex v

. The value true ( f alse) in-

dicates that vertex v

belongs to U (to V − U). For

each edge (v

, v

), there are two clauses (x

∨x

, 1) and

( ¯x

∨ ¯x

, 1). Given a complete assignment, the number

of violated clauses is |E|−S where S is the size of the

cut associated to the assignment.

3 EXTENDED CLAUSES

In this section, we introduce well-known Neighbor-

hood and Extension rules, which are indeed deeply

related one with the other. Then we propose a novel

preprocessor based on the Extension Rule.

3.1 Neighborhood Resolution

The Neigborhood resolution (NRES) (Larrosa et al.,

2008) takes two clauses containing the same literals

C, with an additional literal h that appears in the pos-

itive h and in the negative

h form on each clause and

results in the following:

{(C ∨ h, w

), (C ∨

h, w

) ≡

{(C, m), (C ∨ h, w

⊖ m), (C ∨

h, w

⊖ m)}

where m = min(w

, w

). Note that if weights w

= m we would obtain simply:

{(C ∨ h, m), (C ∨

h, m) ≡ {(C, m)}

The simpliﬁcation capability of neighborhood res-

olution (NRES) is shown in the following example,

where several steps of resolution are applied based on

NRES, sometimes referred to as hyper-resolution.

Example 2. Let F be a WCNF formula with clauses

F = {(x

∨ x

, 1), (x

∨ x

∨ ¯x

, 1), (x

∨ ¯x

∨

, 1), (x

∨ ¯x

, 1), ( ¯x

, 1)}. The application of

the Neighborhood Resolution rule to the ﬁrst and

second clause, and to the third and fouth clause in

F produces the following equivalent formula F

′

{(x

∨ x

, 1), (x

∨ ¯x

, 1), ( ¯x

, 1)} (i.e. F ≡ F

′

). Now,

we can apply again the NRES rule between the ﬁrst

two clauses, resulting in the following equivalent for-

mula F

′′

= {(x

, 1), ( ¯x

, 1)} (i.e. F ≡ F

′

≡ F

′′

Hence, we obtained a single unit clause from 4

ternary clauses after applying the NRES rule three

times, which simpliﬁed substantially the original for-

mula F . But we can go one step further. We can see

, 1), ( ¯x

, 1) as equivalent to (□ ∨ x

, 1), (□ ∨ ¯x

, 1).

Hence, by NRES rule we would obtain (□, 1) making

explicit a cost of 1 for the original formula. Some

authors refers to this last transformation as Com-

plementary Unit clause rule (Niedermeier and Ross-

manith, 2000).

The term Neighborhood Resolution was coined by

(Cha and Iwama, 1996) in the SAT context.

3.2 Extension Rule

Traditionally research work has focused on rules to

converting the input formula into a simpler form by

shortening clauses, reducing the number of clauses,

and generating as many unit or empty clauses as pos-

sible. In this subsection, we introduce a rule called

Extension rule (ER) (Larrosa and Schiex, 2003; Rol-

lon and Larrosa, 2022; Atserias and Lauria, 2019) and

relate it with the Neigborhood resolution rule. Essen-

tially, it takes a single clause (C, w) where C is a set of

literals, and then it creates two new clauses with one

additional and arbitrary literal h and

h respectively.

{(C, w) ≡ {(C ∨ h, w), (C ∨

h, w)}

See an example below to illustrate this concept.

Example 3. Let F be a WCNF formula with clauses

F = {(x

∨ x

, 1)}. The application of the extension

rule to clauses in F produces the following equivalent

formula F

′

= {(x

∨ x

∨ h, 1), (x

∨ x

∨

h, 1)}) with

arbitrary literal h.

The equivalence of the Extension Rule becomes

obvious when examining the cost distributions in a

truth table. Consider the example below:

Example 4. Assume the following formula F and the

result of applying the Extension Rule in F

′

on vari-

able x

. F = {(x

, 3)} and F

′

= {(x

∨ x

, 3), (x

∨

¯x

, 3)}. We can see that the distribution of costs of the

truth table in Table 2 is the same. Note that it also

shows the equivalence for the case of applying NRES

to formula F

′

which would result back into F .

In the context of the Weighted Constraint Satis-

faction Problem, the extension operation (Larrosa and

Schiex, 2003) is applied to transform the problem and

is deeply related to the Extension Rule (ER). For fur-

ther details, refer to the related work section. Next,

we present a formal proof of the Extension Rule’s cor-

rectness.

ICAART 2025 - 17th International Conference on Agents and Artiﬁcial Intelligence

Table 2: Truth table for F = {(x

, 3)} and F

′

= {(x

∨

, 3), (x

∨ ¯x

, 3)}.

{(x

, 3)} {(x

∨ x

, 3), (x

∨ ¯x

, 3)}

f f 3 3

f t 3 3

t f 0 0

t t 0 0

Proof. Similar to (Rollon and Larrosa, 2022), let

be the resulting clauses from Extension rule (C ∨

h, w), (C ∨

h, w), no matter which value is assigned

variable h the resulting clause would be (C, w). If h

is assigned t then clause (C ∨ h, w) is satisﬁed, and

literal

h in (C ∨

h, w) is unsatisﬁed and as such, can

be removed which results in (C, w). Similarly, If h

is assigned f then clause (C ∨

h, w) is satisﬁed, and

literal h in (C ∨ h, w) is unsatisﬁed and as such, can

be removed which results in (C, w). Alternatively, as-

sume any literal in C is satisﬁed, then both (C ∨ h, w)

and (C ∨

h, w) are satisﬁed by the current assignment.

Contrary, assume all literals in C are falsiﬁed by the

current assignment, then we are left with (h, w) and

(

h, w). No matter which value is assigned to h as we

will have to pay cost w in either case.

Since Neighborhood Resolution is based on the

Resolution Rule for Max-SAT which has been proved

to be sound and complete (Larrosa et al., 2008), we

can do a simpler proof based on Neighborhood Reso-

lution:

Proof. By applying Neighborhood Resolution

(NRES) to the extended clauses (C ∨ h, w), (C ∨

h, w),

we would obtain (C, w). Hence, we obtain the

original formula.

The Neighborhood rule can be considered the in-

verse transformation of the extension rule, and vice

versa.

3.3 Extension Preprocessor

Given the original Max-SAT formula see the pseudo-

code for the Extension Preprocessor in Algorithm 1.

As it can be observed, it essentially applies the

Extension Rule to all clauses of the original formula.

The additional literal added to each pair of extended

clauses can be selected in many ways. Refer to the

empirical section for the selected approach.

4 RESULTS

For our experiments, we employed the original (OR)

Max-SAT problem alongside the transformed in-

Algorithm 1: Extension Preprocessor.

Input: A Max-SAT formula F

Output: A Max-SAT formula F

′

which is

equivalent to F

Function ExtensionPreprocessor(F ):

′

←

0 ; // Initialize new formula

foreach clause (C, w) in F do

′

← F

′

∪ {(C ∨ l, w), (C ∨ ¬l, w)} ;

// Being l an arbitrary literal

end

return F

′

stance using the Extension Preprocessor (Ext). The

preprocessor was implemented using .NET Core 7.0,

and its execution time is negligible for the bench-

marks considered. Hence, for clarity of presentation,

we have not included them.

We investigated the impact of the preprocessor on

relevant Stochastic Local Search (SLS) algorithms in

the literature. The following algorithms have been

evaluated which are publicly available in UBCSAT

solver (Tompkins and Hoos, 2004): SAMD (Hansen

and Jaumard, 1990), GSAT (Selman et al., 1992),

Walksat (Selman et al., 1994), Novelty (McAllester

et al., 1997), Tabu Walksat (McAllester et al., 1997),

Novelty+ (Hoos, 1999), Adaptnovelty+ (Hoos, 2002),

IROTS (Smyth et al., 2003), VW2 (Prestwich, 2005),

and adaptg2wsat+p (Li et al., 2007b). Note that

all those algorithms in UBCSAT solver can han-

dle both unweighted and weighted Max-SAT, except

adaptg2wsat+p and VW2.

The benchmarks were selected from recent Max-

SAT Evaluations

and comprise the following:

• M2S. Random Max-2-SAT instances with 120

variables and 1200 to 2600 binary clauses.

• MC7 and MC8. Crafted Max-CUT instances on

bipartite graphs with 140 nodes and 630 edges,

represented as Max-2-SAT instances.

• MCAH. Crafted Max-CUT instances with 140

variables and 1200 to 2600 binary clauses.

• M3SAH. Random Max-3-SAT instances with 110

variables and 700 to 1100 ternary clauses.

• M3SH. Random Max-3-SAT instances instances

with 250/300 variables and 1000/1200 ternary

clauses.

• WM2S. Random Weighted Max-2-SAT with 140

variables and 1200 to 1600 binary clauses.

• WM3SH. Random Weighted Max-3-SAT in-

stances instances with 70 variables and 700 to

1000 ternary clauses.

All experiments were conducted on a PC running

Windows 11 Home, equipped with an AMD Ryzen 5

https://maxsat-evaluations.github.io/

Impact of Extended Clauses on Local Search Solvers for Max-SAT

4500U processor at 2.38 GHz and 8 GB of RAM. The

experiments involved running each Stochastic Local

Search algorithm for each instance 10 times, 10000

iterations for each run, and recording the average of

the returned solutions and average execution time.

We conducted preliminary experiments to select

the arbitrary h literal for each pair of extended clauses

(C ∨ h, w), (C ∨

h, w). We tested three simple meth-

ods: always assigning the literal that appears in the

most clauses, the literal that appears in the fewest

clauses, and a random literal. We realized that assign-

ing the same literal signiﬁcantly impacted execution

performance, as it often required traversing all clauses

within each SLS solver iteration. Therefore, for the

remaining experiments in this section, we assigned a

random literal for each pair of extended clauses.

Find a summary of the comparison in tables 3

to 12. The values in column Diff, show the vari-

ance between the best solution obtained for the origi-

nal problem and the one achieved for the preprocessed

problem. Hence, negative values indicate a deteriora-

tion in the solution resulting from the transformation.

Column OR shows the average execution time for the

original formula, and Ext the average time for the pre-

processed formula.

The different local search algorithms exhibit dis-

tinct behavior to the transformed instances produced

by the preprocessor. Speciﬁcally, Novelty, Novelty+,

AdaptNovelty+, and Tabu WalkSat exhibit notable

improvements with the Extension Preprocessor Ext.

These enhancements are observed in both unweighted

and weighted Max-SAT instances. However, more

substantial improvements are reported for Max-2-

SAT instances compared to Max-3-SAT instances.

Conversely, the Ext preprocessor degrades no-

tably the performance of Walksat, GSAT and SAMD.

For IROTS, VW2, and adaptg2wsat+p, there is some

degradation, but it is less signiﬁcant. Notably, VW2

shows slight improvements in two benchmarks, while

IROTS is the least sensitive algorithm.

Regarding execution time, for Ext preprocessed

instances, it generally takes 1.5 to 2 times longer com-

pared to the OR original instances for most Local

Search algorithms. This is expected since the num-

ber of clauses is doubled. The only exceptions are

IROTS and SAMD, which are even faster with the

transformed instances.

5 PREVIOUS WORK

In the context of SAT several works were proposed to

add additional clauses to help SLS to ﬁnd solutions

(Cha and Iwama, 1996; Lorenz and W

orz, 2020).

Table 3: Solutions for Novelty.

Benchmark Diff OR Ext

M2S 20.87 0.60 1.23

MC7 17.27 0.39 0.79

MC8 17.41 0.39 0.79

MCAH 23.21 0.54 1.13

M3SAH 1.21 0.41 0.68

M3SH 1.50 0.24 0.43

WM2S 90.66 0.68 1.07

WM3SH 5.09 0.64 1.02

Table 4: Solutions for Novelty+.

Benchmark Diff OR Ext

M2S 21.04 0.60 1.23

MC7 16.71 0.39 0.79

MC8 17.28 0.39 0.79

MCAH 23.02 0.54 1.13

M3SAH 1.42 0.41 0.70

M3SH 1.30 0.24 0.44

WM2S 90.46 0.68 1.07

WM3SH 4.66 0.64 1.01

Table 5: Solutions for AdaptNovelty+.

Benchmark Diff OR Ext

M2S 8.78 0.60 1.22

MC7 6.64 0.39 0.79

MC8 6.71 0.39 0.79

MCAH 11.29 0.55 1.14

M3SAH 0.23 0.40 0.68

M3SH -0.02 0.23 0.41

WM2S 30.44 0.65 1.04

WM3SH 0.63 0.62 0.99

Table 6: Solutions for adaptg2wsat+p.

Benchmark Diff OR Ext

M2S -4.03 0.85 2.11

MC7 -2.75 0.42 1.07

MC8 -2.75 0.42 1.07

MCAH -6.55 0.53 1.47

M3SAH -0.58 0.59 1.24

M3SH -0.34 0.36 0.69

Several inference rules based on Max-SAT reso-

lution (i.e. the Neighborhood Resolution rule) were

introduced in (Li et al., 2007a; Larrosa et al., 2008)

and subsquent works which are specially designed to

simplify the formula and usually target binary and

unit clauses. Most of them are based on obtaining

unit clauses from binary ones and then use those to

create new empty clauses making a lower bound ex-

plicit in the formula. Such rules have been shown

to be efﬁcient in different types of algorithms, both

as a preprocessor or embedded in the algorithm, in-

ICAART 2025 - 17th International Conference on Agents and Artiﬁcial Intelligence

Table 7: Solutions for Walksat.

Benchmark Diff OR Ext

M2S -2.87 0.44 0.87

MC7 -5.91 0.29 0.55

MC8 -6.32 0.29 0.56

MCAH -4.81 0.40 0.78

M3SAH -2.28 0.30 0.49

M3SH -4.65 0.18 0.32

WM2S -29.22 0.61 0.90

WM3SH -7.53 0.49 0.75

Table 8: Solutions for Tabu Walksat.

Benchmark Diff OR Ext

M2S 3.94 0.43 0.82

MC7 3.01 0.27 0.53

MC8 3.04 0.29 0.55

MCAH 5.64 0.38 0.74

M3SAH 0.06 0.29 0.47

M3SH 0.24 0.17 0.31

WM2S 14.09 0.55 0.81

WM3SH 0.78 0.47 0.73

Table 9: Solutions for SAMD.

Benchmark Diff OR Ext

M2S -5.40 0.57 0.51

MC7 -10.88 0.39 0.36

MC8 -10.95 0.39 0.36

MCAH -13.65 0.47 0.47

M3SAH -4.67 0.48 0.32

M3SH -6.19 0.50 0.40

WM2S -36.90 0.44 0.50

WM3SH -30.60 0.54 0.48

Table 10: Solutions for VW2.

Benchmark Diff OR Ext

M2S 0.16 0.46 0.90

MC7 -3.28 0.31 0.59

MC8 -3.24 0.31 0.58

MCAH 0.66 0.41 0.81

M3SAH -2.82 0.32 0.52

M3SH -5.77 0.20 0.35

cluding branch-and-bound algorithms (Heras et al.,

2008; Larrosa et al., 2008; Li et al., 2007a; Heras

and Larrosa, 2008; Heras and Ba

neres, 2013; Cherif

et al., 2020), Local Search algorithms (Heras and

neres, 2010; Abram

e and Habet, 2012), and al-

gorithms based on iteratively calling to a SAT oracle

(Heras and Marques-Silva, 2011; Py et al., 2022).

In the context of SLS algorithms, in (Heras and

neres, 2010) a number of preprocessors are intro-

duced based on the resolution rule and tested for sev-

eral local search algorithms. One is based on ap-

Table 11: Solutions for GSAT.

Benchmark Diff OR Ext

M2S -5.29 0.31 0.62

MC7 -4.23 0.15 0.36

MC8 -5.00 0.15 0.35

MCAH -4.13 0.21 0.51

M3SAH -3.08 0.19 0.34

M3SH -6.56 0.15 0.22

WM2S -10.76 0.35 0.51

WM3SH -4.50 0.28 0.47

Table 12: Solutions for IROTS.

Benchmark Diff OR Ext

M2S -0.26 0.67 0.58

MC7 -1.06 0.51 0.37

MC8 -1.02 0.51 0.37

MCAH -1.35 0.59 0.50

M3SAH -0.16 0.56 0.44

M3SH -3.48 0.70 0.39

WM2S -14.65 0.63 0.49

WM3SH -2.79 0.69 0.56

plying the resolution rule until saturation so that a

variable can be eliminated. The second one is based

on hyper-resolution rules applied to problems with

a speciﬁc structure mainly with binary hard clauses

and unit soft clauses. The last one is based on ap-

plying unit propagation to generate new unit clauses

and also new empty clauses. The latter two exhib-

ited notable performance improvements. In (Abram

and Habet, 2012), inference rules were embedded in

a local search solver, resulting in notable improved

performance.

In the context of the Weighted Constraint Satis-

faction Problem (WCSP), the extension operation is

commonly used in various forms of Soft Arc Consis-

tency for WCSP, such as DAC*, FDAC* and EDAC*

(Larrosa and Schiex, 2003; de Givry et al., 2005)

which results in an equivalent problem with an ex-

plicit lower bound. Additionally, in (de Givry et al.,

2003) it is shown how to solve Max-SAT as a WCSP

problem. The extension operation is commonly ap-

plied to transform unary constraints into binary ones

while maintaining the equivalence, but it can also be

applied to constraints of other sizes in WCSP. Hence,

the extension operation in WCSP is deeply related to

the Extension Rule (ER) for Max-SAT. Let us see an

example.

Example 5. Refer to Figure 1. (Weighted) Max-

SAT can be interpreted as a Weighted CSP, where all

variables can take only two values, and (weighted)

clauses are represented as tuples of values that incur

a cost (the weight of the clause) when assigned simul-

taneously. Case a) illustrates a WCSP problem with

Impact of Extended Clauses on Local Search Solvers for Max-SAT

two variables, x

and x

, each having two possible

values (t and f ), and two binary constraints: when

= t and x

= t, a cost of 1 is incurred, and similarly

when x

= t and x

= f . In Max-SAT, this is equiva-

lent to the clauses (¯x

∨ x

, 1) and ( ¯x

∨ ¯x

, 1). Ap-

plying Soft Arc Consistency in a) results in an equiv-

alent problem in b), which reduces two binary con-

straints into a unary constraint x

= t with cost 1, that

is equivalent apply NRES and obtain clause (¯x

, 1) in

Max-SAT. However, applying the extension operation

to b) restores the original equivalent formula c), that

is equivalent to applying the Extension Rule in Max-

SAT context.

Figure 1: WCSP and Max-SAT relationship example.

In a recent work (Rollon and Larrosa, 2022), the

Extension rule was explicitly reintroduced for Max-

SAT and termed the split rule (Atserias and Lau-

ria, 2019). In such work, the extension rule is com-

bined with another rule that requires adding clauses

with negative weight. Its theoretical potential is ex-

perimented exclusively with Pigeon Hole Problem

(PHP) instances and it requires a handcrafted refu-

tation based on the PHP problem structure. In our

paper, we evaluated the application of the extension

(i.e. split) rule in SLS solvers using a new prepro-

cessor. Importantly, the preprocessor is not tied to

any speciﬁc problem or structure. This work repre-

sents the ﬁrst contribution to show that the Extension

Rule can produce improved results from a practical

perspective.

6 CONCLUSIONS

In this paper, we proposed a Max-SAT preproces-

sor based on the Extension Rule (ER) (Atserias and

Lauria, 2019; Rollon and Larrosa, 2022), generat-

ing larger clauses in the resulting formula. Empiri-

cal evaluation with various SLS algorithms showed

notable performance improvements, especially for

Max-2-SAT instances. This ﬁnding is signiﬁcant, as

many problems are naturally encoded as Max-2-SAT,

and some SAT problems can be reduced to Max-2-

SAT (Ans

otegui and Levy, 2021). Thus, achieving

good results in Max-2-SAT serves as a good starting

point. Previous work focused on shortening clauses

and reducing their number, under the assumption that

smaller formulas are better. Our study shows that

larger formulas can improve outcomes for speciﬁc

problems and SLS algorithms. Finally, we explored

the relationships between the Neighborhood and Ex-

tension Rules, and the extension operation in WCSP.

Several potential directions of future work within

this domain exist. Firstly, the development of more

sophisticated heuristics for selecting the arbitrary lit-

eral employed to extend the clauses could be pursued.

Additionally, incorporating a broader range of bench-

marks, particularly those including hard clauses (i.e.

partial Max-SAT), would be valuable to understand

whether the extension rule would beneﬁt hard clauses

as well as soft ones. Furthermore, exploring the po-

tential of hybrid approaches that combine the exten-

sion rule with other known preprocessing techniques

or to consider developing heuristics to selectively ap-

ply the Extension rule, rather than applying it to all

clauses as in current preprocessor. Finally, investi-

gating the circumstances under which other algorithm

families, such as branch and bound or iterative calls

to SAT oracle approaches, might proﬁt from similar

strategies.

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