EVOLVED PREAMBLES FOR MAX-SAT HEURISTICS

Lu´ıs O. Rigo Jr. and Valmir C. Barbosa

Programa de Engenharia de Sistemas e Computac¸˜ao, COPPE, Universidade Federal do Rio de Janeiro

Caixa Postal 68511, 21941-972 Rio de Janeiro, RJ, Brazil

Keywords:

MAX-SAT, MAX-SAT heuristics, Initial truth assignments.

Abstract:

MAX-SAT heuristics normally operate from random initial truth assignments to the variables. We consider

the use of what we call preambles, which are sequences of variables with corresponding single-variable as-

signment actions intended to be used to determine a more suitable initial truth assignment for a given problem

instance and a given heuristic. For a number of well established MAX-SAT heuristics and benchmark in-

stances, we demonstrate that preambles can be evolved by a genetic algorithm such that the heuristics are

outperformed in a signiﬁcant fraction of the cases. The heuristics we consider include the well-known novelty,

walksat-tabu, and adaptnovelty+. Our benchmark instances are those of the 2004 SAT competition and those

of the 2008 MAX-SAT evaluation.

1 INTRODUCTION

Given a set V of Boolean variables and a set of dis-

junctive clauses on literals from V (i.e., variables

or their negations), MAX-SAT asks for a truth as-

signment to the variables that maximizes the num-

ber of clauses that are satisﬁed (i.e., made true by

that assignment). MAX-SAT is NP-hard (Garey and

Johnson, 1979) but can be approximated in polyno-

mial time, though not as close to the optimum as

one wishes. This holds in general (Ausiello et al.,

1999) as well as in the restricted case of three-literal

clauses (Dantsin et al., 2001). MAX-SAT has enjoyed

a paradigmatic status over the years, not only because

of its close relation to SAT, the ﬁrst decision problem

to be proven NP-complete, but also because of its im-

portance to other areas (e.g., constraint satisfaction in

artiﬁcial intelligence (Dechter, 2003)).

Since NP-hardness isa property of worst-case sce-

narios, the difﬁculty of actually solving a speciﬁc

instance of an NP-hard problem varies widely with

both the instance’s size and internal structure. In fact,

in recent years it has become increasingly clear that

small changes in either can lead to signiﬁcant varia-

tion in an algorithm’s performance, possibly even to

a divide between the instance’s being solvable or un-

solvable by that algorithm giventhe computational re-

sources at hand and the time one is willing to spend

(Hartmann and Weigt, 2005). Following some early

groundwork (Rice, 1976), several attempts have been

made at providing theoretical foundations or practical

guidelines for automatically selecting which method

to use given the instance (Russell and Subramanian,

1995; Minton, 1996; Fink, 1998; Gomes and Selman,

2001; Lagoudakis et al., 2001; Leyton-Brown et al.,

2003; Vassilevska et al., 2006; Xu et al., 2008). These

include approaches that have addressed the solution

of NP-complete problems.

Here we investigatea different, though related, ap-

proach to method selection in the case of MAX-SAT

instances. Since all MAX-SAT heuristics require an

initial truth assignment to the variables, and consid-

ering that this is invariably chosen at random, a nat-

ural question seems to be whether it is worth spend-

ing some additional effort to determine an initial as-

signment that is better suited to the instance at hand.

Once we adopt this two-stage template comprising an

initial-assignment selection in tandem with a heuris-

tic, ﬁxing the latter reduces the issue of method selec-

tion to that of identifying a procedure to determine an

appropriate initial assignment. We refer to this pro-

cedure as a preamble to the heuristic. As we demon-

strate in the sequel, for several state-of-the-art heuris-

tics and problem instances the effort to come up with

an appropriate preamble pays off in terms of better

solutions for the same amount of time.

We proceed in the following manner. First, in

Section 2, we deﬁne what preambles are in the case

of MAX-SAT. Then we introduce an evolutionary

method for preamble determination in Section 3 and

givecomputationalresults in Section 4. We close with

concluding remarks in Section 5.

O. Rigo Jr. L. and C. Barbosa V..

EVOLVED PREAMBLES FOR MAX-SAT HEURISTICS.

DOI: 10.5220/0003660400230031

In Proceedings of the International Conference on Evolutionary Computation Theory and Applications (ECTA-2011), pages 23-31

ISBN: 978-989-8425-83-6

 2011 SCITEPRESS (Science and Technology Publications, Lda.)

2 MAX-SAT PREAMBLES

Let n be the number of variables in V. Given a MAX-

SAT instance on V, a preamble p of length ℓ is a

sequence of pairs hv

i,hv

i,...,hv

ℓ

i, each

representing a computational step to be taken as the

preamble is played out. In this sequence, and for

1 ≤ k ≤ ℓ, the kth pair is such that v

∈ V and a

one of 2, 1, or 0, indicating respectively whether to

leave the value of v

unchanged, to act greedily when

choosing a value for v

, or to act contrarily to such

greedy assignment. A preamble need not include all

n variables, and likewise a variable may appear more

than once in it.

The greedy action to which a

sometimes refers

assigns to v

the truth value that maximizes the num-

ber of satisﬁed clauses at that point in the preamble.

Algorithmically, then, playing out p is equivalent to

performing the following steps from some initial truth

assignment to the variables in V:

1. k := 1.

2. If a

= 2, then proceed to Step 5.

3. Given the current values of all other variables,

compute the number of clauses that get satisﬁed

for each of the two possible assignments to v

4. If a

= 1, then set v

to the truth value yielding

the greatest number of satisﬁed clauses. If a

= 0,

then do the opposite. Break ties randomly.

5. k := k + 1. If k ≤ ℓ, then proceed to Step 2.

We use random initial assignments exclusively. A

MAX-SAT preamble, therefore, can be thought of as

isolating such initial randomness from the heuristic

proper that is to follow the preamble. Instead of start-

ing the heuristic at its usual random initial assign-

ment, we start it at the assignment determined by run-

ning the preamble.

It is curious to note that, as deﬁned, a preamble

generalizes the sequence of steps generated by the

simulated annealing method (Kirkpatrick et al., 1983)

when applied to MAX-SAT. In fact, what simulated

annealing does in this case, following one of its varia-

tions (Geman and Geman, 1984; Barbosa, 1993), is to

choose v

by cycling through the members of V and

then let a

be either 1 or 0 with the Boltzmann-Gibbs

probability. At the high initial temperatures the two

outcomes are nearly equally probable, but the near-

zero ﬁnal temperatures imply a

= 1 (i.e., be greedy)

with high probability. The generalization that comes

with our deﬁnition allows for various possibilities of

preamble construction, as the evolutionary procedure

we describe next.

3 METHODS

Given a MAX-SAT instance and heuristic H, our ap-

proach is to evolve the best possible preamble to H.

We do so through a genetic algorithm of the genera-

tional type (Mitchell, 1996). The description that fol-

lows refers to parameter values that were determined

in an initial calibration phase. This phase used the

heuristics gsat (Selman et al., 1992), gwsat (Selman

and Kautz, 1993), hsat (Gent and Walsh, 1993), hwsat

(Gent and Walsh, 1995), gsat-tabu (Mazure et al.,

1997), novelty (McAllester et al., 1997), walksat-

tabu (McAllester et al., 1997), adaptnovelty+ (Hoos,

2002), saps (Hutter et al., 2002), and sapsnr (Tomp-

kins and Hoos, 2004) as heuristic H, and also the in-

stances C880mul, am 8 8, c3540mul, term1mul, and

vdamul (Le Berre and Simon, 2005). Each of the lat-

ter involves variables that number in the order of 10

and clauses numbering in the order of 10

. Moreover,

not all optima are known (cf. Section 4).

The genetic algorithm operates on a population of

50 individuals, each being a preamble to heuristic H.

The ﬁtness of individual p is computed as follows.

First p is run from 10 random truth assignments to

the variables, then H is run from the truth assign-

ment resulting from the run of p that satisﬁed the most

clauses (ties between runs of p are broken randomly).

Let R(p) denote the number of clauses satisﬁed by

this best run of p and S

(p) the number of clauses

satisﬁed after H is run. The ﬁtness of individual p

is the pair hS

(p),R(p)i. Whenever two individuals’

ﬁtnesses are compared, ties are ﬁrst broken lexico-

graphically, then randomly. Selection is always per-

formed from linearly normalized ﬁtnesses, the ﬁttest

individual of the population being 20 times as ﬁt as

the least ﬁt.

For each MAX-SAT instance and each heuristic

H, we let the genetic algorithm run for a ﬁxed amount

of time, during which a new population is repeatedly

produced from the current one and replaces it. The

initial population comprises individuals of maximum

length 1.5n, each created randomly to contain at least

0.4n distinct variables. The process of creating each

new population starts by an elitist step that transfers

the 20% ﬁttest individualsfrom the current population

to the new one. It then repeats the following until the

new population is full.

First a decision is made as to whether crossover

(with probability 0.25) or mutation (with probability

0.75) is to be performed. For crossover two indi-

viduals are selected from the current population and

each is partitioned into three sections for application

of the standard two-point crossover operator. The par-

titioning is done randomly, provided the middle sec-

ECTA 2011 - International Conference on Evolutionary Computation Theory and Applications

tion contains exactly 0.4n distinct variables, which is

always possible by construction of the initial popula-

tion (though at times either of the two extreme sec-

tions may turn out to be empty). The resulting two

individuals (whose lengths are no longer bounded by

1.5n) are added to the new population. For mutation a

single individual is selected from the current popula-

tion and 50% of its pairs are chosen at random. Each

of these, say the kth pair, undergoes either a random

change to both v

and a

(if this will leave the indi-

vidual with at least 0.4n distinct variables) or simply

a random change to a

(otherwise). The mutant is

then added to the new population.

The calibration phase referred to above also

yielded three champion heuristics, viz. novelty,

walksat-tabu, and adaptnovelty+. The results we give

in Section 4 refer exclusively to these, used either in

conjunction with the genetic algorithm as described

above or by themselves. In the latter case each heuris-

tic is run repeatedly, each time from a new random

truth assignment to the variables, until the same ﬁxed

amount of time used for the genetic algorithm has

elapsed. The result reported by the genetic algorithm

refers to the ﬁttest individual in the last population to

have been ﬁlled during that time. As for the heuris-

tic, in order to compare its performance with that of

the genetic algorithm as fairly as possible the result

that is reported is the best one obtained after every 50

repetitions.

All experiments were performed from within the

UBCSAT environment (Tompkins and Hoos, 2005).

Optima, whenever possible, were discovered sepa-

rately via the 2010 release of the MSUnCore code to

solve MAX-SAT exactly (Manquinho et al., 2009).

4 COMPUTATIONAL RESULTS

In our experiments we tackled all 100 instances of

the 2004 SAT competition (Le Berre and Simon,

2005), henceforth referred to as the 2004 dataset, and

all 112 instances of the 2008 MAX-SAT evaluation

(Argelich et al., 2008), henceforth referred to as the

2008 dataset. The time allotted for each instance to

the genetic algorithm or each of the three heuristics

by itself was of 60 minutes, always on identical hard-

ware and software, always with exclusive access to

the system. We report exclusively on the hardest in-

stances from either dataset, here deﬁned to be those

for which MSUnCore found no answer as a result of

being stymied by the available 4 gigabytes of RAM

and the system’s inability to perform further swap-

ping. There are 51 such instances in the 2004 dataset,

11 in the 2008 dataset, totaling 62 instances.

Our results are given in Tables 1 through 6 for H =

novelty (Tables 1 and 4), H = walksat-tabu (Tables 2

and 5), and H = adaptnovelty+ (Tables 3 and 6). They

refer to those of the 62 instances that come from the

2004 dataset (Tables 1 through 3) and those that come

from the 2008 dataset (Tables 4 through 6). Each ta-

ble contains a row for each of the corresponding in-

stances. For each instance the number n of variables

is given, as well as the number of clauses (m) and re-

sults for the genetic algorithm and for the heuristic in

question by itself. These results are the number of sat-

isﬁed clauses and the time at which this solution was

ﬁrst found during the allotted 60 minutes. Missing re-

sults indicate either that no population could be ﬁlled

during this time (in the case of the genetic algorithm)

or that no batch of 50 runs of the heuristic could be

ﬁnished.

Some entries in the tables are highlighted by a

bold typeface to indicate that the genetic algorithm

found a solution strictly better than the one found by

the heuristic when used by itself, or a solution sat-

isfying the same number of clauses but ﬁrst encoun-

tered in a shorter time. In the former case only the

number of satisﬁed clauses is highlighted, in the lat-

ter case the time is highlighted as well. The number

of highlighted instances amounts to the ratios given

in Table 7. Clearly, with the notable exception of

H = walksat-tabu on the 2008 dataset (on which the

use of H alone outperformed the genetic algorithm on

all 11 instances), the genetic algorithm succeeds well

on a signiﬁcant fraction of the instances.

Revising these ratios to contemplate all instances

from both datasets (i.e., include the results omitted

from Tables 1 through 6) yields the ratios in Table 8.

These show that the genetic algorithm fares even bet-

ter when evaluated on all 100 instances of the 2004

dataset. They also show slightly lower ratios for the

genetic algorithm on the 112-instance 2008 dataset

for H = novelty and H = adaptnovelty+. As for

H = walksat-tabu, we see in Table 8 a dramatic in-

crease from the 0.000 of Table 7, indicating that for

this particular H on the complete 2008 dataset the

genetic algorithm does better than the heuristic alone

only on the comparatively easier instances (and then

for a signiﬁcant fraction of them).

5 CONCLUDING REMARKS

Given a heuristic for some problem of combinatorial

optimization, a preamble such as we deﬁned in Sec-

tion 2 for MAX-SAT is a selector of initial conditions.

As such, it aims at isolating the inevitable random-

ness of the initial conditions one normally uses with

EVOLVED PREAMBLES FOR MAX-SAT HEURISTICS

Table 1: Results for H = novelty on the 2004 dataset. Times are given in minutes.

Genetic algorithm novelty alone

Instance n m Num. sat. cl. Time Num. sat. cl. Time

c3540mul 5248 33199 33176 17.965 33180 6.291

c6288mul 9540 61421 61375 21.391 61375 6.706

dalumul 9426 59991 59972 59.961 59973 5.712

frg1mul 3230 20575 20574 0.543 20574 1.390

k2mul 11680 74581 74524 1.729 74516 23.493

x1mul 8760 55571 55570 1.550 55570 5.383

am 6 6 2269 7814 7813 0.263 7813 0.267

am 7 7 4264 14751 14744 55.270 14733 50.688

am 8 8 7361 25538 25331 37.914 25332 36.184

am 9 9 11908 41393 40982 1.827 40970 36.322

li-exam-61 28147 108436 108011 47.997 108041 22.993

li-exam-62 28147 108436 107999 56.149 108006 27.760

li-exam-63 28147 108436 107998 29.129 108003 19.108

li-exam-64 28147 108436 107987 58.590 108013 34.106

li-test4-100 36809 142491 141844 34.560 141856 35.189

li-test4-101 36809 142491 141858 40.733 141865 31.374

li-test4-94 36809 142491 141863 2.556 141843 29.103

li-test4-95 36809 142491 141850 58.823 141868 16.500

li-test4-96 36809 142491 141858 6.038 141848 16.622

li-test4-97 36809 142491 141855 29.113 141852 33.242

li-test4-98 36809 142491 141862 40.632 141850 52.114

li-test4-99 36809 142491 141859 12.537 141855 31.298

gripper10u 2312 18666 18663 0.877 18663 2.014

gripper11u 3084 26019 26017 15.150 26016 6.459

gripper12u 3352 29412 29409 14.485 29409 29.184

gripper13u 4268 38965 38961 9.830 38961 1.315

gripper14u 4584 43390 43386 23.381 43386 36.584

bc56-sensors-1-k391-unsat 561371 1778987 1600252 15.997 1600079 43.783

bc56-sensors-2-k592-unsat 850398 2694319 2366874 48.252 2366923 13.745

bc57-sensors-1-k303-unsat 435701 1379987 1261961 39.150 1262043 14.386

dme-03-1-k247-unsat 261352 773077 736229 49.430 736010 22.759

motors-stuck-1-k407-unsat 654766 2068742 1842393 55.420 1842299 7.294

motors-stuck-2-k314-unsat 505536 1596837 1445274 53.360 1445323 57.764

valves-gates-1-k617-unsat 985042 3113540 2714448 58.357 2714681 30.268

6pipe 15800 394739 394717 42.251 - -

7pipe 23910 751118 - - - -

comb1 5910 16804 16749 43.330 16751 44.024

dp12u11 11137 30792 30785 11.240 30789 17.881

f2clk 50 34678 101319 100629 18.230 100668 29.506

ﬁfo8 300 194762 530713 506270 9.520 506329 49.814

homer17 286 1742 1738 0.126 1738 0.317

homer18 308 2030 2024 0.131 2024 0.331

homer19 330 2340 2332 0.142 2332 0.349

homer20 440 4220 4202 0.170 4202 0.429

k2ﬁx gr 2pinvar w8 3771 270136 269918 47.860 269910 29.096

k2ﬁx gr 2pinvar w9 5028 307674 307563 43.461 307565 42.008

k2ﬁx gr 2pin w8 9882 295998 295657 54.870 295691 5.697

k2ﬁx gr 2pin w9 13176 345426 345230 39.450 345228 24.266

k2ﬁx gr rcs w8 10056 271393 271296 17.291 271292 58.947

sha1 61377 255417 251863 54.40 251927 48.348

sha2 61377 255417 251915 19.47 251873 17.809

ECTA 2011 - International Conference on Evolutionary Computation Theory and Applications

Table 2: Results for H = walksat-tabu on the 2004 dataset. Times are given in minutes.

Genetic algorithm walksat-tabu alone

Instance n m Num. sat. cl. Time Num. sat. cl. Time

c3540mul 5248 33199 33164 24.820 33166 59.816

c6288mul 9540 61421 61392 3.884 61389 23.451

dalumul 9426 59991 59896 2.730 59910 2.546

frg1mul 3230 20575 20570 30.706 20570 6.465

k2mul 11680 74581 74340 6.720 74341 21.223

x1mul 8760 55571 55561 34.556 55562 26.049

am 6 6 2269 7814 7809 2.580 7810 58.420

am 7 7 4264 14751 14732 29.810 14729 0.926

am 8 8 7361 25538 25466 19.600 25473 45.678

am 9 9 11908 41393 41183 34.960 41189 59.298

li-exam-61 28147 108436 107983 14.918 107982 31.107

li-exam-62 28147 108436 107974 41.560 107992 53.727

li-exam-63 28147 108436 107980 48.127 107978 58.207

li-exam-64 28147 108436 107985 3.946 107985 20.522

li-test4-100 36809 142491 141790 39.720 141782 21.592

li-test4-101 36809 142491 141804 19.040 141800 11.141

li-test4-94 36809 142491 141806 2.853 141780 44.018

li-test4-95 36809 142491 141791 35.540 141782 27.593

li-test4-96 36809 142491 141804 24.995 141791 33.641

li-test4-97 36809 142491 141791 26.641 141805 22.784

li-test4-98 36809 142491 141777 26.300 141791 54.528

li-test4-99 36809 142491 141774 37.930 141781 51.666

gripper10u 2312 18666 18662 42.287 18662 58.256

gripper11u 3084 26019 26014 2.383 26014 10.450

gripper12u 3352 29412 29406 2.104 29406 2.788

gripper13u 4268 38965 38959 2.370 38959 50.868

gripper14u 4584 43390 43383 2.520 43382 0.512

bc56-sensors-1-k391-unsat 561371 1778987 1600245 36.329 1599963 43.683

bc56-sensors-2-k592-unsat 850398 2694319 2361372 33.625 2361225 39.390

bc57-sensors-1-k303-unsat 435701 1379987 1264444 33.863 1264039 44.873

dme-03-1-k247-unsat 261352 773077 740068 15.573 739924 53.964

motors-stuck-1-k407-unsat 654766 2068742 1840274 36.547 1839998 36.970

motors-stuck-2-k314-unsat 505536 1596837 1445995 41.870 1445813 34.788

valves-gates-1-k617-unsat 985042 3113540 2705763 55.140 2705761 53.865

6pipe 15800 394739 394727 48.824 - -

7pipe 23910 751118 751102 27.336 - -

comb1 5910 16804 16713 59.406 16717 9.986

dp12u11 11137 30792 30775 22.338 30773 27.777

f2clk 50 34678 101319 100087 49.690 100075 27.279

ﬁfo8 300 194762 530713 509196 2.030 509252 56.291

homer17 286 1742 1738 0.105 1738 0.268

homer18 308 2030 2024 0.112 2024 0.285

homer19 330 2340 2332 0.119 2332 0.309

homer20 440 4220 4202 0.152 4202 0.374

k2ﬁx gr 2pinvar w8 3771 270136 269839 46.921 269855 34.207

k2ﬁx gr 2pinvar w9 5028 307674 307490 31.670 307485 14.594

k2ﬁx gr 2pin w8 9882 295998 295544 12.160 295554 43.311

k2ﬁx gr 2pin w9 13176 345426 345065 3.299 345044 47.063

k2ﬁx gr rcs w8 10056 271393 271301 42.525 271301 48.211

sha1 61377 255417 251374 30.88 251364 9.204

sha2 61377 255417 251390 7.67 251392 16.520

EVOLVED PREAMBLES FOR MAX-SAT HEURISTICS

Table 3: Results for H = adaptnovelty+ on the 2004 dataset. Times are given in minutes.

Genetic algorithm adaptnovelty+ alone

Instance n m Num. sat. cl. Time Num. sat. cl. Time

c3540mul 5248 33199 33162 52.030 33153 30.089

c6288mul 9540 61421 61382 1.893 61382 36.361

dalumul 9426 59991 59930 47.250 59920 44.117

frg1mul 3230 20575 20574 0.527 20574 1.290

k2mul 11680 74581 74417 52.360 74407 16.932

x1mul 8760 55571 55570 1.153 55570 2.477

am 6 6 2269 7814 7813 0.102 7813 0.267

am 7 7 4264 14751 14750 0.277 14750 0.558

am 8 8 7361 25538 25523 44.450 25520 8.450

am 9 9 11908 41393 41296 49.254 41293 53.522

li-exam-61 28147 108436 108037 23.833 108028 54.671

li-exam-62 28147 108436 108045 38.670 108032 17.074

li-exam-63 28147 108436 108031 44.030 108026 49.837

li-exam-64 28147 108436 108030 41.830 108037 49.076

li-test4-100 36809 142491 141903 16.011 141902 35.010

li-test4-101 36809 142491 141894 4.150 141897 15.527

li-test4-94 36809 142491 141899 38.595 141908 37.992

li-test4-95 36809 142491 141900 17.200 141890 57.627

li-test4-96 36809 142491 141902 31.204 141896 52.649

li-test4-97 36809 142491 141897 27.190 141920 33.317

li-test4-98 36809 142491 141898 34.560 141907 52.417

li-test4-99 36809 142491 141902 29.421 141906 12.445

gripper10u 2312 18666 18665 0.161 18665 5.913

gripper11u 3084 26019 26018 18.106 26018 15.914

gripper12u 3352 29412 29410 0.600 29411 56.930

gripper13u 4268 38965 38963 4.270 38963 0.454

gripper14u 4584 43390 43388 23.785 43388 0.456

bc56-sensors-1-k391-unsat 561371 1778987 1623357 17.990 1623430 36.854

bc56-sensors-2-k592-unsat 850398 2694319 2394308 40.210 2393894 6.102

bc57-sensors-1-k303-unsat 435701 1379987 1282208 35.553 1282338 34.813

dme-03-1-k247-unsat 261352 773077 746902 43.755 746935 19.452

motors-stuck-1-k407-unsat 654766 2068742 1866990 12.900 1867266 56.079

motors-stuck-2-k314-unsat 505536 1596837 1467201 55.320 1467123 43.089

valves-gates-1-k617-unsat 985042 3113540 2742641 48.170 2742310 18.078

6pipe 15800 394739 393808 46.970 393771 41.444

7pipe 23910 751118 749636 54.051 - -

comb1 5910 16804 16756 29.750 16759 31.832

dp12u11 11137 30792 30723 15.480 30722 12.257

f2clk 50 34678 101319 100431 6.983 100435 3.063

ﬁfo8 300 194762 530713 516316 55.551 516380 35.441

homer17 286 1742 1738 0.133 1738 0.330

homer18 308 2030 2024 0.138 2024 0.345

homer19 330 2340 2332 0.143 2332 0.367

homer20 440 4220 4202 0.175 4202 0.440

k2ﬁx gr 2pinvar w8 3771 270136 269923 30.530 269930 39.335

k2ﬁx gr 2pinvar w9 5028 307674 307560 47.330 307559 28.284

k2ﬁx gr 2pin w8 9882 295998 295646 55.630 295645 11.987

k2ﬁx gr 2pin w9 13176 345426 345123 38.150 345150 51.661

k2ﬁx gr rcs w8 10056 271393 271282 25.350 271290 39.640

sha1 61377 255417 252622 34.02 252613 52.078

sha2 61377 255417 252615 34.58 252617 22.981

ECTA 2011 - International Conference on Evolutionary Computation Theory and Applications

Table 4: Results for H = novelty on the 2008 dataset. Times are given in minutes.

Genetic algorithm novelty alone

Instance n m Num. sat. cl. Time Num. sat. cl. Time

rsdecoder1 blackbox KESblock 707330 1106376 1030025 5.471 1030001 47.969

rsdecoder4.dimacs 237783 933978 896327 16.880 896441 51.221

rsdecoder-problem.dimacs 38 1198012 3865513 3350748 11.820 3351073 46.902

rsdecoder-problem.dimacs 41 1186710 3829036 3320154 37.064 3320274 36.249

SM MAIN MEM buggy1.dimacs 870975 3812147 3416609 43.833 3416837 37.047

wb 4m8s1.dimacs 463080 1759150 1624751 17.008 1624325 27.448

wb 4m8s4.dimacs 463080 1759150 1624017 4.333 1623800 22.801

wb 4m8s-problem.dimacs 47 2691648 8517027 7159756 56.460 7159458 53.487

wb 4m8s-problem.dimacs 49 2785108 8812799 7401876 43.930 7402080 52.858

wb conmax1.dimacs 277950 1221020 1168273 23.774 1168267 27.155

wb conmax3.dimacs 277950 1221020 1168336 1.808 1168173 56.621

Table 5: Results for H = walksat-tabu on the 2008 dataset. Times are given in minutes.

Genetic algorithm walksat-tabu alone

Instance n m Num. sat. cl. Time Num. sat. cl. Time

rsdecoder1 blackbox KESblock 707330 1106376 1028280 55.013 1028328 33.828

rsdecoder4.dimacs 237783 933978 895865 1.950 895990 16.142

rsdecoder-problem.dimacs 38 1198012 3865513 3334250 34.230 3334450 49.290

rsdecoder-problem.dimacs 41 1186710 3829036 3304293 7.626 3304476 4.832

SM MAIN MEM buggy1.dimacs 870975 3812147 3386126 37.130 3386263 58.493

wb 4m8s1.dimacs 463080 1759150 1611716 56.910 1611839 59.405

wb 4m8s4.dimacs 463080 1759150 1610945 36.070 1611361 56.378

wb 4m8s-problem.dimacs 47 2691648 8517027 7133234 33.360 7133580 44.962

wb 4m8s-problem.dimacs 49 2785108 8812799 7374846 3.900 7374877 32.896

wb conmax1.dimacs 277950 1221020 1156736 35.785 1156771 22.056

wb conmax3.dimacs 277950 1221020 1156718 31.800 1156841 35.277

Table 6: Results for H = adaptnovelty+ on the 2008 dataset. Times are given in minutes.

Genetic algorithm adaptnovelty+ alone

Instance n m Num. sat. cl. Time Num. sat. cl. Time

rsdecoder1 blackbox KESblock 707330 1106376 1042736 58.190 1042849 35.251

rsdecoder4.dimacs 237783 933978 904931 50.378 905006 38.068

rsdecoder-problem.dimacs 38 1198012 3865513 3374308 33.560 3374204 28.959

rsdecoder-problem.dimacs 41 1186710 3829036 3343967 55.770 3343653 12.919

SM MAIN MEM buggy1.dimacs 870975 3812147 3431928 15.243 3431455 18.274

wb 4m8s1.dimacs 463080 1759150 1637425 6.646 1637337 25.506

wb 4m8s4.dimacs 463080 1759150 1636787 15.974 1636718 49.083

wb 4m8s-problem.dimacs 47 2691648 8517027 7189090 14.730 7189244 19.575

wb 4m8s-problem.dimacs 49 2785108 8812799 7432769 20.240 7432463 12.193

wb conmax1.dimacs 277950 1221020 1175783 43.730 1175804 8.269

wb conmax3.dimacs 277950 1221020 1175833 12.660 1175942 14.228

Table 7: Success ratios of the genetic algorithm as per Tables 1 through 6.

Instance set novelty walksat-tabu adaptnovelty+

2004 dataset 0.540 0.667 0.588

2008 dataset 0.545 0.000 0.545

2004 & 2008 datasets combined 0.541 0.548 0.581

EVOLVED PREAMBLES FOR MAX-SAT HEURISTICS

Table 8: Success ratios of the genetic algorithm over all 100 instances of the 2004 dataset and all 112 instances of the 2008

dataset.

Instance set novelty walksat-tabu adaptnovelty+

2004 dataset 0.600 0.700 0.630

2008 dataset 0.518 0.518 0.509

2004 & 2008 datasets combined 0.557 0.604 0.566

such heuristics from the heuristic itself. By doing

so, preambles attempt to poise the heuristic to oper-

ate from more favorable initial conditions.

In this paper we have demonstrated the success

of MAX-SAT preambles when they are discovered,

given the MAX-SAT instance and heuristic of inter-

est, via an evolutionary algorithm. As we showed in

Section 4, for well established benchmark instances

and heuristics the resulting genetic algorithm can out-

perform the heuristics themselves when used alone.

We believe further effort can be proﬁtably spent on at-

tempting similar solutions to other problems that, like

MAX-SAT, can be expressed as an unconstrained op-

timization problem on binary variables. Some of them

are the maximum independent set and minimum dom-

inating set problems on graphs, both admitting well-

known formulations of this type (Barbosa and Gafni,

1989).

ACKNOWLEDGEMENTS

The authors acknowledge partial support from CNPq,

CAPES, and a FAPERJ BBP grant.

REFERENCES

Argelich, J., Li, C. M., Many`a, F., and Planes, J.

(2008). Third Max-SAT evaluation. URL http://

www.maxsat.udl.cat/08/.

Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V.,

Marchetti-Spaccamela, A., and Protasi, M. (1999).

Complexity and Approximation: Combinatorial Op-

timization Problems and their Approximability Prop-

erties. Springer-Verlag, Berlin, Germany.

Barbosa, V. C. (1993). Massively Parallel Models of Com-

putation. Ellis Horwood, Chichester, UK.

Barbosa, V. C. and Gafni, E. (1989). A distributed imple-

mentation of simulated annealing. Journal of Parallel

and Distributed Computing, 6:411–434.

Dantsin, E., Gavrilovich, M., Hirsch, E., and Konev, B.

(2001). MAX SAT approximation beyond the lim-

its of polynomial-time approximation. Annals of Pure

and Applied Logic, 113:81–94.

Dechter, R. (2003). Constraint Processing. Morgan Kauf-

mann, San Francisco, CA.

Fink, E. (1998). How to solve it automatically: Selection

among problem-solving methods. In Proceedings of

the Fourth International Conference on Artiﬁcial In-

telligence Planning Systems, pages 128–136, Menlo

Park, CA. AAAI Press.

Garey, M. R. and Johnson, D. S. (1979). Computers

and Intractability: A Guide to the Theory of NP-

Completeness. W. H. Freeman, New York, NY.

Geman, S. and Geman, D. (1984). Stochastic relaxation,

Gibbs distributions, and the Bayesian restoration of

images. IEEE Transactions on Pattern Analysis and

Machine Intelligence, PAMI-6:721–741.

Gent, I. P. and Walsh, T. (1993). Towards an understand-

ing of hill-climbing procedures for SAT. In Proceed-

ings of the Eleventh National Conference on Artiﬁcial

Intelligence, pages 28–33, Menlo Park, CA. AAAI

Press.

Gent, I. P. and Walsh, T. (1995). Unsatisﬁed variables in lo-

cal search. In Hallam, J., editor, Hybrid Problems, Hy-

brid Solutions, pages 73–85, Amsterdam, The Nether-

lands. IOS Press.

Gomes, C. P. and Selman, B. (2001). Algorithm portfolios.

Artiﬁcial Intelligence, 126:43–62.

Hartmann, A. K. and Weigt, M. (2005). Phase Transi-

tions in Combinatorial Optimization Problems: Ba-

sics, Algorithms and Statistical Mechanics. Wiley-

VCH, Weinheim, Germany.

Hoos, H. H. (2002). An adaptive noise mechanism for

WalkSAT. In Proceedings of the Eighteenth National

Conference on Artiﬁcial Intelligence, pages 655–660,

Menlo Park, CA. AAAI Press.

Hutter, F., Tompkins, D. A. D., and Hoos, H. H. (2002).

Scaling and probabilistic smoothing: Efﬁcient dy-

namic local search for SAT. In van Hentenryck, P., ed-

itor, Principles and Practice of Constraint Program-

ming, volume 2470 of Lecture Notes in Computer

Science, pages 233–248, Berlin, Germany. Springer-

Verlag.

Kirkpatrick, S., Gelatt, C. D., and Vecchi, M. P. (1983). Op-

timization by simulated annealing. Science, 220:671–

680.

Lagoudakis, M. G., Littman, M. L., and Parr, R. E. (2001).

Selecting the right algorithm. In Proceedings of the

2001 AAAI Fall Symposium Series: Using Uncertainty

within Computation, pages 74–75, Menlo Park, CA.

AAAI Press.

Le Berre, D. and Simon, L. (2005). The SAT 2004 com-

petition. In Hoos, H. H. and Mitchell, D. G., edi-

tors, Theory and Applications of Satisﬁability Testing,

volume 3542 of Lecture Notes in Computer Science,

pages 321–344, Berlin, Germany. Springer-Verlag.

Leyton-Brown, K., Nudelman, E., Andrew, G., McFadden,

J., and Shoham, Y. (2003). Boosting as a metaphor for

algorithm design. In Rossi, F., editor, Principles and

Practice of Constraint Programming, volume 2833 of

ECTA 2011 - International Conference on Evolutionary Computation Theory and Applications

Lecture Notes in Computer Science, pages 899–903,

Berlin, Germany. Springer-Verlag.

Manquinho, V., Marques-Silva, J., and Planes, J. (2009).

Algorithms for weighted Boolean optimization. In

Kullmann, O., editor, Theory and Applications of Sat-

isﬁability Testing, volume 5584 of Lecture Notes in

Computer Science, pages 495–508, Berlin, Germany.

Springer-Verlag.

Mazure, B., Sais, L., and Gregoire, E. (1997). Tabu search

for SAT. In Proceedings of the Fourteenth National

Conference on Artiﬁcial Intelligence, pages 281–285,

Menlo Park, CA. AAAI Press.

McAllester, D., Selman, B., and Kautz, H. (1997). Evidence

for invariants in local search. In Proceedings of the

Fourteenth National Conference on Artiﬁcial Intelli-

gence, pages 321–326, Menlo Park, CA. AAAI Press.

Minton, S. (1996). Automatically conﬁguring constraint

satisfaction programs: A case study. Constraints, 1:7–

43.

Mitchell, M. (1996). An Introduction to Genetic Algo-

rithms. The MIT Press, Cambridge, MA.

Rice, J. R. (1976). The algorithm selection problem. In

Rubinoff, M. and Yovits, M. C., editors, Advances

in Computers, volume 15, pages 65–118. Academic

Press, New York, NY.

Russell, S. and Subramanian, D. (1995). Provably bounded-

optimal agents. Journal of Artiﬁcial Intelligence Re-

search, 2:575–609.

Selman, B. and Kautz, H. (1993). Domain-independant ex-

tensions to GSAT: Solving large structured variables.

In Proceedings of the Thirteenth International Joint

Conference on Artiﬁcial Intelligence, pages 290–295,

San Mateo, CA. Morgan Kaufmann.

Selman, B., Levesque, H., and Mitchell, D. (1992). A new

method for solving hard satisﬁability problems. In

Proceedings of the Tenth National Conference on Ar-

tiﬁcial Intelligence, pages 459–465, Menlo Park, CA.

AAAI Press.

Tompkins, D. A. D. and Hoos, H. H. (2004). Warped land-

scapes and random acts of SAT solving. In Proceed-

ings of the Eighth International Symposium on Arti-

ﬁcial Intelligence and Mathematics, Piscataway, NJ.

RUTCOR.

Tompkins, D. A. D. and Hoos, H. H. (2005). UBCSAT:

An implementation and experimentation environment

for SLS algorithms for SAT and MAX-SAT. In Hoos,

H. H. and Mitchell, D. G., editors, Theory and Ap-

plications of Satisﬁability Testing, volume 3542 of

Lecture Notes in Computer Science, pages 306–320,

Berlin, Germany. Springer-Verlag.

Vassilevska, V., Williams, R., and Woo, S. L. M. (2006).

Confronting hardness using a hybrid approach. In

Proceedings of the Seventeenth Annual ACM-SIAM

Symposium on Discrete Algorithms, pages 1–10, New

York, NY. ACM Press.

Xu, L., Hutter, F., Hoos, H. H., and Leyton-Brown, K.

(2008). SATzilla: Portfolio-based algorithm selection

for SAT. Journal of Artiﬁcial Intelligence Research,

32:565–606.

EVOLVED PREAMBLES FOR MAX-SAT HEURISTICS