Statistical mechanics of the random K-SAT model

43 Pages·0.33 MB·English

by  R. Monasson

0.6
0.5
0.4

0.3
0.2
0.1
0
0.25 0.5 0.75 1 1.25 1.5 1.75 2
M/N
1
0.8
0.6

0.4
p=1
p=5
p=10
0.2
0
1.25 1.5 1.75 2 2.25 2.5 2.75 3
M/N
1
0.8
0.12
Ln(2)
0.6
0.08

p=1,10
0.4
0.04
0.2
0
0.5 1 1.5 2 2.5 3
M/N
0.17
0.16
0.15

p>>1
RS-values
p=1
K=2, M/N=3, N=18,20,22,24,26
0.14
0.13
0.01 0.02 0.03 0.04 0.05 0.06
1/N
0.125
0.1
0.075

0.05
p=10
0.025
p=1
0
4.5 5 5.5 6 6.5 7 7.5 8
M/N
0.5
0.4
p=1
0.3

0.2
p=5
0.1
p=10
0
4.6 4.8 5 5.2 5.4 5.6 5.8 6
M/N
0.8
0.7
Ln(2)
0.6
0.5
0.4

0.3
0.2
0.1
0
1 2 3 4 5 6
M/N
25 M/N=Log(2) 2^K
20 K=18
15

10 K=16
5 K=14
K=12
K=10
0
-0.75 -0.5 -0.25 0 0.25 0.5 0.75 1
x
Statistical Mechanics of the Random K-SAT Model
(cid:3) y
R(cid:19)emi Monasson and Riccardo Zecchina
(cid:3)
Laboratoire de Physique Th(cid:19)eorique de l'ENS,
24 rue Lhomond, 75231 Paris cedex 05, France
y
Dipartimento di Fisica, Politecnico di Torino,
C.so Duca degli Abruzzi 24, I-10129 Torino, Italy
Abstract
TheRandomK-Satis(cid:12)abilityProblem,consistinginverifyingtheexistenceof
an assignment of N Boolean variables that satisfy a set of M = (cid:11)N random
logical clauses containing K variables each, is studied using the replica sym-
metricframeworkofdiluteddisorderedsystems. Wepresentanexactiterative
scheme for the replica symmetric functional order parameter together for the
di(cid:11)erent cases of interest K = 2, K (cid:21) 3 and K (cid:29) 1. The calculation of the
number of solutions, which allowed us [Phys. Rev. Lett. 76, 3881 (1996)] to
predict a (cid:12)rst order jump at the threshold where the Boolean expressions be-
come unsatis(cid:12)able with probability one, is thoroughly displayed. In the case
K =2, the (rigorously known) critical value ((cid:11) =1) of the number of clauses
per Boolean variable is recovered while for K (cid:21) 3 we show that the system
exhibitsareplicasymmetrybreakingtransition. Theannealedapproximation
is proven to be exact for large K.
PACS Numbers : 05.20 - 64.60 - 87.10
Typeset using REVTEX
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I. INTRODUCTION
The emergent collective behaviours observed in a variety of models of statisticalmechan-
icsandinparticularinfrustrateddisorderedsystems, have beenrecognizedtoplayarelevant
role in apparently distant (cid:12)elds such as theoretical computer science, discrete mathematics
and complex systems theory [1{5]. Computationally hard problems, characterized (in worst
cases) by exponential running time scaling of their algorithms or memory requirements, the
so called NP{complete problems [6], are known to be in one{to{one correspondence with
the ground state properties of spin{glass like models (see [1] and references therein). As
a consequence, tools and concepts of statistical physics have shed some new light onto the
notion of the typical complexity of NP-complete problems and have lead to the de(cid:12)nition of
new search algorithms as the simulated annealing algorithm, based on the introduction of
an arti(cid:12)cial temperature and some cooling procedures [7].
Very recently, other techniques inspired from statistical mechanics, namely (cid:12)nite size
scaling analysis, have been applied [8] also to the study of universal behaviour in the com-
putational cost (running time) of some classes of algorithms in the course of searching for
solutions of random realizationsof the prototype of NP{complete problems, the satis(cid:12)ability
(SAT) problem we shall discuss.
More generally, phase transition concepts are starting to play a relevant role in theoret-
ical computer science [4], where the analysis of general search methods applied to various
classes of hard computational problems, characterized by a large number of relevant vari-
ables and generated according to some probability distributions, is of crucial importance in
building a theory for the typical{case complexity. NP{complete decision problems which
are computationally hard in the worst case appear not to be really so in the typical case,
except in critical regions of their parameter space (with a polynomial{exponential pattern)
where almost all instances of the problems become computationally hard to solve. Far from
criticality, the problems are either under- or over-constrained and both the stochastic search
procedures and the systematic ones are capable of (cid:12)nding solutions in polynomial times.
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