403 research outputs found
Performance of Sequential Local Algorithms for the Random NAE--SAT Problem
We formalize the class of âsequential local algorithms" and show that these algorithms fail to find satisfying assignments on random instances of the âNot-All-Equal--SATâ (NAE--SAT) problem if the number of message passing iterations is bounded by a function moderately growing in the number of variables and if the clause-to-variable ratio is above for sufficiently large . Sequential local algorithms are those that iteratively set variables based on some local information and/or local randomness and then recurse on the reduced instance. Our model captures some weak abstractions of natural algorithms such as Survey Propagation (SP)-guided as well as Belief Propagation (BP)-guided decimation algorithms---two widely studied message-passing--based algorithms---when the number of message-passing rounds in these algorithms is restricted to be growing only moderately with the number of variables. The approach underlying our paper is based on an intricate geometry of the solution space of a random NAE--SAT problem. We show that above the threshold, the overlap structure of -tuples of nearly (in an appropriate sense) satisfying assignments exhibit a certain behavior expressed in the form of some constraints on pairwise distances between the assignments for appropriately chosen positive integer . We further show that if a sequential local algorithm succeeds in finding a satisfying assignment with probability bounded away from zero, then one can construct an -tuple of solutions violating these constraints, thus leading to a contradiction. Along with [D. Gamarnik and M. Sudan, Ann. Probab., to appear], where a similar approach was used in a (somewhat simpler) setting of nonsequential local algorithms, this result is the first work that directly links the overlap property of random constraint satisfaction problems to the computational hardness of finding satisfying assignments.National Science Foundation (U.S.) (CMMI-1335155
Comparing Beliefs, Surveys and Random Walks
Survey propagation is a powerful technique from statistical physics that has
been applied to solve the 3-SAT problem both in principle and in practice. We
give, using only probability arguments, a common derivation of survey
propagation, belief propagation and several interesting hybrid methods. We then
present numerical experiments which use WSAT (a widely used random-walk based
SAT solver) to quantify the complexity of the 3-SAT formulae as a function of
their parameters, both as randomly generated and after simplification, guided
by survey propagation. Some properties of WSAT which have not previously been
reported make it an ideal tool for this purpose -- its mean cost is
proportional to the number of variables in the formula (at a fixed ratio of
clauses to variables) in the easy-SAT regime and slightly beyond, and its
behavior in the hard-SAT regime appears to reflect the underlying structure of
the solution space that has been predicted by replica symmetry-breaking
arguments. An analysis of the tradeoffs between the various methods of search
for satisfying assignments shows WSAT to be far more powerful that has been
appreciated, and suggests some interesting new directions for practical
algorithm development.Comment: 8 pages, 5 figure
Random subcubes as a toy model for constraint satisfaction problems
We present an exactly solvable random-subcube model inspired by the structure
of hard constraint satisfaction and optimization problems. Our model reproduces
the structure of the solution space of the random k-satisfiability and
k-coloring problems, and undergoes the same phase transitions as these
problems. The comparison becomes quantitative in the large-k limit. Distance
properties, as well the x-satisfiability threshold, are studied. The model is
also generalized to define a continuous energy landscape useful for studying
several aspects of glassy dynamics.Comment: 21 pages, 4 figure
The decimation process in random k-SAT
Let F be a uniformly distributed random k-SAT formula with n variables and m
clauses. Non-rigorous statistical mechanics ideas have inspired a message
passing algorithm called Belief Propagation Guided Decimation for finding
satisfying assignments of F. This algorithm can be viewed as an attempt at
implementing a certain thought experiment that we call the Decimation Process.
In this paper we identify a variety of phase transitions in the decimation
process and link these phase transitions to the performance of the algorithm
On the cavity method for decimated random constraint satisfaction problems and the analysis of belief propagation guided decimation algorithms
We introduce a version of the cavity method for diluted mean-field spin
models that allows the computation of thermodynamic quantities similar to the
Franz-Parisi quenched potential in sparse random graph models. This method is
developed in the particular case of partially decimated random constraint
satisfaction problems. This allows to develop a theoretical understanding of a
class of algorithms for solving constraint satisfaction problems, in which
elementary degrees of freedom are sequentially assigned according to the
results of a message passing procedure (belief-propagation). We confront this
theoretical analysis to the results of extensive numerical simulations.Comment: 32 pages, 24 figure
Aspects of Statistical Physics in Computational Complexity
The aim of this review paper is to give a panoramic of the impact of spin
glass theory and statistical physics in the study of the K-sat problem. The
introduction of spin glass theory in the study of the random K-sat problem has
indeed left a mark on the field, leading to some groundbreaking descriptions of
the geometry of its solution space, and helping to shed light on why it seems
to be so hard to solve.
Most of the geometrical intuitions have their roots in the
Sherrington-Kirkpatrick model of spin glass. We'll start Chapter 2 by
introducing the model from a mathematical perspective, presenting a selection
of rigorous results and giving a first intuition about the cavity method. We'll
then switch to a physical perspective, to explore concepts like pure states,
hierarchical clustering and replica symmetry breaking.
Chapter 3 will be devoted to the spin glass formulation of K-sat, while the
most important phase transitions of K-sat (clustering, condensation, freezing
and SAT/UNSAT) will be extensively discussed in Chapter 4, with respect their
complexity, free-entropy density and the Parisi 1RSB parameter.
The concept of algorithmic barrier will be presented in Chapter 5 and
exemplified in detail on the Belief Propagation (BP) algorithm. The BP
algorithm will be introduced and motivated, and numerical analysis of a
BP-guided decimation algorithm will be used to show the role of the clustering,
condensation and freezing phase transitions in creating an algorithmic barrier
for BP.
Taking from the failure of BP in the clustered and condensed phases, Chapter
6 will finally introduce the Cavity Method to deal with the shattering of the
solution space, and present its application to the development of the Survey
Propagation algorithm.Comment: 56 pages, 14 figure
- âŠ