Performance of Sequential Local Algorithms for the Random NAE-KK-SAT Problem

We formalize the class of “sequential local algorithms" and show that these algorithms fail to find satisfying assignments on random instances of the “Not-All-Equal-$K$-SAT” (NAE-$K$-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 $(1+o_K(1)){2^{K-1}\over K}\ln^2 K$ for sufficiently large $K$. 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-$K$-SAT problem. We show that above the $(1+o_K(1)){2^{K-1}\over K}\ln^2 K$ threshold, the overlap structure of $m$-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 $m$ assignments for appropriately chosen positive integer $m$. 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 $m$-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

Approaching the Rate-Distortion Limit with Spatial Coupling, Belief propagation and Decimation

We investigate an encoding scheme for lossy compression of a binary symmetric source based on simple spatially coupled Low-Density Generator-Matrix codes. The degree of the check nodes is regular and the one of code-bits is Poisson distributed with an average depending on the compression rate. The performance of a low complexity Belief Propagation Guided Decimation algorithm is excellent. The algorithmic rate-distortion curve approaches the optimal curve of the ensemble as the width of the coupling window grows. Moreover, as the check degree grows both curves approach the ultimate Shannon rate-distortion limit. The Belief Propagation Guided Decimation encoder is based on the posterior measure of a binary symmetric test-channel. This measure can be interpreted as a random Gibbs measure at a "temperature" directly related to the "noise level of the test-channel". We investigate the links between the algorithmic performance of the Belief Propagation Guided Decimation encoder and the phase diagram of this Gibbs measure. The phase diagram is investigated thanks to the cavity method of spin glass theory which predicts a number of phase transition thresholds. In particular the dynamical and condensation "phase transition temperatures" (equivalently test-channel noise thresholds) are computed. We observe that: (i) the dynamical temperature of the spatially coupled construction saturates towards the condensation temperature; (ii) for large degrees the condensation temperature approaches the temperature (i.e. noise level) related to the information theoretic Shannon test-channel noise parameter of rate-distortion theory. This provides heuristic insight into the excellent performance of the Belief Propagation Guided Decimation algorithm. The paper contains an introduction to the cavity method