Performance of the Survey Propagation-guided decimation algorithm for the random NAE-K-SAT problem

We show that the Survey Propagation-guided decimation algorithm fails to find satisfying assignments on random instances of the "Not-All-Equal-$K$-SAT" problem if the number of message passing iterations is bounded by a constant independent of the size of the instance and the clause-to-variable ratio is above $(1+o_K(1)){2^{K-1}\over K}\log^2 K$ for sufficiently large $K$. Our analysis in fact applies to a broad class of algorithms described as "sequential local algorithms". Such algorithms iteratively set variables based on some local information and then recurse on the reduced instance. Survey Propagation-guided as well as Belief Propagation-guided decimation algorithms - two widely studied message passing based algorithms, fall under this category of algorithms provided the number of message passing iterations is bounded by a constant. Another well-known algorithm falling into this category is the Unit Clause algorithm. Our work constitutes the first rigorous analysis of the performance of the SP-guided decimation algorithm. The approach underlying our paper is based on an intricate geometry of the solution space of random NAE-$K$-SAT problem. We show that above the $(1+o_K(1)){2^{K-1}\over K}\log^2 K$ threshold, the overlap structure of $m$-tuples of satisfying assignments exhibit a certain clustering behavior expressed in the form of constraints on distances between the $m$ assignments, for appropriately chosen $m$. 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 (citation), this result is the first work which directly links the clustering property of random constraint satisfaction problems to the computational hardness of finding satisfying assignments.

[1]  J. Michael Steele,et al.  The Objective Method: Probabilistic Combinatorial Optimization and Local Weak Convergence , 2004 .

[2]  David Gamarnik,et al.  Randomized Greedy Algorithms for Independent Sets and Matchings in Regular Graphs: Exact Results and Finite Girth Corrections , 2008, Combinatorics, Probability and Computing.

[3]  Amin Coja-Oghlan A Better Algorithm for Random k-SAT , 2010, SIAM J. Comput..

[4]  Leonid A. Levin,et al.  Average Case Complete Problems , 1986, SIAM J. Comput..

[5]  David Gamarnik,et al.  Maximum Weight Independent Sets and Matchings in Sparse Random Graphs. Exact Results Using the Local Weak Convergence Method , 2004, APPROX-RANDOM.

[6]  Andrea Montanari,et al.  Gibbs states and the set of solutions of random constraint satisfaction problems , 2006, Proceedings of the National Academy of Sciences.

[7]  Andrea Montanari,et al.  Reconstruction and Clustering in Random Constraint Satisfaction Problems , 2011, SIAM J. Discret. Math..

[8]  Guilhem Semerjian,et al.  On the cavity method for decimated random constraint satisfaction problems and the analysis of belief propagation guided decimation algorithms , 2009, ArXiv.

[9]  Martin J. Wainwright,et al.  A new look at survey propagation and its generalizations , 2004, SODA '05.

[10]  Riccardo Zecchina,et al.  Entropy landscape and non-Gibbs solutions in constraint satisfaction problems , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[11]  Riccardo Zecchina,et al.  Survey propagation: An algorithm for satisfiability , 2002, Random Struct. Algorithms.

[12]  Michael Krivelevich,et al.  Two‐coloring random hypergraphs , 2002, Random Struct. Algorithms.

[13]  Madhu Sudan,et al.  Limits of local algorithms over sparse random graphs , 2013, ITCS.

[14]  Amin Coja-Oghlan A Better Algorithm for Random k-SAT , 2009, ICALP.

[15]  Bart Selman,et al.  Survey Propagation Revisited , 2007, UAI.

[16]  Konstantinos Panagiotou,et al.  Catching the k-NAESAT threshold , 2011, STOC '12.

[17]  E. Friedgut,et al.  Sharp thresholds of graph properties, and the -sat problem , 1999 .

[18]  S. Kak Information, physics, and computation , 1996 .

[19]  Amin Coja-Oghlan,et al.  On independent sets in random graphs , 2010, SODA '11.

[20]  D. Gamarnik,et al.  Counting without sampling: Asymptotics of the log-partition function for certain statistical physics models , 2008 .

[21]  M. Mézard,et al.  Reconstruction on Trees and Spin Glass Transition , 2005, cond-mat/0512295.

[22]  B. Szegedy,et al.  Limits of local-global convergent graph sequences , 2012, 1205.4356.

[23]  Krzysztof Onak,et al.  Constant-Time Approximation Algorithms via Local Improvements , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[24]  Bálint Virág,et al.  Local algorithms for independent sets are half-optimal , 2014, ArXiv.

[25]  M. Mézard,et al.  Analytic and Algorithmic Solution of Random Satisfiability Problems , 2002, Science.

[26]  Amin Coja-Oghlan,et al.  On belief propagation guided decimation for random k-SAT , 2010, SODA '11.