Closing the Gap Between Short and Long XORs for Model Counting

Many recent algorithms for approximate model counting are based on a reduction to combinatorial searches over random subsets of the space defined by parity or XOR constraints. Long parity constraints (involving many variables) provide strong theoretical guarantees but are computationally difficult. Short parity constraints are easier to solve but have weaker statistical properties. It is currently not known how long these parity constraints need to be. We close the gap by providing matching necessary and sufficient conditions on the required asymptotic length of the parity constraints. Further, we provide a new family of lower bounds and the first non-trivial upper bounds on the model count that are valid for arbitrarily short XORs. We empirically demonstrate the effectiveness of these bounds on model counting benchmarks and in a Satisfiability Modulo Theory (SMT) application motivated by the analysis of contingency tables in statistics.

[1]  Yuguo Chen,et al.  Sequential Monte Carlo Methods for Statistical Analysis of Tables , 2005 .

[2]  Sharad Malik,et al.  On computing minimal independent support and its applications to sampling and counting , 2015, Constraints.

[3]  Supratik Chakraborty,et al.  A Scalable and Nearly Uniform Generator of SAT Witnesses , 2013, CAV.

[4]  Pedro M. Domingos,et al.  Probabilistic theorem proving , 2011, UAI.

[5]  Bart Selman,et al.  Low-density Parity Constraints for Hashing-Based Discrete Integration , 2014, ICML.

[6]  Henry A. Kautz,et al.  Solving Bayesian Networks by Weighted Model Counting , 2005 .

[7]  Robin K. S. Hankin,et al.  Exact Tests for Two-Way Contingency Tables with Structural Zeros , 2008 .

[8]  Luca Trevisan,et al.  Lecture Notes on Computational Complexity , 2004 .

[9]  Henry A. Kautz,et al.  Performing Bayesian Inference by Weighted Model Counting , 2005, AAAI.

[10]  M. Kendall.  Statistical Methods for Research Workers , 1937, Nature.

[11]  Leslie G. Valiant,et al.  Fast probabilistic algorithms for hamiltonian circuits and matchings , 1977, STOC '77.

[12]  Bart Selman,et al.  Optimization With Parity Constraints: From Binary Codes to Discrete Integration , 2013, UAI.

[13]  Yuguo Chen,et al.  Conditional Inference on Tables With Structural Zeros , 2007 .

[14]  Leslie G. Valiant,et al.  The Complexity of Enumeration and Reliability Problems , 1979, SIAM J. Comput..

[15]  Theo Tryfonas,et al.  Frontiers in Artificial Intelligence and Applications , 2009 .

[16]  Claude Castelluccia,et al.  Extending SAT Solvers to Cryptographic Problems , 2009, SAT.

[17]  Yehuda Lindell.  Introduction to Coding Theory Lecture Notes , 2009 .

[18]  Jörg Hoffmann,et al.  Short XORs for Model Counting: From Theory to Practice , 2007, SAT.

[19]  Vibhav Gogate,et al.  Approximate Counting by Sampling the Backtrack-free Search Space , 2007, AAAI.

[20]  Bart Selman,et al.  Model Counting: A New Strategy for Obtaining Good Bounds , 2006, AAAI.

[21]  Leslie G. Valiant,et al.  The Complexity of Computing the Permanent , 1979, Theor. Comput. Sci..

[22]  Colin Cooper,et al.  On the rank of random matrices , 2000, Random Struct. Algorithms.

[23]  G. S. Tseitin.  On the Complexity of Derivation in Propositional Calculus , 1983 .

[24]  Thomas G. Dietterich,et al.  Collective Graphical Models , 2011, NIPS.

[25]  Supratik Chakraborty,et al.  A Scalable Approximate Model Counter , 2013, CP.

[26]  Chris Calabro,et al.  The exponential complexity of satisfiability problems , 2009 .

[27]  Guy Van den Broeck,et al.  Hashing-Based Approximate Probabilistic Inference in Hybrid Domains , 2015, UAI.

[28]  Martin E. Dyer,et al.  Sampling contingency tables , 1997, Random Struct. Algorithms.

[29]  Nikolaj Bjørner,et al.  Z3: An Efficient SMT Solver , 2008, TACAS.

[30]  Dimitris Achlioptas,et al.  Stochastic Integration via Error-Correcting Codes , 2015, UAI.

[31]  Bart Selman,et al.  Taming the Curse of Dimensionality: Discrete Integration by Hashing and Optimization , 2013, ICML.

[32]  Larry J. Stockmeyer,et al.  On Approximation Algorithms for #P , 1985, SIAM J. Comput..

[33]  Toniann Pitassi,et al.  Combining Component Caching and Clause Learning for Effective Model Counting , 2004, SAT.

[34]  Evimaria Terzi,et al.  What do row and column marginals reveal about your dataset? , 2013, NIPS.

[35]  Matthew Richardson,et al.  Markov logic networks , 2006, Machine Learning.

[36]  Ashish Sabharwal,et al.  Leveraging Belief Propagation, Backtrack Search, and Statistics for Model Counting , 2008, ISAIM.

[37]  Alan M. Frieze,et al.  Random graphs , 2006, SODA '06.