Phase Transition Behavior of Cardinality and XOR Constraints

The runtime performance of modern SAT solvers is deeply connected to the phase transition behavior of CNF formulas. While CNF solving has witnessed significant runtime improvement over the past two decades, the same does not hold for several other classes such as the conjunction of cardinality and XOR constraints, denoted as CARD-XOR formulas. The problem of determining the satisfiability of CARD-XOR formulas is a fundamental problem with a wide variety of applications ranging from discrete integration in the field of artificial intelligence to maximum likelihood decoding in coding theory. The runtime behavior of random CARD-XOR formulas is unexplored in prior work. In this paper, we present the first rigorous empirical study to characterize the runtime behavior of 1-CARD-XOR formulas. We show empirical evidence of a surprising phase-transition that follows a non-linear tradeoff between CARD and XOR constraints.

[1]  Dong Yang,et al.  Applying constraint satisfaction approach to solve product configuration problems with cardinality-based configuration rules , 2013, J. Intell. Manuf..

[2]  Supratik Chakraborty,et al.  Algorithmic Improvements in Approximate Counting for Probabilistic Inference: From Linear to Logarithmic SAT Calls , 2016, IJCAI.

[3]  Moshe Y. Vardi,et al.  The Hard Problems Are Almost Everywhere For Random CNF-XOR Formulas , 2017, IJCAI.

[4]  Carsten Sinz,et al.  Towards an Optimal CNF Encoding of Boolean Cardinality Constraints , 2005, CP.

[5]  Jacques Stern,et al.  The hardness of approximate optima in lattices, codes, and systems of linear equations , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[6]  Olivier Roussel,et al.  A Translation of Pseudo Boolean Constraints to SAT , 2006, J. Satisf. Boolean Model. Comput..

[7]  Rina Dechter,et al.  Experimental Evaluation of Preprocessing Algorithms for Constraint Satisfaction Problems , 1994, Artif. Intell..

[8]  M. Piccioni,et al.  Importance sampling for families of distributions , 1999 .

[9]  Mate Soos,et al.  BIRD: Engineering an Efficient CNF-XOR SAT Solver and Its Applications to Approximate Model Counting , 2019, AAAI.

[10]  Nadia Creignou,et al.  Satisfiability Threshold for Random XOR-CNF Formulas , 1999, Discret. Appl. Math..

[11]  Bart Selman,et al.  Model Counting , 2021, Handbook of Satisfiability.

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

[13]  Roland H. C. Yap,et al.  Arc Consistency on n-ary Monotonic and Linear Constraints , 2000, CP.

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

[15]  Alexander Vardy,et al.  List decoding of polar codes , 2011, 2011 IEEE International Symposium on Information Theory Proceedings.

[16]  Toniann Pitassi,et al.  Algorithms and complexity results for #SAT and Bayesian inference , 2003, 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings..

[17]  Sharad Malik,et al.  On Solving the Partial MAX-SAT Problem , 2006, SAT.

[18]  F. MacWilliams,et al.  The Theory of Error-Correcting Codes , 1977 .

[19]  Daniel Kroening,et al.  Property-Driven Fence Insertion Using Reorder Bounded Model Checking , 2014, FM.

[20]  Kevin P. Murphy,et al.  Machine learning - a probabilistic perspective , 2012, Adaptive computation and machine learning series.

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

[22]  Elwyn R. Berlekamp,et al.  On the inherent intractability of certain coding problems (Corresp.) , 1978, IEEE Trans. Inf. Theory.

[23]  Moshe Y. Vardi,et al.  Counting-Based Reliability Estimation for Power-Transmission Grids , 2017, AAAI.

[24]  Sanjit A. Seshia,et al.  Distribution-Aware Sampling and Weighted Model Counting for SAT , 2014, AAAI.

[25]  Mark Jerrum,et al.  The Markov chain Monte Carlo method: an approach to approximate counting and integration , 1996 .

[26]  Nadia Creignou,et al.  Smooth and sharp thresholds for random k-XOR-CNF satisfiability , 2003, RAIRO Theor. Informatics Appl..

[27]  Adnan Darwiche,et al.  A Lightweight Component Caching Scheme for Satisfiability Solvers , 2007, SAT.

[28]  Tobias Philipp,et al.  PBLib - A Library for Encoding Pseudo-Boolean Constraints into CNF , 2015, SAT.

[29]  David Chase,et al.  Code Combining - A Maximum-Likelihood Decoding Approach for Combining an Arbitrary Number of Noisy Packets , 1985, IEEE Transactions on Communications.

[30]  S Kirkpatrick,et al.  Critical Behavior in the Satisfiability of Random Boolean Expressions , 1994, Science.

[31]  Peter Gritzmann,et al.  On the computational complexity of reconstructing lattice sets from their X-rays , 1999, Discret. Math..

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

[33]  Carmel Domshlak,et al.  Probabilistic Planning via Heuristic Forward Search and Weighted Model Counting , 2007, J. Artif. Intell. Res..

[34]  Bruce A. Reed,et al.  Mick gets some (the odds are on his side) (satisfiability) , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

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

[36]  Amin Coja-Oghlan,et al.  Algorithmic Barriers from Phase Transitions , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[37]  Moshe Y. Vardi,et al.  Combining the k-CNF and XOR Phase-Transitions , 2016, IJCAI.

[38]  Larry J. Stockmeyer,et al.  The complexity of approximate counting , 1983, STOC.

[39]  Olivier Bailleux,et al.  Efficient CNF Encoding of Boolean Cardinality Constraints , 2003, CP.