Correlations between Horn fractions, satisfiability and solver performance for fixed density random 3-CNF instances

Abstract An enhanced concept of sub-optimal reverse Horn fraction of a CNF-formula was introduced in [18]. It was shown that this fraction is very useful in effectively (almost) separating 3-colorable random graphs with fixed node-edge density from the non-3-colorable ones. A correlation between this enhanced sub-optimal reverse Horn fraction and satisfiability of random 3-SAT instances with a fixed density was observed. In this paper, we present experimental evidence that this correlation scales to larger-sized instances and that it extends to solver performances as well, both of complete and incomplete solvers. Furthermore, we give a motivation for various phases in the algorithm aHS, establishing the enhanced sub-optimal reverse Horn fraction, and we present clear evidence for the fact that the observed correlations are stronger than correlations between satisfiability and sub-optimal MAXSAT-fractions established similarly to the enhanced sub-optimal reverse Horn fraction. The latter observation is noteworthy because the correlation between satisfiability and the optimal MAXSAT-fraction is obviously 100%.

[1]  David S. Johnson,et al.  Some Simplified NP-Complete Graph Problems , 1976, Theor. Comput. Sci..

[2]  Armando Tacchella,et al.  Theory and Applications of Satisfiability Testing: 6th International Conference, Sat 2003, Santa Margherita Ligure, Italy, May 5-8 2003: Selected Revised Papers (Lecture Notes in Computer Science, 2919) , 2004 .

[3]  Hachemi Bennaceur,et al.  Characterizing SAT Problems with the Row Convexity Property , 2002, CP.

[4]  Edward A. Hirsch,et al.  UnitWalk: A new SAT solver that uses local search guided by unit clause elimination , 2005, Annals of Mathematics and Artificial Intelligence.

[5]  Gilles Dequen,et al.  A backbone-search heuristic for efficient solving of hard 3-SAT formulae , 2001, IJCAI.

[6]  Rémi Monasson,et al.  Determining computational complexity from characteristic ‘phase transitions’ , 1999, Nature.

[7]  Hans van Maaren,et al.  Solving Satisfiability Problems Using Elliptic Approximations. A Note on Volumes and Weights , 2004, Annals of Mathematics and Artificial Intelligence.

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

[9]  Hans van Maaren,et al.  Hidden Threshold Phenomena for Fixed-Density SAT-formulae , 2003, SAT.

[10]  Simona Cocco,et al.  Phase transitions and complexity in computer science: an overview of the statistical physics approach to the random satisfiability problem , 2002 .

[11]  Robert E. Tarjan,et al.  A Linear-Time Algorithm for Testing the Truth of Certain Quantified Boolean Formulas , 1979, Inf. Process. Lett..

[12]  Endre Boros.  Maximum Renamable Horn sub-CNFs , 1999, Discret. Appl. Math..

[13]  Stephen A. Cook,et al.  The complexity of theorem-proving procedures , 1971, STOC.

[14]  Naomi Nishimura,et al.  Detecting Backdoor Sets with Respect to Horn and Binary Clauses , 2004, SAT.

[15]  Brian Borchers,et al.  A Two-Phase Exact Algorithm for MAX-SAT and Weighted MAX-SAT Problems , 1998, J. Comb. Optim..

[16]  Jean H. Gallier,et al.  Linear-Time Algorithms for Testing the Satisfiability of Propositional Horn Formulae , 1984, J. Log. Program..

[17]  Stefan Porschen,et al.  Worst Case Bounds for some NP-Complete Modified Horn-SAT Problems , 2004, SAT.

[18]  Peter L. Hammer,et al.  Variable and Term Removal From Boolean Formulae , 1997, Discret. Appl. Math..