Approximate Counting for ComplexWeighted Boolean Constraint Satisfaction Problems
Abstract
Constraint satisfaction problems (or CSPs) have been extensively studied in, for instance, artificial intelligence, database theory, graph theory, and statistical physics. From a practical viewpoint, it is beneficial to approximately solve those CSPs. When one tries to approximate the total number of truth assignments that satisfy all Booleanvalued constraints for (unweighted) Boolean CSPs, there is a known trichotomy theorem by which all such counting problems are neatly classified into exactly three categories under polynomialtime (randomized) approximationpreserving reductions. In contrast, we obtain a dichotomy theorem of approximate counting for complexweighted Boolean CSPs, provided that all complexvalued unary constraints are freely available to use. It is the expressive power of free unary constraints that enables us to prove such a stronger, complete classification theorem. This discovery makes a step forward in the quest for the approximationcomplexity classification of all counting CSPs. To deal with complex weights, we employ proof techniques of factorization and arity reduction along the line of solving Holant problems. Moreover, we introduce a novel notion of Tconstructibility that naturally induces approximationpreserving reducibility. Our result also gives an approximation analogue of the dichotomy theorem on the complexity of exact counting for complexweighted Boolean CSPs.
 Publication:

arXiv eprints
 Pub Date:
 July 2010
 arXiv:
 arXiv:1007.0391
 Bibcode:
 2010arXiv1007.0391Y
 Keywords:

 Computer Science  Computational Complexity;
 68Q15;
 68Q17;
 68W20;
 68W25;
 68W40
 EPrint:
 A4, 10 point, 25 pages. This version significantly improves its conference version that appeared in the Proceedings of the 8th Workshop on Approximation and Online Algorithms (WAOA 2010), Lecture Notes in Computer Science, SpringerVerlag, Vol.6534, pp.261272, 2011