On the Tractability of Smooth Constraint Satisfaction Problems

• Published in: Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems pp. 304 - 319
• Main Author: Satish Kumar, T. K.
• Format: Book Chapter
• Language: English
• Published: Berlin, Heidelberg Springer Berlin Heidelberg
• Series: Lecture Notes in Computer Science
• Subjects: Binary Constraint; Constraint Network; Constraint Satisfaction Problem; Space Complexity; True Assignment
• DOI: 10.1007/11493853_23
• ISBN: 9783540261520; 3540261524
• ISSN: 0302-9743

We identify a property of constraints called smoothness, and present an extremely simple randomized algorithm for solving smooth constraints. The complexity of the algorithm is much less than the lower bound for establishing path-consistency, and because smoothness is shown to be identical to connected row-convexity (CRC) for the case of binary constraints, the time and space complexity of solving CRC constraints is improved. Central to our algorithm is the relationship of smooth constraints to random walks on directed graphs. We also provide simple deterministic algorithms to test for the smoothness of a given CSP under given domain orderings of the variables. Finally, we show that some other known tractable constraint languages, like the set of implicational constraints, and the set of binary integer linear constraints, are special cases of smooth constraints, and can therefore be solved much more efficiently than the traditional time and space complexities attached with them.