Fast parallel constraint satisfaction

Abstract

A Constraint Satisfaction Problem (CSP) involves searching for an assignment of values to a given set of variables so that the values assigned satisfy a given set of constraints. The general CSP is NP-complete. To confront the intractability of the general CSP, relaxation procedures have been devised: instead of searching for a globally consistent assignment of values to the variables, try to restrict the domain of values of each variable in a way that ensures local consistency only. The relaxation procedures are efficient, but have been proved to be inherently sequential. In this paper, we define a class of CSPs for which a global solution can be found by a fast parallel algorithm. No relaxation preprocessing is needed for the parallel algorithm to work. The result is motivated from the problem of labelling a 2-D line drawing of a 3-D object by the Clowes-Huffman-Malik labelling scheme—an important application of CSP in computer vision. For such a labelling CSP, the constraint graph can be general, but the constraint relations are usually of the type we call implicational. It is shown here that a CSP with this type of constraint relations (and no restrictions on its graph) can be solved by an efficient (i.e., with polynomial time complexity) sequential algorithm. Also, it is shown that it can be solved by a fast parallel algorithm that executes in time O(log3 n) with O((m+n 3)/log n) processors on a Exlusive-Read-Exclusive-Write Parallel Random Access Machine (n is the number of variables and m is the number of constrain relations—the constraint relations may have arity more than two).KeywordsBipartite GraphParallel AlgorithmLine DrawingConstraint Satisfaction ProblemTransitive ClosureThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.