Path Consistency on Triangulated Constraint Graphs

Christian Bliek ILOG 1681 Route des Dolines 06560 Valbonne, France [email protected] Djamila Sam-Haroud Arti cial Intelligence Laboratory Swiss Federal Institute of Technology 1015 Lausanne, Switzerland [email protected] Abstract Among the local consistency techniques used in the resolution of constraint satisfaction problems (CSPs), path consistency (PC) has received a great deal of attention. A constraint graph G is PC if for any valuation of a pair of variables that satisfy the constraint in G between them, one can nd values for the intermediate variables on any other path in G between those variables so that all the constraints along that path are satis ed. On complete graphs, Montanari showed that PC holds if and only if each path of length two is PC. By convention, it is therefore said that a CSP is PC if the completion of its constraint graph is PC. In this paper, we show that Montanari's theorem extends to triangulated graphs. One can therefore enforce PC on sparse graphs by triangulating instead of completing them. The advantage is that with triangulation much less universal constraints need to be added. We then compare the pruning capacity of the two approaches. We show that when the constraints are convex, the pruning capacity of PC on triangulated graphs and their completion are identical on the common edges. Furthermore, our experiments show that there is little di erence for general non-convex problems.