Witnessing Solution Counting in Tree-Structured Methods for CSPs

Counting the exact number of solutions of a Constraint Satisfaction Problem (CSP) is an important but difficult task. To overcome this difficulty, the techniques proposed in the literature organize the search process along a tree decomposition of the CSP, where all the extensions of a given partial solution over different branches of the tree are first independently counted in each branch before their numbers can be multiplied. We observe that this count is zero when any of the branches has no solution. We propose witnessbased search, which first ensures the existence of a solution (i.e., witness) in each branch before starting the counting. We empirically establish the benefits of our technique in the context of the BTD and AND/OR search graphs.