Threshold Treewidth and Hypertree Width
This work addresses the computational inefficiency in CSP solving for researchers and practitioners by providing a novel parameterization that enables fixed-parameter tractability, though it is incremental as it builds on existing width measures.
The authors tackled the problem of high polynomial-time complexity in solving Constraint Satisfaction Problems (CSP) when treewidth or hypertree width is bounded, by introducing threshold treewidth and hypertree width, which incorporate computational cost information into decompositions, resulting in fixed-parameter algorithms for CSP and other problems, with experimental evaluations showing efficiency.
Treewidth and hypertree width have proven to be highly successful structural parameters in the context of the Constraint Satisfaction Problem (CSP). When either of these parameters is bounded by a constant, then CSP becomes solvable in polynomial time. However, here the order of the polynomial in the running time depends on the width, and this is known to be unavoidable; therefore, the problem is not fixed-parameter tractable parameterized by either of these width measures. Here we introduce an enhancement of tree and hypertree width through a novel notion of thresholds, allowing the associated decompositions to take into account information about the computational costs associated with solving the given CSP instance. Aside from introducing these notions, we obtain efficient theoretical as well as empirical algorithms for computing threshold treewidth and hypertree width and show that these parameters give rise to fixed-parameter algorithms for CSP as well as other, more general problems. We complement our theoretical results with experimental evaluations in terms of heuristics as well as exact methods based on SAT/SMT encodings.