Threshold Treewidth and Hypertree Width

TL;DR

This paper introduces threshold-based enhancements to treewidth and hypertree width, enabling fixed-parameter algorithms for CSP and related problems, supported by theoretical analysis and empirical evaluations.

Contribution

It proposes a novel threshold-based approach to treewidth and hypertree width, improving fixed-parameter tractability for CSP and providing algorithms and heuristics for their computation.

Findings

Threshold measures improve fixed-parameter algorithms for CSP.

Efficient algorithms for computing threshold treewidth and hypertree width.

Empirical results validate the effectiveness of the proposed methods.

Abstract

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.