Treewidth-Aware Quantifier Elimination and Expansion for QCSP

Abstract

Research on the constraint satisfaction problem (CSP) is an active field both in theory and practice. An extension of CSP even considers universal and existential quantification of domain variables, known as QCSP, for which also solvers exist. The number of alternations between existential and universal quantifications is usually called quantifier rank. While QCSP is PSPACE-complete for bounded domain size, one can consider additional structural restrictions that make the problem tractable, for example, bounded treewidth. Chen  [14] showed that one can solve QCSP instances of size n, quantifier rank $\ell$, bounded treewidth k, and bounded domain size d in $\text {poly}(n)\cdot \text {tower}(\ell -1,d^k)$, where tower are exponentials of height $\ell -1$ with $d^k$ on top. We follow up on Chen’s result and develop a treewidth-aware quantifier elimination technique that reduces the rank without an exponential blow-up in the instance size at the cost of exponentially increasing the treewidth. With this reduction at hand we show that one cannot significantly improve the blow-up of the treewidth assuming the exponential time hypothesis (ETH). Further, we lift a recently introduced technique for treewidth-decreasing quantifier expansion from the Boolean domain to QCSP.

The work has been supported by the Austrian Science Fund (FWF), Grants Y698 and P32830, and the Vienna Science and Technology Fund, Grant WWTF ICT19-065. We would like to thank the anonymous reviewers for very detailed feedback and their suggestions. Special appreciation goes to Andreas Pfandler for early discussions.