Weighted Model Counting with Twin-Width

TL;DR

This paper introduces the concept of signed twin-width for CNF formulas and demonstrates that bounded signed twin-width enables fixed-parameter tractable algorithms for weighted model counting with bounds, supported by empirical analysis.

Contribution

It defines signed twin-width for CNF formulas and proves fixed-parameter tractability of BWMC parameterized by this measure plus the bound, extending twin-width applications.

Findings

BWMC is fixed-parameter tractable with signed twin-width and bound k.

Dropping the bound k or using standard twin-width makes the problem intractable.

Empirical evaluation shows the effectiveness of signed twin-width on various CNF classes.

Abstract

Bonnet et al. (FOCS 2020) introduced the graph invariant twin-width and showed that many NP-hard problems are tractable for graphs of bounded twin-width, generalizing similar results for other width measures, including treewidth and clique-width. In this paper, we investigate the use of twin-width for solving the propositional satisfiability problem (SAT) and propositional model counting. We particularly focus on Bounded-ones Weighted Model Counting (BWMC), which takes as input a CNF formula $F$ along with a bound $k$ and asks for the weighted sum of all models with at most $k$ positive literals. BWMC generalizes not only SAT but also (weighted) model counting. We develop the notion of "signed" twin-width of CNF formulas and establish that BWMC is fixed-parameter tractable when parameterized by the certified signed twin-width of $F$ plus $k$. We show that this result is tight: it is neither possible to drop the bound $k$ nor use the vanilla twin-width instead if one wishes to retain fixed-parameter tractability, even for the easier problem SAT. Our theoretical results are complemented with an empirical evaluation and comparison of signed twin-width on various classes of CNF formulas.