[Paper Review] Better algorithms for satisfiability problems for formulas of bounded rank-width

Robert Ganian, Petr Hliněný+1 more|arXiv (Cornell University)|Jun 29, 2010

Complexity and Algorithms in Graphs18 references4 citations

TL;DR

This paper presents parameterized algorithms for #SAT and Max-SAT that run in single-exponential time with respect to the rank-width of the formula's incidence graph. By leveraging rank-width decompositions and dynamic programming over subspaces in GF(2)^t, the approach achieves exponential runtime improvements over prior clique-width-based methods, especially for formulas where rank-width is significantly smaller than clique-width.

ABSTRACT

We provide a parameterized polynomial algorithm for the propositional model counting problem #SAT, the runtime of which is single-exponential in the rank-width of a formula. Previously, analogous algorithms have been known -- e.g.~[Fischer, Makowsky, and Ravve] -- with a single-exponential dependency on the clique-width of a formula. Our algorithm thus presents an exponential runtime improvement (since clique-width reaches up to exponentially higher values than rank-width), and can be of practical interest for small values of rank-width. We also provide an algorithm for the MAX-SAT problem along the same lines.

Motivation & Objective

• To develop efficient parameterized algorithms for propositional model counting (#SAT) and Max-SAT on formulas with bounded rank-width.
• To overcome the limitations of clique-width-based algorithms, which suffer from exponential approximation errors and lack efficient computation of the width parameter.
• To exploit the structural advantages of rank-width, which is bounded if and only if clique-width is bounded, but can be exponentially smaller in value.
• To provide algorithms with runtime linear in formula size and single-exponential in rank-width, avoiding large hidden constants.
• To demonstrate that recent algebraic techniques for bounded rank-width graphs can be effectively extended to SAT-related problems.

Proposed method

• The algorithm uses a rank-decomposition of the formula's incidence graph, with rank-width t, and constructs a dynamic programming table over subspaces of GF(2)^t.
• Each node in the decomposition maintains a table storing the minimum defect (unsatisfied clauses) for partial assignments with specific subspaces of positive and negative variable contributions.
• The algorithm combines tables from child nodes using composition operators derived from the labeling parse trees, enforcing consistency via linear transformations and duality constraints.
• For #SAT, the algorithm computes the number of satisfying assignments by summing over all valid configurations in the root table with empty constraint sets.
• For Max-SAT, the algorithm finds the minimum defect (number of unsatisfied clauses) by minimizing over all possible subspace configurations at the root.
• The runtime is bounded by O(t³ · 2^{3t(t+1)/2} · |φ|), with t being the rank-width and |φ| the size of the formula.

Experimental results

Research questions

• RQ1 Can #SAT and Max-SAT be solved in single-exponential time with respect to rank-width, offering a practical improvement over clique-width-based approaches?
• RQ2 How does the runtime of rank-width-based algorithms compare to that of clique-width-based algorithms in the worst case, especially when rank-width is exponentially smaller than clique-width?
• RQ3 Is it possible to design dynamic programming algorithms for SAT problems using the algebraic structure of rank-width decompositions and linear algebra over GF(2)?
• RQ4 Can the computation of rank-width be efficiently integrated into SAT solving pipelines, given that it admits FPT algorithms for exact computation?
• RQ5 To what extent do techniques from parameterized graph algorithms on bounded rank-width graphs generalize to SAT problems?

Key findings

• The proposed algorithms for #SAT and Max-SAT run in time O(t³ · 2^{3t(t+1)/2} · |φ|), which is single-exponential in the rank-width t and linear in the formula size |φ|.
• The algorithm achieves exponential runtime improvement over previous clique-width-based methods, since rank-width can be exponentially smaller than clique-width for the same class of formulas.
• Unlike clique-width, rank-width can be computed exactly in FPT time, avoiding the exponential approximation errors inherent in clique-width approximation algorithms.
• The method successfully extends algebraic techniques from bounded rank-width graph algorithms to SAT problems, demonstrating a smooth translation of structural insights.
• The approach is practically viable for formulas with small rank-width, as it avoids large hidden constants and leverages efficient rank-decomposition computation.
• The results show that rank-width is a superior parameter than clique-width for SAT problems, both in theoretical runtime and practical feasibility.