[Paper Review] Phase Selection Heuristics for Satisfiability Solvers

Jingchao Chen|arXiv (Cornell University)|Jun 7, 2011

Constraint Satisfaction and Optimization6 references4 citations

TL;DR

This paper proposes integrating the computationally expensive ACE (Approximation of Combined Lookahead Evaluation) phase selection heuristic into conflict-driven SAT solvers via a selective, portfolio-style approach. By applying ACE only when beneficial and switching to lighter heuristics like Jeroslow-Wang otherwise, the new solver MPhaseSAT significantly improves performance on application instances—solving previously unsolvable problems and outperforming top solvers like PrecoSAT and CryptoMiniSat in that category.

ABSTRACT

In general, a SAT Solver based on conflict-driven DPLL consists of variable selection, phase selection, Boolean Constraint Propagation, conflict analysis, clause learning and its database maintenance. Optimizing any part of these components can enhance the performance of a solver. This paper focuses on optimizing phase selection. Although the ACE (Approximation of the Combined lookahead Evaluation) weight is applied to a lookahead SAT solver such as March, so far, no conflict-driven SAT solver applies successfully the ACE weight, since computing the ACE weight is time-consuming. Here we apply the ACE weight to partial phase selection of conflict-driven SAT solvers. This can be seen as an improvement of the heuristic proposed by Jeroslow-Wang (1990). We incorporate the ACE heuristic and the existing phase selection heuristics in the new solver MPhaseSAT, and select a phase heuristic in a way similar to portfolio methods. Experimental results show that adding the ACE heuristic can improve the conflict-driven solvers. Particularly on application instances, MPhaseSAT with the ACE heuristic is significantly better than MPhaseSAT without the ACE heuristic, and even can solve a few SAT instances that remain unsolvable so far.

Motivation & Objective

• To improve phase selection in conflict-driven SAT solvers, which remains underexplored despite its critical role.
• To address the challenge of applying the ACE heuristic—previously too costly for conflict-driven solvers—by using it selectively.
• To develop a hybrid phase selection strategy that combines ACE with existing heuristics (e.g., Jeroslow-Wang, RSAT) using instance classification.
• To evaluate whether selective use of ACE can enhance solver performance without degrading it on unsuitable instances.
• To demonstrate that phase selection heuristics can be significantly improved through adaptive, context-aware selection of heuristics.

Proposed method

• Introduce a new SAT solver, MPhaseSAT, which integrates multiple phase selection heuristics, including ACE and Jeroslow-Wang.
• Apply the ACE heuristic only under specific conditions determined by a lightweight instance classifier, avoiding its high computational cost on unsuitable instances.
• Use a portfolio-like method to dynamically select the best phase heuristic per instance based on a small set of features.
• Classify instances based on empirical behavior on application benchmarks, enabling selective ACE usage without full lookahead overhead.
• Maintain backward compatibility by falling back to standard heuristics (e.g., Jeroslow-Wang, RSAT) when ACE is not beneficial.
• Leverage the Glucose-style clause database reduction strategy for efficient clause learning and maintenance.

Experimental results

Research questions

• RQ1 Can the ACE heuristic, previously used only in lookahead solvers, be effectively adapted for conflict-driven SAT solvers despite its high computational cost?
• RQ2 Does selectively applying ACE based on instance characteristics improve overall solver performance compared to using only standard heuristics?
• RQ3 To what extent does the hybrid phase selection strategy enhance solving capability on real-world (application) SAT instances?
• RQ4 Can the integration of ACE reduce the number of decisions required to solve hard SAT instances, particularly those previously unsolved?
• RQ5 How does the performance of the new solver MPhaseSAT compare to top solvers like PrecoSAT and CryptoMiniSat across different SAT instance categories?

Key findings

• MPhaseSAT with the ACE heuristic solved a set of previously unsolvable SAT instances, particularly in the application category.
• On the application category of SAT 2009, MPhaseSAT II (with ACE) solved 178 out of 281 instances, significantly outperforming MPhaseSAT I (176), PrecoSAT (149), and CryptoMiniSat (143).
• The average runtime per solved instance for MPhaseSAT II was only slightly higher than MPhaseSAT I, indicating that ACE does not degrade performance on unsuitable instances.
• MPhaseSAT II approached the performance of the virtual best solver on the application category, primarily due to training on application instances and effective heuristic selection.
• On the crafted category, the improvement from ACE was modest but consistent, with MPhaseSAT II solving 166 out of 187 instances compared to 166 solved by the virtual best solver.
• The scatter plots of runtimes (Figures 1 and 2) show that MPhaseSAT II’s performance is largely consistent with MPhaseSAT I and PrecoSAT, with most points near the diagonal, confirming no negative impact from ACE on non-ideal cases.