Too much information: Why CDCL solvers need to forget learned clauses

Krüger, Tom; Lorenz, Jan-Hendrik; Wörz, Florian

doi:10.1371/journal.pone.0272967

Cited by 3 publications

(2 citation statements)

References 45 publications

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“…Furthermore, it could be a worthwhile pursuit to theoretically investigate the distribution of other SLS solving paradigms, such as configuration checking solvers, e. g., CCAnr [CLS15], or WalkSAT [SKC94]. For the case of CDCL solvers we have reported results in [KLW22].…”

Section: Discussionmentioning

confidence: 99%

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Toward an Understanding of Long-tailed Runtimes of SLS Algorithms

Lorenz

Wörz

2022

ACM J. Exp. Algorithmics

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The satisfiability problem (SAT) is one of the most famous problems in computer science. Traditionally, its NP-completeness has been used to argue that SAT is intractable. However, there have been tremendous practical advances in recent years that allow modern SAT solvers to solve instances with millions of variables and clauses. A particularly successful paradigm in this context is stochastic local search (SLS). In most cases, there are different ways of formulating the underlying SAT problem. While it is known that the precise formulation of the problem has a significant impact on the runtime of solvers, finding a helpful formulation is generally non-trivial. The recently introduced GapSAT solver [Lorenz and Wörz 2020] demonstrated a successful way to improve the performance of an SLS solver on average by learning additional information which logically entails from the original problem. Still, there were also cases in which the performance slightly deteriorated. This justifies in-depth investigations into how learning logical implications affects runtimes for SLS algorithms. In this work, we propose a method for generating logically equivalent problem formulations, generalizing the ideas of GapSAT . This method allows a rigorous mathematical study of the effect on the runtime of SLS SAT solvers. Initially, we conduct empirical investigations. If the modification process is treated as random, Johnson SB distributions provide a perfect characterization of the hardness. Since the observed Johnson SB distributions approach lognormal distributions, our analysis also suggests that the hardness is long-tailed. As a second contribution, we theoretically prove that restarts are useful for long-tailed distributions. This implies that incorporating additional restarts can further refine all algorithms employing above mentioned modification technique. Since the empirical studies compellingly suggest that the runtime distributions follow Johnson SB distributions, we also investigate this property on a theoretical basis. We succeed in proving that the runtimes for the special case of Schöning’s random walk algorithm [Schöning 2002] are approximately Johnson SB distributed.

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Section: Discussionmentioning

confidence: 99%

“…The runtime distributions of CDCL solvers was studies in [KLW22]. The authors empirically demonstrated that Weibull mixture distributions can accurately describe the multimodal distributions found.…”

Section: Previous Work On Runtime Distributionsmentioning

confidence: 99%

Toward an Understanding of Long-tailed Runtimes of SLS Algorithms

Lorenz

Wörz

2022

ACM J. Exp. Algorithmics

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Limiting the memory consumption of caching for detecting subproblem dominance in constraint problems

Medema,

Breeman,

Lazovik

2024

Constraints

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Solving constraint satisfaction problems often involves a large amount of redundant exploration stemming from the existence of subproblems whose information can be reused for other subproblems. Subproblem dominance is a general notion of reusability that arises when one subproblem imposes more constraints on the remaining part of the search than another subproblem and allows the search to reuse the solutions of the dominating subproblem for the dominated subproblem. The search can exploit subproblem dominance by storing the subproblems that have been explored in a cache and abandoning the current subproblem whenever the cache contains a subproblem that dominates it. While using caching makes it possible to solve problems where subproblem dominance arises orders of magnitude faster, storing all of these subproblems can require a substantial amount of memory, making it impractical in many cases. This paper analyses the dominance between different subproblems for various constraint problems, revealing that only a relatively small number of subproblems dominate other subproblems. Based on these findings, two types of strategies are proposed for reducing the number of subproblems stored in the cache: limiting the number of subproblems that can be stored in the cache and periodically cleaning up the cache. An experimental evaluation demonstrates that these strategies provide an effective instrument for reducing the memory consumption of caching, allowing it to be used on a larger scale. However, there is a trade-off between saving memory and reducing redundant exploration, as removing subproblems from the cache may prevent dominance from being detected for certain subproblems.

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Too much information: Why CDCL solvers need to forget learned clauses

2022

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Conflict-driven clause learning (CDCL) is a remarkably successful paradigm for solving the satisfiability problem of propositional logic. Instead of a simple depth-first backtracking approach, this kind of solver learns the reason behind occurring conflicts in the form of additional clauses. However, despite the enormous success of CDCL solvers, there is still only a limited understanding of what influences the performance of these solvers in what way. Considering different measures, this paper demonstrates, quite surprisingly, that clause learning (without being able to get rid of some clauses) can not only help the solver but can oftentimes deteriorate the solution process dramatically. By conducting extensive empirical analysis, we furthermore find that the runtime distributions of CDCL solvers are multimodal. This multimodality can be seen as a reason for the deterioration phenomenon described above. Simultaneously, it also gives an indication of why clause learning in combination with clause deletion is virtually the de facto standard of SAT solving, in spite of this phenomenon. As a final contribution, we show that Weibull mixture distributions can accurately describe the multimodal distributions. Thus, adding new clauses to a base instance has an inherent effect of making runtimes long-tailed. This insight provides an explanation as to why the technique of forgetting clauses is useful in CDCL solvers apart from the optimization of unit propagation speed.

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Too much information: Why CDCL solvers need to forget learned clauses

Cited by 3 publications

References 45 publications

Toward an Understanding of Long-tailed Runtimes of SLS Algorithms

Toward an Understanding of Long-tailed Runtimes of SLS Algorithms

Limiting the memory consumption of caching for detecting subproblem dominance in constraint problems

Too much information: Why CDCL solvers need to forget learned clauses

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