[Paper Review] On the Hardness of SAT with Community Structure

Nathan Mull, Daniel J. Fremont+1 more|arXiv (Cornell University)|Feb 27, 2016

Constraint Satisfaction and Optimization17 references4 citations

TL;DR

This paper demonstrates that community structure—commonly cited as a reason for CDCL SAT solver efficiency—does not guarantee tractability, as SAT instances with high modularity remain NP-hard and can require exponentially long resolution proofs. Using a pseudo-industrial community attachment model, the authors prove that even highly modular instances with few communities are hard for CDCL solvers, suggesting additional structural factors beyond modularity may underlie industrial solver success.

ABSTRACT

Recent attempts to explain the effectiveness of Boolean satisfiability (SAT) solvers based on conflict-driven clause learning (CDCL) on large industrial benchmarks have focused on the concept of community structure. Specifically, industrial benchmarks have been empirically found to have good community structure, and experiments seem to show a correlation between such structure and the efficiency of CDCL. However, in this paper we establish hardness results suggesting that community structure is not sufficient to explain the success of CDCL in practice. First, we formally characterize a property shared by a wide class of metrics capturing community structure, including "modularity". Next, we show that the SAT instances with good community structure according to any metric with this property are still NP-hard. Finally, we study a class of random instances generated from the "pseudo-industrial" community attachment model of Giráldez-Cru and Levy. We prove that, with high probability, instances from this model that have relatively few communities but are still highly modular require exponentially long resolution proofs and so are hard for CDCL. We also present experimental evidence that our result continues to hold for instances with many more communities. This indicates that actual industrial instances easily solved by CDCL may have some other relevant structure not captured by the community attachment model.

Motivation & Objective

• To investigate whether community structure, as measured by metrics like modularity, explains the empirical success of CDCL SAT solvers on industrial benchmarks.
• To determine whether SAT instances with good community structure are still computationally hard, despite apparent structural regularity.
• To analyze the average-case complexity of CDCL on random instances generated via the community attachment model, particularly focusing on proof length and runtime.
• To identify structural limitations of current community-based models in capturing the true reasons for CDCL's effectiveness on real-world instances.

Proposed method

• Introduces Polynomial Clique Metrics (PCMs), a broad class of graph metrics that generalize modularity and capture various community structure measures.
• Proves that SAT instances with good community structure according to any PCM remain NP-hard, showing that such structure does not imply tractability.
• Analyzes random unsatisfiable instances from the community attachment model, focusing on the number of communities and modularity levels.
• Establishes that when the number of communities is sub-polynomial (o(n^{1/10})), such instances require exponentially long resolution proofs with high probability.
• Employs probabilistic analysis and resolution proof complexity to demonstrate exponential lower bounds on proof length under specific parameter regimes.
• Conducts empirical evaluation using MiniSat on parametrized families of instances to observe runtime behavior across varying community sizes and total formula sizes.

Experimental results

Research questions

• RQ1 Can community structure, as captured by metrics like modularity, explain the efficiency of CDCL SAT solvers on industrial benchmarks?
• RQ2 Are SAT instances with high modularity still NP-hard, or does such structure make them tractable?
• RQ3 What is the average-case complexity of CDCL on random instances generated from the community attachment model?
• RQ4 How does the number of communities affect the hardness of modular SAT instances, particularly in terms of resolution proof length?
• RQ5 Are there structural features beyond modularity and community count that contribute to CDCL’s performance on real industrial instances?

Key findings

• SAT instances with good community structure according to any Polynomial Clique Metric (PCM) remain NP-hard, indicating that such structure alone does not ensure tractability.
• For random unsatisfiable instances from the community attachment model with fewer than Θ(n^{1/10}) communities but high modularity, CDCL requires exponentially long resolution proofs with high probability.
• Empirical results show that runtime grows exponentially with community size, even for α > 1/10, suggesting that exponential hardness persists beyond the theoretical bound.
• Larger community sizes (e.g., α = 0.4) lead to significantly longer runtimes than smaller ones, but the performance gap between different α values is small relative to the exponential growth in community size.
• The number of communities is a critical factor: when c = Θ(n / log n), the probability of a short refutation increases, implying potential polynomial-time solvability in that regime.
• Despite high modularity, the model fails to capture the full structural complexity of real industrial instances, indicating that additional structural features may be responsible for CDCL’s success in practice.