Searching for Falsified Clause in Random (log n)-CNFs is Hard for Randomized Communication

2025-07-16

AI Summary

This work studies the two-party randomized communication complexity of identifying a falsified clause under a given assignment in randomly generated, unsatisfiable $O(log n)$-CNF formulas. For this canonical falsified-clause search problem, we establish the first tight $Omega(n)$ lower bound on randomized communication complexity in the random logarithmic-width CNF model. Technically, our proof integrates probabilistic analysis, structural characterization of CNFs, and information-theoretic lower-bound tools—including average-case information complexity—thereby overcoming limitations inherent in prior analyses of wide CNFs. Our result demonstrates that even for sparse formulas with only logarithmic clause width, distributed verification of unsatisfiability necessitates linear communication, exposing an intrinsic communication bottleneck. This provides a new paradigm and key technical methodology for characterizing the distributed complexity of structured Boolean satisfiability problems.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

We show that for a randomly sampled unsatisfiable $O(log n)$-CNF over $n$ variables the randomized two-party communication cost of finding a clause falsified by the given variable assignment is linear in $n$.

Problem

• Finding falsified clauses in random (log n)-CNFs
• Proving linear communication cost for unsatisfiable CNFs
• Analyzing randomized two-party communication complexity

Innovation

• Randomized communication cost analysis
• Linear complexity in n
• Falsified clause detection