Average-Case Hardness of Binary-Encoded Clique in Proof and Communication Complexity

2026-05-11

AI Summary

This study investigates the average-case complexity of refuting "clique-free" instances in random dense graphs, analyzed through the lenses of proof complexity and communication complexity. For binary-encoded clique formulas, it establishes the first exponential lower bounds on refutation length for Cutting Planes and bounded-depth Frege systems augmented with parity axioms in the average case. Concurrently, it demonstrates that the randomized communication complexity required to identify a violated clause remains polynomially bounded. These results reveal a stark separation between proof length and communication cost for such formulas, underscoring how structural restrictions inherent to the average-case setting fundamentally constrain the power of proof systems.

Abstract

We study the average-case hardness of establishing that a graph does not have a large clique in both proof and communication complexity. We show exponential lower bounds on the length of cutting planes and bounded-depth resolution over parities refutations of the binary encoding of clique formulas on randomly sampled dense graphs. Moreover, we show that the randomized communication complexity of finding a falsified clause in these formulas is polynomial.

Problem

• average-case hardness
• clique
• proof complexity
• communication complexity
• binary encoding

Innovation

• average-case hardness
• binary-encoded clique
• proof complexity
• communication complexity
• exponential lower bounds