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

TL;DR

This paper demonstrates exponential proof complexity lower bounds and polynomial communication complexity for the binary-encoded clique problem in average-case dense graphs.

Contribution

It provides new exponential lower bounds for proof systems and polynomial bounds for communication complexity in the average-case setting.

Findings

Exponential lower bounds on cutting planes and resolution refutations.

Polynomial randomized communication complexity for clause falsification.

Results apply to dense graphs with binary-encoded clique formulas.

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.