Clique problem, cutting plane proofs and communication complexity

TL;DR

This paper explores the connection between the clique problem, cutting plane proofs, and communication complexity, demonstrating that a logarithmic amount of communication suffices for certain graph-based tasks.

Contribution

It establishes a bound on communication complexity for a graph problem related to the clique problem and cutting plane proofs, advancing understanding of these computational aspects.

Findings

O(log n) bits of communication suffice for the problem on n-vertex graphs

The result links communication complexity to the length of cutting plane proofs

Provides insights into the structure of graphs related to the clique problem

Abstract

Motivated by its relation to the length of cutting plane proofs for the Maximum Biclique problem, we consider the following communication game on a given graph G, known to both players. Let K be the maximal number of vertices in a complete bipartite subgraph of G, which is not necessarily an induced subgraph if G is not bipartite. Alice gets a set A of vertices, and Bob gets a disjoint set B of vertices such that |A|+|B|>K. The goal is to find a nonedge of G between A and B. We show that O(\log n) bits of communication are enough for every n-vertex graph.