Clique versus Independent Set

TL;DR

This paper investigates the Clique versus Independent Set problem in communication complexity, exploring the existence of polynomial CS-separators for various graph classes, and establishing their equivalence to significant conjectures and problems in graph theory and CSPs.

Contribution

It proves the almost sure existence of polynomial CS-separators for random graphs, extends results to H-free and path-free graphs, and links the problem to the Alon-Saks-Seymour Conjecture and CSP solution covering.

Findings

Polynomial CS-separator exists almost surely for random graphs.

Polynomial CS-separators are found for H-free graphs with H as a split graph.

The existence of polynomial CS-separators is equivalent to the polynomial Alon-Saks-Seymour Conjecture.

Abstract

Yannakakis' Clique versus Independent Set problem (CL-IS) in communication complexity asks for the minimum number of cuts separating cliques from stable sets in a graph, called CS-separator. Yannakakis provides a quasi-polynomial CS-separator, i.e. of size $O(n^{\log n})$, and addresses the problem of finding a polynomial CS-separator. This question is still open even for perfect graphs. We show that a polynomial CS-separator almost surely exists for random graphs. Besides, if H is a split graph (i.e. has a vertex-partition into a clique and a stable set) then there exists a constant $c_H$ for which we find a $O(n^{c_H})$ CS-separator on the class of H-free graphs. This generalizes a result of Yannakakis on comparability graphs. We also provide a $O(n^{c_k})$ CS-separator on the class of graphs without induced path of length k and its complement. Observe that on one side, $c_H$ is of order $O(|H| \log |H|)$ resulting from Vapnik-Chervonenkis dimension, and on the other side, $c_k$ is exponential. One of the main reason why Yannakakis' CL-IS problem is fascinating is that it admits equivalent formulations. Our main result in this respect is to show that a polynomial CS-separator is equivalent to the polynomial Alon-Saks-Seymour Conjecture, asserting that if a graph has an edge-partition into k complete bipartite graphs, then its chromatic number is polynomially bounded in terms of k. We also show that the classical approach to the stubborn problem (arising in CSP) which consists in covering the set of all solutions by $O(n^{\log n})$ instances of 2-SAT is again equivalent to the existence of a polynomial CS-separator.