Unambiguous DNFs and Alon-Saks-Seymour

TL;DR

This paper constructs an unambiguous k-DNF formula with optimal complexity bounds and applies this to achieve near-optimal solutions for the longstanding Alon--Saks--Seymour problem, impacting graph theory and complexity theory.

Contribution

It introduces an unambiguous k-DNF formula with tight bounds and uses it to resolve a major open problem in graph theory, providing improved complexity separations.

Findings

Unambiguous k-DNF formula requiring CNF width $ ilde{ ext{Omega}}(k^2)$

Near-optimal solution to the Alon--Saks--Seymour problem

Implications for query and communication complexity separations

Abstract

We exhibit an unambiguous k-DNF formula that requires CNF width $\tilde{\Omega}(k^2)$, which is optimal up to logarithmic factors. As a consequence, we get a near-optimal solution to the Alon--Saks--Seymour problem in graph theory (posed in 1991), which asks: How large a gap can there be between the chromatic number of a graph and its biclique partition number? Our result is also known to imply several other improved separations in query and communication complexity.