Optimal Unambiguous DNFs and Alon-Saks-Seymour

2026-08-03

AI Summary

This work resolves the problem of constructing an optimal counterexample to the Alon–Saks–Seymour conjecture, thereby establishing optimal communication complexity lower bounds for the Clique versus Independent Set problem. By designing an unambiguous DNF of width $O(n)$ yet 0-certificate complexity $\Omega(n^2)$, and leveraging a lifting theorem with constant-size gadgets, the authors achieve a lossless transformation from certificate complexity separations to communication complexity separations. Key contributions include the first optimal refutation of the conjecture, a proof of an optimal quartic separation between certificate complexity and approximate degree, improvements over prior results by multiple double-logarithmic factors, and the derivation of sample compression lower bounds of $\Omega(\sqrt{\log c})$ for several concept classes, advancing both query complexity and computational learning theory.

Abstract

We construct unambiguous DNFs having width $O(n)$ but $0$-certificate complexity $Ω(n^2)$. By utilizing the special structure of these DNFs, we prove a lifting theorem with a constant-sized gadget that lifts the DNF to a communication problem, while losslessly translating the separation in certificate complexity to a separation in communication complexity. This leads to an optimal refutation of the Alon-Saks-Seymour conjecture, as well as an optimal communication lower bound for the Clique versus Independent Set problem, improving the previous results of Balodis, Ben-David, Göös, Jain and Kothari (FOCS 2021, SICOMP 2023) by several doubly logarithmic factors. As further applications of our construction to query complexity and learning theory, we exhibit: (a) a family of Boolean functions that has an optimal quartic separation between certificate complexity and approximate degree, and (b) a sample compression lower bound of $Ω(\sqrt{\log c})$ for multiclass concept classes over $c$ labels.

Problem

• Alon-Saks-Seymour conjecture
• communication complexity
• certificate complexity
• DNF
• Clique versus Independent Set

Innovation

• unambiguous DNF
• certificate complexity
• lifting theorem
• Alon-Saks-Seymour conjecture
• communication complexity