Optimal Unambiguous DNFs and Alon-Saks-Seymour

Chirag Pabbaraju

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arXiv · 2608.02533

Optimal Unambiguous DNFs and Alon-Saks-Seymour

Chirag Pabbaraju

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Abstract

We construct unambiguous DNFs having width $O(n)$ but $0$-certificate complexity $\Omega(n^2)$. 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, G\"{o}\"{o}s, 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 $\Omega(\sqrt{\log c})$ for multiclass concept classes over $c$ labels.

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Chirag Pabbaraju. 2026-08-03. Optimal Unambiguous DNFs and Alon-Saks-Seymour. https://arxiv.org/abs/2608.02533

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