Optimal Unambiguous DNFs and Alon-Saks-Seymour

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, 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 $\Omega(\sqrt{\log c})$ for multiclass concept classes over $c$ labels.

Computational Complexity (cs.CC); Discrete Mathematics (cs.DM); Machine Learning (cs.LG)

arXiv:2608.02533 [cs.CC]

(or arXiv:2608.02533v1 [cs.CC] for this version)

https://doi.org/10.48550/arXiv.2608.02533 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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