最优无歧义DNF与Alon-Saks-Seymour
Optimal Unambiguous DNFs and Alon-Saks-Seymour
- Stanford University（斯坦福大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文构造了宽度$O(n)$、0-证书复杂度$\u03a9(n^2)$的无歧义DNF，利用其结构证明提升定理，反驳Alon-Saks-Seymour猜想、得到团与独立集问题最优通信下界，还应用于查询与学习理论，获证书复杂度与近似度的最优四次方分离及多类概念类样本压缩下界。
AI中文摘要：
我们构造了宽度为$O(n)$但0-证书复杂度为$\u03a9(n^2)$的无歧义DNF。通过利用这些DNF的特殊结构，我们证明了一个带有常数大小构件的提升定理，该定理将DNF提升为通信问题，同时无损地将证书复杂度中的分离转化为通信复杂度中的分离。这一结果实现了Alon-Saks-Seymour猜想的最优反驳，同时为团与独立集问题（Clique versus Independent Set）带来了最优通信下界，相比Balodis、Ben-David、Göös、Jain和Kothari（FOCS 2021，SICOMP 2023）的先前结果提升了若干双对数因子。将我们的构造进一步应用于查询复杂度和学习理论，我们展示了：(a)一类布尔函数，其证书复杂度与近似度之间存在最优四次方分离；(b)针对含c个标签的多类概念类，样本压缩下界为$\u03a9(\u221a{\log c})$。
英文摘要：
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.
