There is a large and important collection of Ramsey-type combinatorial problems, closely related to central problems in complexity theory, that can be formulated in terms of the asymptotic growth of the size of the maximum independent sets in powers of a fixed small (directed or undirected) hypergraph, also called the Shannon capacity. An important instance of this is the corner problem studied in the context of multiparty communication complexity in the Number On the Forehead (NOF) model. Versions of this problem and the NOF connection have seen much interest (and progress) in recent works of Linial, Pitassi and Shraibman (ITCS 2019) and Linial and Shraibman (CCC 2021). We introduce and study a general algebraic method for lower bounding the Shannon capacity of directed hypergraphs via combinatorial degenerations, a combinatorial kind of "approximation" of subgraphs that originates from the study of matrix multiplication in algebraic complexity theory (and which play an important role there) but which we use in a novel way. Using the combinatorial degeneration method, we make progress on the corner problem by explicitly constructing a corner-free subset in $F_2^n \times F_2^n$ of size $\Omega(3.39^n/poly(n))$, which improves the previous lower bound $\Omega(2.82^n)$ of Linial, Pitassi and Shraibman (ITCS 2019) and which gets us closer to the best upper bound $4^{n - o(n)}$. Our new construction of corner-free sets implies an improved NOF protocol for the Eval problem. In the Eval problem over a group $G$, three players need to determine whether their inputs $x_1, x_2, x_3 \in G$ sum to zero. We find that the NOF communication complexity of the Eval problem over $F_2^n$ is at most $0.24n + O(\log n)$, which improves the previous upper bound $0.5n + O(\log n)$.
翻译:大量且重要的 Ramsey 类型 $ 的组合问题, 与复杂理论中的核心问题密切相关, 可以用固定的小( 方向或非方向) 高压压压力中最大独立机体大小的无效果增长来制定, 也称为 香农能力。 其中一个重要的例子就是在数字前头( NOF) 模型中多党沟通复杂性的背景下研究的角落问题。 这个问题的版本和 NOF 连接在Linial、 Pitassi 和 Shraibman ( ITS 2019) 以及 Linal 和 Shraibbman ( CCC 2021) 的近期作品中看到了很多兴趣( 和进展) 。 我们引入并研究一个通用的平面 O& 平面 平面 Oral_ 平面法, 我们从前的平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面, 平面平面平面平面平面平面平面平面平面平面平面平面, 平面平面平平面平平平平平平平平平平平面, 平面平面平面平平平, 平面平面平面平面平面平面平面平面,平面平面平面平面平面平面,平面平面平面平面平面平面平面平面,平面,平面平面,平面平面平面平面平面平面平面平面平面,平面平面平面,平面平面平面,平面平面平面平面平面平面平面平面平,平,平面平面平,平面平面平,平,平面平面平面平,平面平面平,平面平面平面平,平面平平面平,平,平面平面平面平面平面平面平面平面平面平面平,平,平面平面平平平面平,平面平面平