超图中阿贝尔覆盖的范围
The Reach of Abelian Covers in Hypergraphs
浏览论文内容
中文总结 AI 辅助
本文引入并研究超图中比偶覆盖更严格的阿贝尔覆盖和加泰罗尼亚覆盖，利用格论、代数拓扑和幂零群理论刻画其出现条件，并证明元数3的CSP具有线性非冗余性，从而获得近最优算法。
中文摘要 AI 辅助
超图中的覆盖常被研究以捕捉超边之间的各种依赖形式。例如，偶覆盖——检查每个顶点是否出现在偶数条超边中——近年来在局部可解码码的研究中取得了很大成功。受近期关于约束满足问题（CSPs）非冗余性研究工作的启发，我们引入并研究了两个比偶覆盖更严格的超图覆盖新族：阿贝尔覆盖和加泰罗尼亚覆盖。阿贝尔覆盖与偶覆盖类似，区别在于算术现在在整数上而非模2上进行，从而允许我们捕捉任意阿贝尔群上的依赖。加泰罗尼亚覆盖通过仅允许超边序列中的局部抵消来捕捉非阿贝尔群的行为。我们证明了关于阿贝尔覆盖和加泰罗尼亚覆盖的三个主要结果。首先，利用格论工具，我们证明任何具有n个顶点和n log(r)条超边的r-均匀超图都有一个阿贝尔覆盖。其次，利用代数拓扑工具，我们证明在任何3-均匀超图中，阿贝尔覆盖和加泰罗尼亚覆盖是等价的；从而表明在3-均匀超图中，加泰罗尼亚覆盖在O(n)条超边后出现。最后，利用幂零群理论，我们证明存在一个4-均匀超图，它有阿贝尔覆盖但没有加泰罗尼亚覆盖。这些结果共同精确刻画了阿贝尔覆盖在推导超图依赖方面的范围。作为我们的主要应用，我们证明任何具有无限域Mal'tsev扩展的元数为3的CSP具有线性非冗余性。这意味着对于这类CSP族，存在近最优的流式、稀疏化和核化算法。此前，这样的结果仅对更简单的元数为2的CSP已知。
英文摘要
Covers in hypergraphs are frequently studied to capture various forms of dependence between hyperedges. For example, even covers--which check if each vertex appears in an even number of hyperedges--have found much success recently in the study of locally decodable codes. Inspired by a recently-emerging line of work on the non-redundancy of constraint satisfaction problems (CSPs), we introduce and study two novel families of covers of hypergraphs which are stricter than even covers: \emph{Abelian} covers and Catalan covers. Abelian covers are similar to even covers, except that arithmetic is now done over the integers rather than modulo 2, allowing us to capture dependences over arbitrary Abelian groups. Catalan covers capture the behavior of non-Abelian groups by only allowing local cancellations in a sequence of hyperedges. We prove three main results about Abelian and Catalan covers. First, using tools from lattice theory, we show that any $r$-uniform hypergraph with $n$ vertices and $n \log(r)$ hyperedges has an Abelian cover. Second, using tools from algebraic topology, we show that in any $3$-uniform hypergraph, Abelian covers and Catalan covers are equivalent; thereby showing that Catalan covers emerge after $O(n)$ hyperedges in $3$-uniform hypergraphs. Finally, using the theory of nilpotent groups, we show that there exists a $4$-uniform hypergraph which has an Abelian cover but not a Catalan cover. Collectively, these results exactly characterize the reach that Abelian covers have in deducing dependences in hypergraphs. As our primary application, we show that any arity-$3$ CSP with an infinite-domain Mal'tsev extension has linear non-redundancy. This implies near optimal streaming, sparsification, and kernelization algorithms for this family of CSPs. Previously, such a result was only known for the much simpler case of arity-$2$ CSPs.
发表机构
- University of California, Berkeley（加州大学伯克利分校）
- Simons Institute for the Theory of Computing（西蒙斯计算理论研究所）
- Harvard University（哈佛大学）
机构由 AI 辅助整理，请以论文原文为准。
