Tight Approximation Bounds for Vertex Cover on Dense k-Partite Hypergraphs

子模超图类中保持割的几乎紧界

Almost-Tight Bounds on Preserving Cuts in Classes of Submodular Hypergraphs

本文通过在不同超图稀疏化变体之间建立归约，并利用简单超图类模拟复杂超图类的割，给出了保持割的空间复杂度的几乎紧上界和下界。

超图的最优高效部集分解

Optimal and Efficient Partite Decompositions of Hypergraphs

本文研究d-均匀超图的完全d-部超图边划分问题，得到顶点所属部集数的更优上界，解决多个长期悬而未决的公开问题，同时给出固定密度图的高效最优二团划分构造及相关算法应用。

Towards Tight Bounds for Spectral Sparsification of Hypergraphs

Cut and spectral sparsification of graphs have numerous applications, including e.g. speeding up algorithms for cuts and Laplacian solvers. These powerful notions have recently been extended to hypergraphs, which are much richer and may offer new applications. However, the current bounds on the size of hypergraph sparsifiers are not as tight as the corresponding bounds for graphs. Our first result is a polynomial-time algorithm that, given a hypergraph on $n$ vertices with maximum hyperedge size $r$, outputs an $ε$-spectral sparsifier with $O^*(nr)$ hyperedges, where $O^*$ suppresses $(ε^{-1} …

Approximate Counting of Matchings in $(3,3)$-Hypergraphs

We design a fully polynomial time approximation scheme (FPTAS) for counting the number of matchings (packings) in arbitrary 3-uniform hypergraphs of maximum degree three, referred to as $(3,3)$-hypergraphs. It is the first polynomial time approximation scheme for that problem, which includes also, as a special case, the 3D Matching counting problem for 3-partite $(3,3)$-hypergraphs. The proof technique of this paper uses the general correlation decay technique and a new combinatorial analysis of the underlying structures of the intersection graphs. The proof method could be also of independent…

Tight Algorithms for Vertex Cover with Hard Capacities on Multigraphs and Hypergraphs

In this paper we give a f-approximation algorithm for the minimum unweighted Vertex Cover problem with Hard Capacity constraints (VCHC) on f-hypergraphs. This problem generalizes standard vertex cover for which the best known approximation ratio is also f and cannot be improved assuming the unique game conjecture. Our result is therefore essentially the best possible. This improves over the previous 2.155 (for f=2) and 2f-approximation algorithms by Cheung, Goemans and Wong (CGW). At the heart of our approach is to apply iterative rounding to the problem with ideas coming from several previous…

有向超图的近紧谱稀疏化：一种简单迭代采样算法

Nearly Tight Spectral Sparsification of Directed Hypergraphs by a Simple Iterative Sampling Algorithm

针对有向超图谱稀疏化，提出首个构造 $O^*(n^2)$ 大小 $\varepsilon$-谱稀疏化器的迭代采样算法，达到最优界，并给出非平凡下界，同时扩展至无向情形。