On the Complexity of Nondeterministically Testable Hypergraph Parameters

The paper proves the equivalence of the notions of nondeterministic and deterministic parameter testing for uniform dense hypergraphs of arbitrary order. It generalizes the result previously known only for the case of simple graphs. By a similar method we establish also the equivalence between nondeterministic and deterministic hypergraph property testing, answering the open problem in the area. We introduce a new notion of a cut norm for hypergraphs of higher order, and employ regularity techniques combined with the ultralimit method.Comment: 33 page

Covering graphs by monochromatic trees and Helly-type results for hypergraphs

How many monochromatic paths, cycles or general trees does one need to cover all vertices of a given $r$-edge-coloured graph $G$? These problems were introduced in the 1960s and were intensively studied by various researchers over the last 50 years. In this paper, we establish a connection between this problem and the following natural Helly-type question in hypergraphs. Roughly speaking, this question asks for the maximum number of vertices needed to cover all the edges of a hypergraph $H$ if it is known that any collection of a few edges of $H$ has a small cover. We obtain quite accurate bounds for the hypergraph problem and use them to give some unexpected answers to several questions about covering graphs by monochromatic trees raised and studied by Bal and DeBiasio, Kohayakawa, Mota and Schacht, Lang and Lo, and Gir\~ao, Letzter and Sahasrabudhe.Comment: 20 pages including references plus 2 pages of an Appendi

Unsplittable coverings in the plane

A system of sets forms an {\em $m$-fold covering} of a set $X$ if every point of $X$ belongs to at least $m$ of its members. A $1$-fold covering is called a {\em covering}. The problem of splitting multiple coverings into several coverings was motivated by classical density estimates for {\em sphere packings} as well as by the {\em planar sensor cover problem}. It has been the prevailing conjecture for 35 years (settled in many special cases) that for every plane convex body $C$, there exists a constant $m=m(C)$ such that every $m$-fold covering of the plane with translates of $C$ splits into $2$ coverings. In the present paper, it is proved that this conjecture is false for the unit disk. The proof can be generalized to construct, for every $m$, an unsplittable $m$-fold covering of the plane with translates of any open convex body $C$ which has a smooth boundary with everywhere {\em positive curvature}. Somewhat surprisingly, {\em unbounded} open convex sets $C$ do not misbehave, they satisfy the conjecture: every $3$-fold covering of any region of the plane by translates of such a set $C$ splits into two coverings. To establish this result, we prove a general coloring theorem for hypergraphs of a special type: {\em shift-chains}. We also show that there is a constant $c>0$ such that, for any positive integer $m$, every $m$-fold covering of a region with unit disks splits into two coverings, provided that every point is covered by {\em at most} $c2^{m/2}$ sets

Efficient Flow-based Approximation Algorithms for Submodular Hypergraph Partitioning via a Generalized Cut-Matching Game

In the past 20 years, increasing complexity in real world data has lead to the study of higher-order data models based on partitioning hypergraphs. However, hypergraph partitioning admits multiple formulations as hyperedges can be cut in multiple ways. Building upon a class of hypergraph partitioning problems introduced by Li & Milenkovic, we study the problem of minimizing ratio-cut objectives over hypergraphs given by a new class of cut functions, monotone submodular cut functions (mscf's), which captures hypergraph expansion and conductance as special cases. We first define the ratio-cut improvement problem, a family of local relaxations of the minimum ratio-cut problem. This problem is a natural extension of the Andersen & Lang cut improvement problem to the hypergraph setting. We demonstrate the existence of efficient algorithms for approximately solving this problem. These algorithms run in almost-linear time for the case of hypergraph expansion, and when the hypergraph rank is at most $O(1)$. Next, we provide an efficient $O(\log n)$-approximation algorithm for finding the minimum ratio-cut of $G$. We generalize the cut-matching game framework of Khandekar et. al. to allow for the cut player to play unbalanced cuts, and matching player to route approximate single-commodity flows. Using this framework, we bootstrap our algorithms for the ratio-cut improvement problem to obtain approximation algorithms for minimum ratio-cut problem for all mscf's. This also yields the first almost-linear time $O(\log n)$-approximation algorithms for hypergraph expansion, and constant hypergraph rank. Finally, we extend a result of Louis & Makarychev to a broader set of objective functions by giving a polynomial time $O\big(\sqrt{\log n}\big)$-approximation algorithm for the minimum ratio-cut problem based on rounding $\ell_2^2$-metric embeddings.Comment: Comments and feedback welcom

Extremal sets with forbidden configurations and the independence ratio of geometric hypergraphs

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