An official website of the United States government
Abstract Theregularity methodwas pioneered by Szemerédi for graphs and is an important tool in extremal combinatorics. Over the last two decades, several extensions to hypergraphs were developed which were based on seemingly different notions ofquasirandomhypergraphs. We consider the regularity lemmata for three‐uniform hypergraphs of Frankl and Rödl and of Gowers, and present a new proof that the concepts behind these approaches are equivalent.
more »
« less
An algorithmic hypergraph regularity lemma
Abstract Szemerédi 's Regularity Lemma is a powerful tool in graph theory. It asserts that all large graphs admit bounded partitions of their edge sets, most classes of which consist of uniformly distributed edges. The original proof of this result was nonconstructive, and a constructive proof was later given by Alon, Duke, Lefmann, Rödl, and Yuster. Szemerédi's Regularity Lemma was extended to hypergraphs by various authors. Frankl and Rödl gave one such extension in the case of 3‐uniform hypergraphs, which was later extended tok‐uniform hypergraphs by Rödl and Skokan. W.T. Gowers gave another such extension, using a different concept of regularity than that of Frankl, Rödl, and Skokan. Here, we give a constructive proof of a regularity lemma for hypergraphs.
more »
« less
- Award ID(s):
- 1700280
- PAR ID:
- 10047914
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Random Structures & Algorithms
- Volume:
- 52
- Issue:
- 2
- ISSN:
- 1042-9832
- Page Range / eLocation ID:
- p. 301-353
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Abstract A classical result of Erdős and, independently, of Bondy and Simonovits [3] says that the maximum number of edges in ann-vertex graph not containingC2k, the cycle of length 2k, isO(n1+1/k). Simonovits established a corresponding supersaturation result forC2k’s, showing that there exist positive constantsC,cdepending only onksuch that everyn-vertex graphGwithe(G)⩾Cn1+1/kcontains at leastc(e(G)/v(G))2kcopies ofC2k, this number of copies tightly achieved by the random graph (up to a multiplicative constant). In this paper we extend Simonovits' result to a supersaturation result ofr-uniform linear cycles of even length inr-uniform linear hypergraphs. Our proof is self-contained and includes ther= 2 case. As an auxiliary tool, we develop a reduction lemma from general host graphs to almost-regular host graphs that can be used for other supersaturation problems, and may therefore be of independent interest.more » « less
-
Abstract In this paper, we consider a randomized greedy algorithm for independent sets inr‐uniformd‐regular hypergraphsGonnvertices with girthg. By analyzing the expected size of the independent sets generated by this algorithm, we show that, whereconverges to 0 asg → ∞for fixeddandr, andf(d, r) is determined by a differential equation. This extends earlier results of Garmarnik and Goldberg for graphs [8]. We also prove that when applying this algorithm to uniform linear hypergraphs with bounded degree, the size of the independent sets generated by this algorithm concentrate around the mean asymptotically almost surely.more » « less
-
null (Ed.)Abstract The leftover hash lemma (LHL) is used in the analysis of various lattice-based cryptosystems, such as the Regev and Dual-Regev encryption schemes as well as their leakage-resilient counterparts. The LHL does not hold in the ring setting, when the ring is far from a field, which is typical for efficient cryptosystems. Lyubashevsky et al . (Eurocrypt ’13) proved a “regularity lemma,” which can be used instead of the LHL, but applies only for Gaussian inputs. This is in contrast to the LHL, which applies when the input is drawn from any high min-entropy distribution. Our work presents an approach for generalizing the “regularity lemma” of Lyubashevsky et al . to certain conditional distributions. We assume the input was sampled from a discrete Gaussian distribution and consider the induced distribution, given side-channel leakage on the input. We present three instantiations of our approach, proving that the regularity lemma holds for three natural conditional distributions.more » « less
-
Abstract The concept of branch decomposition was first introduced by Robertson and Seymour in their proof of the Graph Minors Theorem, and can be seen as a measure of the global connectivity of a graph. Since then, branch decomposition and branchwidth have been used for computationally solving combinatorial optimization problems modeled on graphs and matroids. General branchwidth is the extension of branchwidth to any symmetric submodular function defined over a finite set. General branchwidth encompasses graphic branchwidth, matroidal branchwidth, and rankwidth. A tangle basis is related to a tangle, a notion also introduced by Robertson and Seymour; however, a tangle basis is more constructive in nature. It was shown in [I. V. Hicks. Graphs, branchwidth, and tangles! Oh my!Networks, 45:55‐60, 2005] that a tangle basis of orderkis coextensive to a tangle of orderk. In this paper, we revisit the construction of tangle bases computationally for other branchwidth parameters and show that the tangle basis approach is still competitive for computing optimal branch decompositions for general branchwidth.more » « less
