Fractional decompositions of dense hypergraphs

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A seminal result of Rödl (the Rödl nibble) asserts that the edges of the complete r-uniform hypergraph Knr can be packed, almoet completely, with copies of Kkr, where k is fixed. This result is considered one of the most fruitful applications of the probabilistic method. It was not known whether the same result also holds in a dense hypergraph setting. In this paper we prove that it does. We prove that for every r-uniform hypergraph H0, there exists a constant α = α(H0) < 1 such that every r-uniform hypergraph H in which every (r - 1)-set is contained in at least on edges has an H0-packing that covers |E(H)|(1 - on(1)) edges. Our method of proof is via fractional decompositions. Let H0 be a fixed hypergraph. A fractional H0-decomposition of a hypergraph H is an assignment of nonnegative real weights to the copies of H0 in H such that for each edge e ε E(H), the sum of the weights of copies of H0 containing e is precisely one. Let k and r be positive integers with k > r > 2. We prove that there exists a constant α = α(k, r) < 1 such that every r-uniform hypergraph with n (sufficiently large) vertices in which every (r - 1)-set is contained in at least on edges has a fractional Kk r-decomposition. We then apply a recent result of Rödl, Schacht, Siggers and Tokushige to obtain our integral packing result. The proof makes extensive use of probabilistic arguments and additional combinatorial ideas.

Original languageEnglish
Pages (from-to)482-493
Number of pages12
JournalLecture Notes in Computer Science
StatePublished - 2005
Event8th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2005 and 9th International Workshop on Randomization and Computation, RANDOM 2005 - Berkeley, CA, United States
Duration: 22 Aug 200524 Aug 2005

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science (all)


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