Nonnegative k-sums, fractional covers, and probability of small deviations

Noga Alon, Hao Huang, Benny Sudakov

Research output: Contribution to journalArticle

21 Scopus citations

Abstract

More than twenty years ago, Manickam, Miklós, and Singhi conjectured that for any integers n, k satisfying n≥4k, every set of n real numbers with nonnegative sum has at least (n-1k-1) k-element subsets whose sum is also nonnegative. In this paper we discuss the connection of this problem with matchings and fractional covers of hypergraphs, and with the question of estimating the probability that the sum of nonnegative independent random variables exceeds its expectation by a given amount. Using these connections together with some probabilistic techniques, we verify the conjecture for n≥33k 2. This substantially improves the best previously known exponential lower bound n≥e ckloglogk. In addition we prove a tight stability result showing that for every k and all sufficiently large n, every set of n reals with a nonnegative sum that does not contain a member whose sum with any other k-1 members is nonnegative, contains at least (n-1k-1)+(n-k-1k-1)-1 subsets of cardinality k with nonnegative sum.

Original languageEnglish (US)
Pages (from-to)784-796
Number of pages13
JournalJournal of Combinatorial Theory. Series B
Volume102
Issue number3
DOIs
StatePublished - May 1 2012
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

Keywords

  • Fractional cover
  • Hypergraph matching
  • Nonnegative k-sum

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