TY - JOUR

T1 - Nonnegative k-sums, fractional covers, and probability of small deviations

AU - Alon, Noga

AU - Huang, Hao

AU - Sudakov, Benny

N1 - Funding Information:
E-mail addresses: [email protected] (N. Alon), [email protected] (H. Huang), [email protected] (B. Sudakov). 1 Research supported in part by an ERC Advanced grant and by a USA–Israeli BSF grant. 2 Research supported in part by NSF grant DMS-1101185, NSF CAREER award DMS-0812005 and by USA–Israeli BSF grant.

PY - 2012/5

Y1 - 2012/5

N2 - 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.

AB - 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.

KW - Fractional cover

KW - Hypergraph matching

KW - Nonnegative k-sum

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U2 - 10.1016/j.jctb.2011.12.002

DO - 10.1016/j.jctb.2011.12.002

M3 - Article

AN - SCOPUS:84857631038

SN - 0095-8956

VL - 102

SP - 784

EP - 796

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

IS - 3

ER -