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

TL;DR

This paper proves a conjecture about the minimum number of nonnegative k-sums in large sets of real numbers with nonnegative total, using probabilistic and hypergraph techniques, and establishes a stability result.

Contribution

It verifies the Manickam-Miklós-Singhi conjecture for n ≥ 33k^2 and provides a tight stability theorem for large n, improving previous bounds significantly.

Findings

Verified the conjecture for n ≥ 33k^2

Established a tight stability result for large n

Improved bounds from exponential to polynomial in k

Abstract

More than twenty years ago, Manickam, Mikl\'{o}s, and Singhi conjectured that for any integers $n, k$ satisfying $n \geq 4k$, every set of $n$ real numbers with nonnegative sum has at least $\binom{n-1}{k-1}$ $k$-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 $n \geq 33k^2$. This substantially improves the best previously known exponential lower bound $n \geq e^{ck \log\log k}$. 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 $\binom{n-1}{k-1}+\binom{n-k-1}{k-1}-1$ subsets of cardinality $k$ with nonnegative sum.