A linear programming approach to the Manickam-Miklós-Singhi conjecture

A Branch-and-Cut Strategy for the Manickam-Miklos-Singhi Conjecture

The Manickam-Miklós-Singhi Conjecture states that when n ≥ 4k, every multiset of n real numbers with nonnegative total sum has at least ( n−1 k−1 ) k-subsets with nonnegative sum. We develop a branch-and-cut strategy using a linear programming formulation to show that verifying the conjecture for fixed values of k is a finite problem. To improve our search, we develop a zero-error randomized pr...

A linear bound on the Manickam-Miklós-Singhi conjecture

Suppose that we have a set of numbers x1, . . . , xn which have nonnegative sum. How many subsets of k numbers from {x1, . . . , xn} must have nonnegative sum? Manickam, Miklós, and Singhi conjectured that for n ≥ 4k the answer is ( n−1 k−1 ) . This conjecture is known to hold when n is large compared to k. The best known bounds are due to Alon, Huang, and Sudakov who proved the conjecture when...

The Minimum Number of Nonnegative Edges in Hypergraphs

An r-unform n-vertex hypergraph H is said to have the Manickam-Miklós-Singhi (MMS) property if for every assignment of weights to its vertices with nonnegative sum, the number of edges whose total weight is nonnegative is at least the minimum degree of H. In this paper we show that for n > 10r, every r-uniform n-vertex hypergraph with equal codegrees has the MMS property, and the bound on n is ...

A note on the Manickam-Miklós-Singhi conjecture for vector spaces

New results related to a conjecture of Manickam and Singhi

In 1998 Manickam and Singhi conjectured that for every positive integer d and every n ≥ 4d, every set of n real numbers whose sum is nonnegative contains at least ( n−1 d−1 ) subsets of size d whose sums are nonnegative. In this paper we establish new results related to this conjecture. We also prove that the conjecture of Manickam and Singhi does not hold for n = 2d + 2.