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

The Manickam–Miklós–Singhi conjecture states that when n ≥ 4 k , 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 branching 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 propagation algorithm. Using implementations of these algorithms, we verify a stronger form of the conjecture for all k ≤ 7 .