A Branch-and-Cut Strategy for the Manickam-Miklos-Singhi Conjecture
نویسندگان
چکیده
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 propagation algorithm. Using implementations of these algorithms, we verify a stronger form of the conjecture for all k ≤ 7.
منابع مشابه
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.
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ورودعنوان ژورنال:
- CoRR
دوره abs/1302.3636 شماره
صفحات -
تاریخ انتشار 2013