On the complexity of Hilbert refutations for partition. (January 2015)

• Record Type: Journal Article
• Authors: Margulies, S.; Onn, S.; Pasechnik, D.V.
• Is Part Of: Journal of symbolic computation. Volume 66(2015)
• Page Start: 70
• Page End: 83
• Publication Date: 2015-01
• Subjects: Hilbert's Nullstellensatz -- Linear algebra -- Partition; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2013.06.005
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: Given a set of integers W, thePartition problem determines whether W can be divided into two disjoint subsets with equal sums. We model thePartition problem as a system of polynomial equations, and then investigate the complexity of a Hilbert's Nullstellensatz refutation, or certificate, that a given set of integers is not partitionable. We provide an explicit construction of a minimum-degree certificate, and then demonstrate that thePartition problem is equivalent to the determinant of a carefully constructed matrix called the partition matrix. In particular, we show that the determinant of the partition matrix is a polynomial that factors into an iteration over all possible partitions of W .