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arXiv · 1208.3346

On the Complexity of Hilbert Refutations for Partition

Susan Margulies·Shmuel Onn·Dmitrii Pasechnik
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Abstract

Given a set of integers W, the Partition problem determines whether W can be divided into two disjoint subsets with equal sums. We model the Partition 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 the Partition 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.

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Susan Margulies, Shmuel Onn, Dmitrii Pasechnik. 2012-08-16. On the Complexity of Hilbert Refutations for Partition. https://doi.org/10.1016/j.jsc.2013.06.005

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