Branching proofs of infeasibility in low density subset sum problems

TL;DR

This paper introduces a polynomial-time certificate of infeasibility for low-density subset sum problems using hyperplane branching, leveraging diophantine approximation and near-parallel vectors, applicable to almost all right sides.

Contribution

It presents a novel polynomial-time method to certify infeasibility in subset sum problems for low density vectors, expanding the understanding of problem structure and solution certificates.

Findings

Certificates of infeasibility are computable in polynomial time for low density vectors.

Near-parallel vectors serve as effective branching directions regardless of density.

Most integer right sides can be covered by intervals where infeasibility is proven by hyperplane branching.

Abstract

We prove that the subset sum problem has a polynomial time computable certificate of infeasibility for all $a$ weight vectors with density at most $1/(2n)$ and for almost all integer right hand sides. The certificate is branching on a hyperplane, i.e. by a methodology dual to the one explored by Lagarias and Odlyzko; Frieze; Furst and Kannan; and Coster et. al. The proof has two ingredients. We first prove that a vector that is near parallel to $a$ is a suitable branching direction, regardless of the density. Then we show that for a low density $a$ such a near parallel vector can be computed using diophantine approximation, via a methodology introduced by Frank and Tardos. We also show that there is a small number of long intervals whose disjoint union covers the integer right hand sides, for which the infeasibility is proven by branching on the above hyperplane.