Branching proofs of infeasibility in low density subset sum problems

We prove that the subset sum problem ax = x {0, 1}n (SUB) has a polynomial time computable certificate of infeasibility for all a 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 [6]; Frieze [3]; Furst and Kannan [4]; and Coster et. al. in [1]. We proof has two ingredients. We first prove that a vector that is near parallel to a is a suitable branching direction. Then we show that such a near parallel vector can be computed using diophantine approximation, via a methodology introduced by Frank and Tardos in [2]. 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 of (SUB) is proven by branching on the above hyperplane.

• Type: Journal
• Year: 2008
• Where: CORR
• Authors: Gábor Pataki, Mustafa Tural
• Keywords: CORR 2008, Education, Right Hand, Subset Sum Problem, Time Computable Certificate