Subset Sum Instances in ZFC Limbo

TL;DR

This paper demonstrates a large class of subset sum problems solvable in polynomial time, with the proof relying on a set-theoretic independence result, placing it in ZFC limbo.

Contribution

It introduces a new class of subset sum problems solvable efficiently, proved using an independent set-theoretic theorem, highlighting limitations of ZFC in computational complexity.

Findings

Existence of many subset sum problems solvable in polynomial time

Proof relies on the ZFC independent Jump Free Theorem

Mathematics used is elementary and accessible at undergraduate level

Abstract

Our main result, Theorem 2.5, shows the existence of a vast infinity of subset sum problems solvable in polynomial time. The only proof we have of this result uses the ZFC independent Jump Free Theorem of Harvey Friedman, thus putting Theorem 2.5 in what we call ZFC limbo. The mathematics we use is elementary and at the level of a good undergraduate course in combinatorics or design and analysis of algorithms. The statement of the Jump Free Theorem is also easily understood at this level (but not the proof).