Does Subset Sum Admit Short Proofs?

TL;DR

This paper explores whether Subset Sum can be solved efficiently with short certificates, examining its certification complexity and related problems, and establishing new lower bounds for nondeterministic computation.

Contribution

It introduces a systematic study of certification complexity for parameterized problems, identifying equivalence classes and developing techniques for lower bounds.

Findings

Subset Sum has equivalent certification complexity to certain integer programming problems.

New lower bounds for nondeterministic computation in problems like Subset Sum in permutation groups.

Establishment of hardness results linking Subset Sum to 3Coloring in bounded-pathwidth graphs.

Abstract

We investigate the question whether Subset Sum can be solved by a polynomial-time algorithm with access to a certificate of length poly(k) where k is the maximal number of bits in an input number. In other words, can it be solved using only few nondeterministic bits? This question has motivated us to initiate a systematic study of certification complexity of parameterized problems. Apart from Subset Sum, we examine problems related to integer linear programming, scheduling, and group theory. We reveal an equivalence class of problems sharing the same hardness with respect to having a polynomial certificate. These include Subset Sum and Boolean Linear Programming parameterized by the number of constraints. Secondly, we present new techniques for establishing lower bounds in this regime. In particular, we show that Subset Sum in permutation groups is at least as hard for nondeterministic computation as 3Coloring in bounded-pathwidth graphs.