Succinct Certificates for Almost All Subset Sum Problems

Abstract

Given n natural numbers $a_1 , \cdots ,a_n $ and a target integer b, the SubsetSum problem is to determine whether some subset of the $a_i $; sums to b. That is, to recognize members of the following set: \[ {\text{SubsetSum}} = \left\{ \left\langle {a_1 , \cdots ,a_n ;b} \right\rangle |a_i \in {\bf N} b \in {\bf Z},{\text{ and }}\exists x \in \{ 0,1\} ^n {\text{ such that }} a \cdot x = b \right\} . \] For a given vector $a = (a_1 , \cdots ,a_n )$ and integer b, if a subset of the $a_i $ sums to b, then listing which subset provides a short proof that $\langle {a;b} \rangle \in {\text{SubsetSum}}$. However, in general there are no short (polynomial-length) proofs of nonmembership unless NP equals coNP.The main result in this paper provides a proof system that contains polynomial-length nonmembership proofs for a vast majority of the problem instances that do not belong to SubsetSum.