Near-linear-time Subset Sum

Determine whether the one-dimensional Subset Sum problem, given an integer target and input integers of absolute value at most N, can be solved in O(N) time (or, in the paper’s more precise formulation, in \widetilde O(n+N) time for n explicitly listed items).

Background

The paper studies pseudopolynomial algorithms for Subset Sum in terms of the number n of explicitly listed items and the maximum absolute input value N. Prior algorithms achieve increasingly improved bounds, and the paper obtains a deterministic O_\varepsilon(n+N^{1+\varepsilon}) algorithm for every fixed \varepsilon>0.

This result leaves open the exact linear-dependence goal: removing the arbitrarily small polynomial overhead N^\varepsilon and obtaining \widetilde O(n+N) time. The question is the central unresolved benchmark for the one-dimensional problem, even though the paper establishes a near-linear approximation to that bound.