Almost linear-time algorithm for the dense subset-sum problem

This paper describes a new approach for solving the subset-sum problem. It is useful for solving other NP-hard problems. The limits and potential of this approach are discussed. The approach yields an algorithm for solving the dense version of the subset-sum problem. It runs in time O(ℓ log ℓ), where ℓ is the bound on the size of the elements. But for dense enough inputs and target numbers near the middle sum, it runs in time O(m), where m is the number of elements. Consequently, it improves the previously best algorithms by at least one order of magnitude and sometimes by two. The algorithm yields a characterization of the set of subset sums as a collection of arithmetic progressions with the same difference. This characterization is derived by elementary number-theoretic and algorithmic techniques. Such a characterization was first obtained by using analytic number theory and yielded inferior algorithms.