An 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( log ), where 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.