On Near-Linear-Time Algorithms for Dense Subset Sum

In the Subset Sum problem we are given a set of n positive integers X and a target t and are asked whether some subset of X sums to t. Natural parameters for this problem that have been studied in the literature are n and t as well as the maximum input number mxX and the sum of all input numbers ΣX . In this paper we study the dense case of Subset Sum, where all these parameters are polynomial in n. In this regime, standard pseudo-polynomial algorithms solve Subset Sum in polynomial time n O(1) .

Our main question is: When can dense Subset Sum be solved in near-linear time O(n)? We provide an essentially complete dichotomy by designing improved algorithms and proving conditional lower bounds, thereby determining essentially all settings of the parameters n, t, mxX , ΣX for which dense Subset Sum is in time O(n). For notational convenience we assume without loss of generality that t ≥ mxX (as larger numbers can be ignored) and t ≤ ΣX /2 (using symmetry). Then our dichotomy reads as follows:

By reviving and improving an additive-combinatorics-based approach by Galil and Margalit [SICOMP'91], we show that Subset Sum is in near-linear time O(n) if t mxX ΣX /n 2 . We prove a matching conditional lower bound: If Subset Sum is in near-linear time for any setting with t mxX ΣX /n 2 , then the Strong Exponential Time Hypothesis and the Strong k-Sum Hypothesis fail. We also generalize our algorithm from sets to multi-sets, albeit with non-matching upper and lower bounds.