5 research outputs found
On Near-Linear-Time Algorithms for Dense Subset Sum
In the Subset Sum problem we are given a set of positive integers and a target and are asked whether some subset of sums to . Natural parameters for this problem that have been studied in the literature are and as well as the maximum input number and the sum of all input numbers . In this paper we study the dense case of Subset Sum, where all these parameters are polynomial in . In this regime, standard pseudo-polynomial algorithms solve Subset Sum in polynomial time . Our main question is: When can dense Subset Sum be solved in near-linear time ? We provide an essentially complete dichotomy by designing improved algorithms and proving conditional lower bounds, thereby determining essentially all settings of the parameters for which dense Subset Sum is in time . For notational convenience we assume without loss of generality that (as larger numbers can be ignored) and (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 if . - We prove a matching conditional lower bound: If Subset Sum is in near-linear time for any setting with , 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