Faster algorithms for k-subset sum and variations

Abstract

We present new, faster pseudopolynomial time algorithms for the k-Subset Sum problem, defined as follows: given a set Z of n positive integers and k targets $$t_1, \ldots , t_k$$ t 1 , … , t k , determine whether there exist k disjoint subsets $$Z_1,\dots ,Z_k \subseteq Z$$ Z 1 , ⋯ , Z k ⊆ Z , such that $$\Sigma (Z_i) = t_i$$ Σ ( Z i ) = t i , for $$i = 1, \ldots , k$$ i = 1 , … , k . Assuming $$t = \max \{ t_1, \ldots , t_k \}$$ t = max { t 1 , … , t k } is the maximum among the given targets, a standard dynamic programming approach based on Bellman’s algorithm can solve the problem in $$O(n t^k)$$ O ( n t k ) time. We build upon recent advances on Subset Sum due to Koiliaris and Xu, as well as Bringmann, in order to provide faster algorithms for k-Subset Sum. We devise two algorithms: a deterministic one of time complexity $${\tilde{O}}(n^{k / (k+1)} t^k)$$ O ~ ( n k / ( k + 1 ) t k ) and a randomised one of $${\tilde{O}}(n + t^k)$$ O ~ ( n + t k ) complexity. Additionally, we show how these algorithms can be modified in order to incorporate cardinality constraints enforced on the solution subsets. We further demonstrate how these algorithms can be used in order to cope with variations of k-Subset Sum, namely Subset Sum Ratio, k-Subset Sum Ratio and Multiple Subset Sum.

Approximating subset sum ratio via partition computations

“…These problems are tightly connected to Subset Sum, which has seen impressive advances recently, due to Koiliaris and Xu [23] who gave a deterministic Õ( √ nt) algorithm, where n is the number of input elements and t is the target, and Bringmann [10] who gave a Õ(n +t) randomized algorithm, which is essentially optimal under SETH [1]. See also [3] for an extension of these algorithms to a more general setting. Jin and Wu subsequently proposed a simpler randomized algorithm [19] achieving the same bounds as [10], which however seems to only solve the decision version of the problem.…”