Faster Algorithms for $k$-Subset Sum and Variations

TL;DR

This paper introduces faster algorithms for the k-Subset Sum problem, improving computational efficiency using recent advances, and extends these methods to related problem variations with practical constraints.

Contribution

It presents two novel algorithms for k-Subset Sum with improved time complexities, building on recent research, and adapts them for problem variants with additional constraints.

Findings

Deterministic algorithm with $ ilde{O}(n^{k/(k+1)} t^k)$ time complexity.

Randomized algorithm with $ ilde{O}(n + t^k)$ time complexity.

Algorithms can be modified for problems with cardinality constraints and related 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 $t_1, \ldots, t_k$, determine whether there exist $k$ disjoint subsets $Z_1,\dots,Z_k \subseteq Z$, such that $\Sigma(Z_i) = t_i$, for $i = 1, \ldots, k$. Assuming $t = \max \{ t_1, \ldots, t_k \}$ is the maximum among the given targets, a standard dynamic programming approach based on Bellman's algorithm [Bell57] can solve the problem in $O(n t^k)$ time. We build upon recent advances on Subset Sum due to Koiliaris and Xu [Koil19] and Bringmann [Brin17] in order to provide faster algorithms for $k$-Subset Sum. We devise two algorithms: a deterministic one of time complexity $\tilde{O}(n^{k / (k+1)} t^k)$ and a randomised one of $\tilde{O}(n + t^k)$ 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.