Faster Algorithms for $k$-Subset Sum and Variations

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. 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.