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