Approximating Subset Sum Ratio via Subset Sum Computations

We present a new FPTAS for the Subset Sum Ratio problem, which, given a set of integers, asks for two disjoint subsets such that the ratio of their sums is as close to 1 as possible. Our scheme makes use of exact and approximate algorithms for the closely related Subset Sum problem, hence any progress over those—such as the recent improvement due to Bringmann and Nakos [SODA 2021]—carries over to our FPTAS. Depending on the relationship between the size of the input set n and the error margin $$\varepsilon $$ , we improve upon the best currently known algorithm of Melissinos and Pagourtzis [COCOON 2018] of complexity $$\mathcal {O} (n^4 / \varepsilon )$$ . In particular, the exponent of n in our proposed scheme may decrease down to 2, depending on the Subset Sum algorithm used. Furthermore, while the aforementioned state of the art complexity, expressed in the form $$\mathcal {O} ((n + 1 / \varepsilon )^c)$$ , has constant $$c = 5$$ , our results establish that $$c < 5$$ .