Approximation Schemes for k-Subset Sum Ratio and k-way Number Partitioning Ratio
The Subset Sum Ratio problem (SSR) asks, given a multiset $A$ of positive integers, to find two disjoint subsets of $A$ such that the largest-to-smallest ratio of their sums is minimized. In this paper we study the $k$-version of SSR, namely $k$-Subset Sum Ratio ($k$-SSR), which asks to minimize the largest-to-smallest ratio of sums of $k$ disjoint subsets of $A$. We develop an approximation scheme for $k$-SSR running in $O({n^{2k}}/{\varepsilon^{k-1}})$ time, where $n=|A|$ and $\varepsilon$ is the error parameter. To the best of our knowledge, this is the first FPTAS for $k$-SSR for fixed $k>2$. We also study the $k$-way Number Partitioning Ratio ($k$-PART) problem, which differs from $k$-SSR in that the $k$ subsets must constitute a partition of $A$; this problem in fact corresponds to the objective of minimizing the largest-to-smallest sum ratio in the family of Multiway Number Partitioning problems. We present a more involved FPTAS for $k$-PART, also achieving $O({n^{2k}}/{\varepsilon^{k-1}})$ time complexity. Notably, $k$-PART is also equivalent to the Minimum Envy-Ratio problem with identical valuation functions, which has been studied in the context of fair division of indivisible goods. Thus, for the case of identical valuations, our FPTAS represents a significant improvement over the $O(n^{4k^2+1}/\varepsilon^{2k^2})$ bound obtained by Nguyen and Rothe's FPTAS for Minimum Envy-Ratio with general additive valuations. Lastly, we propose a second FPTAS for $k$-SSR, which employs carefully designed calls to the first one; the new scheme has a time complexity of $\widetilde{O}(n/{\varepsilon^{3k-1}})$, thus being much faster when $n\gg 1/ \varepsilon$.