Subset Sum Ratio is the following optimization problem: Given a set of $n$ positive numbers $I$, find disjoint subsets $X,Y \subseteq I$ minimizing the ratio $\max\{\Sigma(X)/\Sigma(Y),\Sigma(Y)/\Sigma(X)\}$, where $\Sigma(Z)$ denotes the sum of all elements of $Z$. Subset Sum Ratio is an optimization variant of the Equal Subset Sum problem. It was introduced by Woeginger and Yu in '92 and is known to admit an FPTAS [Bazgan, Santha, Tuza '98]. The best approximation schemes before this work had running time $O(n^4/\varepsilon)$ [Melissinos, Pagourtzis '18], $\tilde O(n^{2.3}/\varepsilon^{2.6})$ and $\tilde O(n^2/\varepsilon^3)$ [Alonistiotis et al. '22]. In this work, we present an improved approximation scheme for Subset Sum Ratio running in time $O(n / \varepsilon^{0.9386})$. Here we assume that the items are given in sorted order, otherwise we need an additional running time of $O(n \log n)$ for sorting. Our improved running time simultaneously improves the dependence on $n$ to linear and the dependence on $1/\varepsilon$ to sublinear. For comparison, the related Subset Sum problem admits an approximation scheme running in time $O(n/\varepsilon)$ [Gens, Levner '79]. If one would achieve an approximation scheme with running time $\tilde O(n / \varepsilon^{0.99})$ for Subset Sum, then one would falsify the Strong Exponential Time Hypothesis [Abboud, Bringmann, Hermelin, Shabtay '19] as well as the Min-Plus-Convolution Hypothesis [Bringmann, Nakos '21]. We thus establish that Subset Sum Ratio admits faster approximation schemes than Subset Sum. This comes as a surprise, since at any point in time before this work the best known approximation scheme for Subset Sum Ratio had a worse running time than the best known approximation scheme for Subset Sum.

翻译：子集和比值问题（Subset Sum Ratio）是如下优化问题：给定一组 $n$ 个正数 $I$，寻找不相交子集 $X,Y \subseteq I$，使得比值 $\max\{\Sigma(X)/\Sigma(Y),\Sigma(Y)/\Sigma(X)\}$ 最小化，其中 $\Sigma(Z)$ 表示 $Z$ 中所有元素的和。子集和比值问题是等值子集和问题（Equal Subset Sum problem）的优化变体，由 Woeginger 和 Yu 于 1992 年提出，已知其存在完全多项式时间近似方案（FPTAS）[Bazgan, Santha, Tuza '98]。在本工作之前，最优近似方案的时间复杂度分别为 $O(n^4/\varepsilon)$ [Melissinos, Pagourtzis '18]、$\tilde O(n^{2.3}/\varepsilon^{2.6})$ 和 $\tilde O(n^2/\varepsilon^3)$ [Alonistiotis et al. '22]。本文提出了一个改进的子集和比值近似方案，运行时间为 $O(n / \varepsilon^{0.9386})$。这里假设输入项已排序，否则需要额外 $O(n \log n)$ 的排序时间。我们的改进方案同时将关于 $n$ 的依赖优化至线性，将关于 $1/\varepsilon$ 的依赖优化至亚线性。作为对比，相关子集和问题（Subset Sum）已知存在运行时间为 $O(n/\varepsilon)$ 的近似方案 [Gens, Levner '79]。若在子集和问题中实现运行时间为 $\tilde O(n / \varepsilon^{0.99})$ 的近似方案，则将证伪强指数时间假设（Strong Exponential Time Hypothesis）[Abboud, Bringmann, Hermelin, Shabtay '19] 以及最小加卷积假设（Min-Plus-Convolution Hypothesis）[Bringmann, Nakos '21]。因此我们证明子集和比值问题可比子集和问题拥有更快的近似方案。这一结果令人意外，因为在本工作之前的任何时间点，子集和比值问题已知最优近似方案的运行时间均劣于子集和问题已知最优近似方案。