  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.
