On the sum–product ratio problem and its applications

We study the so-called sum–product ratio problem, under which the numerator is the sum of numbers in a subset and the denominator is the product of numbers in the same subset. Unlike the sum–sum ratio problem, which corresponds to the assortment problem under the multinomial logit model, the sum–product ratio problem is generally NP-complete. We develop a fully polynomial-time approximation scheme and discuss several potential applications and useful extensions.