A Note on Optimizing the Ratio of Monotone Supermodular Functions

TL;DR

This paper demonstrates fundamental limitations in approximating the ratio of two monotone supermodular functions, showing that no bounded approximation ratio can be achieved with polynomial queries.

Contribution

It provides a theoretical result establishing the impossibility of bounded approximation ratios for ratio optimization of monotone supermodular functions.

Findings

No bounded approximation ratio with polynomial queries for monotone supermodular ratio optimization.

The result applies to both minimization and maximization problems.

Highlights inherent computational hardness in supermodular ratio problems.

Abstract

We show that for the problem of minimizing (or maximizing) the ratio of two supermodular functions, no bounded approximation ratio can be achieved via polynomial number of queries, if the two supermodular functions are both monotone non-decreasing or non-increasing.