Fractional 0-1 programming and submodularity

TL;DR

This paper characterizes when single-ratio functions in fractional 0-1 programming are submodular, enabling efficient greedy solutions for complex NP-hard problems like assortment optimization and facility location.

Contribution

It provides a complete characterization of submodularity conditions in fractional 0-1 programs and discusses practical applications where these conditions hold.

Findings

Submodularity conditions are fully characterized under mild assumptions.

Greedy algorithms can find near-optimal solutions in certain practical cases.

Applications include assortment optimization and facility location problems.

Abstract

In this note we study multiple-ratio fractional 0--1 programs, a broad class of NP-hard combinatorial optimization problems. In particular, under some relatively mild assumptions we provide a complete characterization of the conditions, which ensure that a single-ratio function is submodular. Then we illustrate our theoretical results with the assortment optimization and facility location problems, and discuss practical situations that guarantee submodularity in the considered application settings. In such cases, near-optimal solutions for multiple-ratio fractional 0--1 programs can be found via simple greedy algorithms.