A Gilmore-Gomory-Type Construction of Integer Programming Value Functions

TL;DR

This paper explores how adding decision variables sequentially impacts the value function in integer programming, using a Gilmore-Gomory approach to analyze the structure of level sets and their relation to solutions.

Contribution

It introduces a novel framework for analyzing IP value functions via parametrized functions and the concept of maximal connected subsets of level sets.

Findings

Characterization of level set structures in IPs

Relation between level set volumes and optimal solutions

Use of Gilmore-Gomory approach for parametrized value functions

Abstract

In this paper, we analyze how sequentially introducing decision variables into an integer program (IP) affects the value function and its level sets. We use a Gilmore-Gomory approach to find parametrized IP value functions over a restricted set of variables. We introduce the notion of maximal connected subsets of level sets - volumes in which changes to the constraint right-hand side have no effect on the value function - and relate these structures to IP value functions and optimal solutions.