Permutatorial Optimization via the Permutahedron

TL;DR

This paper introduces a polyhedral approach to solve permutatorial optimization problems, particularly for linear objectives, using the permutahedron to determine optimal deployment sequences.

Contribution

It formalizes permutatorial problems with combinatorial and continuous subproblems and extends polyhedral methods to these problems leveraging the permutahedron.

Findings

Efficient solutions for linear permutatorial problems using polyhedral methods.

Application of the permutahedron to model and solve deployment sequencing.

Framework applicable to incremental deployment scenarios.

Abstract

A water company decides to expand its network with a set of water lines, but it cannot build them all at once. However, it starts reaping benefits from a partial expansion. In what order should the company build the lines? We formalize a class of permutatorial problems with combinatorial/continuous subproblems capturing applications of incremental deployment. We show that, for additive/linear objective functions, efficient polyhedral methods for the subproblems extend to the permutatorial problem. Our main technical ingredient is the permutahedron.