Fixed-charge transportation problems on trees

TL;DR

This paper studies fixed-charge transportation problems on trees, proving NP-hardness in general but providing a pseudo-polynomial dynamic programming solution for trees, and proposing a new formulation for general graphs.

Contribution

It introduces a dynamic programming approach for trees and a novel unary expansion formulation for general graphs, advancing solution methods for fixed-charge transportation problems.

Findings

NP-hardness of the problem on general graphs

Pseudo-polynomial solvability on trees using dynamic programming

A new unary expansion formulation advantageous for computation

Abstract

We consider a class of fixed-charge transportation problems over graphs. We show that this problem is strongly NP-hard, but solvable in pseudo-polynomial time over trees using dynamic programming. We also show that the LP formulation associated to the dynamic program can be obtained from extended formulations of single-node flow polytopes. Given these results, we present a unary expansion-based formulation for general graphs that is computationally advantageous when compared to a standard formulation, even if its LP relaxation is not stronger.