# Path Cover and Path Pack Inequalities for the Capacitated Fixed-Charge Network Flow Problem

## TL;DR

This paper introduces new path cover and path pack inequalities for the capacitated fixed-charge network flow problem, enhancing solution methods through strong, computationally efficient cuts.

## Contribution

It derives novel path-based inequalities that generalize existing flow inequalities, with explicit characterization and facet conditions, improving optimization in network flow models.

## Key findings

- Inequalities are strong and easy to compute.
- Effective as cuts in branch-and-cut algorithms.
- Significant computational improvements demonstrated.

## Abstract

Capacitated fixed-charge network flows are used to model a variety of problems in telecommunication, facility location, production planning and supply chain management. In this paper, we investigate capacitated path substructures and derive strong and easy-to-compute \emph{path cover and path pack inequalities}. These inequalities are based on an explicit characterization of the submodular inequalities through a fast computation of parametric minimum cuts on a path, and they generalize the well-known flow cover and flow pack inequalities for the single-node relaxations of fixed-charge flow models. We provide necessary and sufficient facet conditions. Computational results demonstrate the effectiveness of the inequalities when used as cuts in a branch-and-cut algorithm.