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

@article{Atamtrk2017PathCA, title={Path Cover and Path Pack Inequalities for the Capacitated Fixed-Charge Network Flow Problem}, author={Alper Atamt{\"u}rk and Simge K{\"u}ç{\"u}kyavuz and Birce Tezel}, journal={SIAM J. Optim.}, year={2017}, volume={27}, pages={1943-1976} }

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 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…

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## References

SHOWING 1-10 OF 19 REFERENCES

### Fixed-charge transportation on a path: optimization, LP formulations and separation

- MathematicsMath. Program.
- 2013

A polyhedral analysis of the polynomially solvable special case in which the associated bipartite graph is a path, which describes a new class of inequalities that are called “path-modular” inequalities and shows that these inequalities suffice to describe the convex hull of the set of feasible solutions.

### A branch‐and‐cut algorithm for the single‐commodity, uncapacitated, fixed‐charge network flow problem

- BusinessNetworks
- 2003

We present a branch‐and‐cut algorithm to solve the single‐commodity, uncapacitated, fixed‐charge network flow problem, which includes the Steiner tree problem, uncapacitated lot‐sizing problems, and…

### Lot-size models with backlogging: Strong reformulations and cutting planes

- BusinessMath. Program.
- 1988

An implicit description of the convex hull of solutions is given, and it is shown how the problem of finding a violated cutting plane can be solved as a linear program.

### Lot Sizing with Inventory Bounds and Fixed Costs: Polyhedral Study and Computation

- BusinessOper. Res.
- 2005

Two models of the polyhedral structure of the lot-sizing problem with inventory bounds and fixed costs are investigated and facet-defining inequalities that make use of the inventory bounds explicitly are identified.

### A study of the lot-sizing polytope

- BusinessMath. Program.
- 2004

A polynomial–time combinatorial separation algorithm is given for the inequalities of the lot-sizing polytope that cut off all fractional extreme points of its linear programming relaxation, as well as liftings from those facets.