# Short combinatorial proof that the DFJ polytope is contained in the MTZ polytope for the Asymmetric Traveling Salesman Problem

@article{Velednitsky2017ShortCP, title={Short combinatorial proof that the DFJ polytope is contained in the MTZ polytope for the Asymmetric Traveling Salesman Problem}, author={Mark Velednitsky}, journal={ArXiv}, year={2017}, volume={abs/1805.06997} }

Abstract For the Asymmetric Traveling Salesman Problem (ATSP), it is known that the Dantzig–Fulkerson–Johnson (DFJ) polytope is contained in the Miller–Tucker–Zemlin (MTZ) polytope. The analytic proofs of this fact are quite long. Here, we present a proof which is combinatorial and significantly shorter by relating the formulation to distances in a modified graph.

## Topics from this paper

## 9 Citations

Accelerating the Miller-Tucker-Zemlin model for the asymmetric traveling salesman problem

- Computer ScienceExpert Syst. Appl.
- 2020

This approach can help practitioners to solve real-life problems to near optimality using a standard optimization solver and may be useful to solve a variety of routing problems that use MTZ-type of subtour elimination constraints.

Routing for unmanned aerial vehicles: Touring dimensional sets

- Computer Science, MathematicsEur. J. Oper. Res.
- 2022

In this paper we deal with an extension of the crossing postman problem to design Hamiltonian routes that have to visit different shapes of dimensional elements (neighborhoods or polygonal chains)…

Weighted-Sum Approach for Bi-objective Optimization of Fleet Size with Environmental Aspects

- Computer ScienceApplications of Management Science
- 2018

Weighted-sum approach optimization models are formulated with the use of mixed-integer programming and show some balance between fleet size, truck types, and utilization of fuel, carbon emission, and production of noise.

Multi-Criteria Optimization for Fleet Size with Environmental Aspects

- Computer Science
- 2017

Computational results based on formulated optimization models could help logistics managers lead the initiative in environmental conservation by saving fuel and consequently minimizing pollution.

The Buy-Online-Pick-Up-in-Store Retailing Model: Optimization Strategies for In-Store Picking and Packing

- MedicineAlgorithms
- 2021

Effective strategies for the Buy-Online-Pick-up-in-Store paradigm are put forward that can be easily implemented by stores with different topologies and improvements in efficiency in terms of time spent during the picking process are estimated.

The impact of airspace regulations on unmanned aerial vehicles in last-mile operation

- Environmental ScienceTransportation Research Part D: Transport and Environment
- 2020

Abstract Utilizing autonomous unmanned aerial vehicles (drones) in the last-mile delivery of parcels is regarded as the ultimate disruptive technology that might significantly reduce the GHG…

A I ] 2 3 M ay 2 02 1 A review of approaches to modeling applied vehicle routing problems A Preprint

- 2021

Routing decisions for Buddhist pilgrimage: an elitist genetic algorithm approach

- International Journal of System Assurance Engineering and Management
- 2021

## References

SHOWING 1-7 OF 7 REFERENCES

The asymmetric travelling salesman problem and a reformulation of the Miller-Tucker-Zemlin constraints

- Mathematics, Computer ScienceEur. J. Oper. Res.
- 1999

The main result of this paper is the derivation of a compact formulation whose LP relaxation is characterized by a set of circuit inequalities given by Grotschel and Padberg, an improved and disaggregated version of a well-known model for the ATSP based on the subtour elimination constraints.

Requiem for the Miller-Tucker-Zemlin subtour elimination constraints?

- Mathematics, Computer ScienceEur. J. Oper. Res.
- 2014

This paper presents a systematic way of deriving inequalities that are more complicated than the MTZ and DL inequalities and that, in a certain way, “generalize” the underlying idea of the original inequalities.

Improvements and extensions to the Miller-Tucker-Zemlin subtour elimination constraints

- Computer Science, MathematicsOper. Res. Lett.
- 1991

This paper shows how the subtour elimination constraints developed by Miller, Tucker and Zemlin for the traveling salesman problem can be improved and extended to various types of vehicle routing…

Teaching Integer Programming Formulations Using the Traveling Salesman Problem

- Computer Science, MathematicsSIAM Rev.
- 2003

A simple computational exercise to compare weak and strong integer programming formulations of the traveling salesman problem, where students can optimally solve instances with up to 70 cities in a few minutes by adding cuts from the stronger formulation to the weaker, but simpler one.

An analytical comparison of different formulations of the travelling salesman problem

- Mathematics, Computer ScienceMath. Program.
- 1991

This result is used to analytically compare various formulations of the asymmetric travelling salesman problem to the standard formulation due to Dantzig, Fulkerson and Johnson which are shown to be “weaker formulations” in a precise setting.

Integer Programming Formulation of Traveling Salesman Problems

- Mathematics, Computer ScienceJACM
- 1960

The present paper provides yet another example of the versatility of integer programming as a mathematical modeling device by representing a generalization of the well-known “Travelling Salesman Problem” in integer programming terms.

CLASSIFICATION OF TRAVELING SALESMAN PROBLEM FORMULATIONS

- Mathematics, Computer Science
- 1988

The purpose of this paper is to clarify the relations between these formulations and with other classical formulations for the traveling salesman problem.