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A new integer programming formulation of the graphical traveling salesman problem

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Abstract

In the Traveling Salesman Problem (TSP), a salesman wants to visit a set of cities and return home. There is a cost \(c_{ij}\) of traveling from city i to city j, which is the same in either direction for the Symmetric TSP. The objective is to visit each city exactly once, minimizing total travel costs. In the Graphical TSP, a city may be visited more than once, which may be necessary on a sparse graph. We present a new integer programming formulation for the Graphical TSP requiring only two classes of polynomial-sized constraints while addressing an open question proposed by Denis Naddef. We generalize one of these classes, and present promising preliminary computational results.

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Notes

  1. Note that the cut separating the two cycles has x-value equal to its y-value which is 1, and hence this solution does not satisfy the subtour elimination constraints.

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Authors and Affiliations

  1. Computer Science Department, University of New Mexico, Albuquerque, NM, 87131, USA

    Robert Carr

  2. Tepper School of Business, Carnegie Mellon University, Pittsburgh, USA

    R. Ravi

  3. Business, Computer Science, and Mathematics Department, Bryn Athyn College, Bryn Athyn, PA, 19009-0717, USA

    Neil Simonetti

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  1. Robert Carr
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  2. R. Ravi
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  3. Neil Simonetti
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Correspondence to Neil Simonetti.

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This material is based upon work supported in part by the U. S. Office of Naval Research under award numbers N00014-18-1-2099 and N00014-21-1-2243.

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Carr, R., Ravi, R. & Simonetti, N. A new integer programming formulation of the graphical traveling salesman problem. Math. Program. 197, 877–902 (2023). https://doi.org/10.1007/s10107-022-01849-w

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