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Symmetric Travelling Salesman Problem

Some New Algorithmic Possibilities

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Mathematical Programming and Game Theory

Part of the book series: Indian Statistical Institute Series ((INSIS))

Abstract

The Symmetric Travelling Salesman Problem (STSP) has many different formulations. The most famous formulation is the standard formulation by Dantzig, Fulkerson and Johnson (Oper Res 2(4):393–410, 1954, [22]), which has \(n(n - 1)\) variables and \(2n - 1 + n - 1\) constraint. The focus of this research is on the multistage insertion formulation (Arthanari, Mathematical Programming - The State of the Art, 1983, [5]). The MI formulation is as tight as the standard formulation but only has \(n^3\) variables and \(n(n - 1)/2 + (n - 3)\) constraints. Integer gaps found in computational studies comparing different formulations indicate the superior performance of the MI-formulation. However, these comparisons have used generic LP solvers to solve the MI-formulation of STSP instances. Due to memory restrictions, problem sizes below 300 are only considered by them (Haerian, New insights on the multistage insertion formulation of the traveling salesman problem-polytopes, experiments, and algorithm, 2011, [30]) In this chapter, we outline a four-phase approach to solve this problem based on Arthanari’s (Atti della Accademia Peloritana dei Pericolanti- Classe di Scienze Fisiche, Matematiche e Naturali, 2017, [10]) finding that the MI relaxation is a special type of hypergraph minimum cost flow problem. Phase 1 adapts the HySimplex methods from Cambini et al. (Math Program 78(2):195–217, 1997, [15]) for the MI-relaxation problem of the STSP. Phase 2 is the implementation of a prototype from the algorithms in phase 1. Phase 3 consists of optimizing the prototype from phase 2. Phase 4 consists of computational experiments. This chapter focuses on phase 1 of this research.

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Notes

  1. 1.

    We use HC to represent a Hamiltonian cycle instead of H because in later sections, we use H to represent a hypergraph.

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Author information

Authors and Affiliations

  1. Department of ISOM, Faculty of Business and Economics, University of Auckland, Auckland, New Zealand

    Tiru Arthanari & Kun Qian

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  1. Tiru Arthanari
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  2. Kun Qian
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Correspondence to Tiru Arthanari .

Editor information

Editors and Affiliations

  1. Indian Statistical Institute, New Delhi, India

    S.K. Neogy

  2. Indian Statistical Institute, New Delhi, India

    Ravindra B. Bapat

  3. Indian Statistical Institute, New Delhi, India

    Dipti Dubey

Appendix A—Hypergraph Algorithms

Appendix A—Hypergraph Algorithms

The four algorithms used to solve the minimum cost flow problem on the hypergraph are presented here.

figure a
figure b
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figure d
figure e

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Arthanari, T., Qian, K. (2018). Symmetric Travelling Salesman Problem. In: Neogy, S., Bapat, R., Dubey, D. (eds) Mathematical Programming and Game Theory. Indian Statistical Institute Series. Springer, Singapore. https://doi.org/10.1007/978-981-13-3059-9_5

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