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Symmetric Travelling Salesman Problem

Some New Algorithmic Possibilities
  • Tiru ArthanariEmail author
  • Kun Qian
Chapter
Part of the Indian Statistical Institute Series book 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.

Keywords

Symmetrical travelling salesman problem Pedigree polytope Hypergraph simplex Multistage insertion formulation Compact formulations 

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Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Department of ISOM, Faculty of Business and EconomicsUniversity of AucklandAucklandNew Zealand

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