#### Title

### Symmetric Traveling Salesman Problem: Some New Insight.

#### Date of Submission

February 2011

#### Date of Award

2-28-2012

#### Institute Name (Publisher)

Indian Statistical Institute

#### Document Type

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

#### Subject Name

Mathematics

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

#### Supervisor

Arthanari, T. S.

#### Abstract (Summary of the Work)

This thesis is a study on a compact formulation of the symmetric traveling salesman problem STSP. Arthanari(1982) posed the STSP as a multistage decision problemn. We call this formulation as the Multistage-insertion(MI) formulation. We study properties of this formulation in detail. We also ob- tain the linear description of the projection of the MI polytope and prove its equivalence to the classic subtour elizination polytope, SEP. We discuss the equivalence of the M1 formulatiou to the Cycle-shrink, (CS), formulation proposed by Carr(1996). Both the MI and CS formaulations are compact formulations which use fewer umber of constraints. We also study structure of snall SEP and MI polytopes. The gist of the thesis is presented below in a chepter-wise summary.Chapter-1 is an introductory chapter in which we present a brief introduction to concepts from combinatorial optimisation problems, graph theory, poly- hedral combinatorics and linear programming which are used in the thesis.Chapter-2 is an introduction to the Traveling Salesman Problea( TSP). We discuss various formulations of the TSP such as the classic Dantzig, Fulk- erson and Johnson (DFJ), Bellmanns dynamic progranming formulation, Miller, Tucker , Zellin( MTZ) and Gavish, Graves formulation. The STSP polytope is defined and we introduce different facets of this polytope. This chapter has a brief introduction to the graphical traveling salesman problem and discuses the separation problem for the STSP polytope.In chapter-3 we present the Multistage-insertion formulation of the symmet- ric traveling salesinan problem given by Arthanari(1982). We have a (n - 3) stage decision problem,in which in stage (k - 3), 4 S k< n,we decide on where to insert k. We give the formulation and state properties. There is a 1-1 correspondence between n-tours and the integer feasible solutions to the MI problem. The vector of slack variables in the MI problem is the edge- tour incidence vector. We define two polytopes ((72) and U(n) and show U(n) to be at least as tight as the subtour elimination polytope SEP(n). U(n) is the orthogonal projection of the MI polytope, ((n). We briefly state the Mi formulation for the asyÄ±nmetric traveling salesman problem, A7SP, and give some properties.In chapter-4, we obtain the linear description of U(r2) and show U(n) is equiv- alent to SEP(n). The linear description of U(72) is obtained by applying the results of Padberg and Sung(1991). We explicitly work out generators for U(n),n = 6. The results are given in Appendix -I.In chapter-5 we discuss another polynomial sized formulation proposed by Carr(1996) called the Cycle-shrink ,(CS). The CS is equivalent to SEP(n). We show it is equivalent to MI.In chapter-6 we study sÄ±nall subtour polytopes for n <7. We present some criteria to characterise hamiltonian cycles. We use the results to give a ne- essary and sufficient condition for a feasible solution to MI to lie within STSP(n). We apply these resuits to sinall polytopes and give computa- tional results. Appendix-II gives these results.Chapter-7 surmmarises all the results presented in the thesis and addresses Problem for further research.

#### Control Number

ISILib-TH

#### Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

#### DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/2146

#### Recommended Citation

Usha, M. Dr., "Symmetric Traveling Salesman Problem: Some New Insight." (2012). *Doctoral Theses*. 221.

https://digitalcommons.isical.ac.in/doctoral-theses/221

## Comments

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28842998