# A Near Optimal Approach for Symmetric Traveling Salesman Problem in Euclidean Space

### Wenhong Tian, Chaojie Huang, Xinyang Wang

#### Abstract

The traveling salesman problem (TSP) is one of the most challenging NP-hard problems. It has widely applications in various disciplines such as physics, biology, computer science and so forth. The best known approximation algorithm for Symmetric TSP (STSP) whose cost matrix satisfies the triangle inequality (called ?STSP) is Christofides algorithm which was proposed in 1976 and is a 3/2 approximation. Since then no proved improvement is made and improving upon this bound is a funda- mental open question in combinatorial optimization. In this paper, for the first time, we propose Trun- cated Generalized Beta distribution (TGB) for the probability distribution of optimal tour lengths in a TSP. We then introduce an iterative TGB approach to obtain quality-proved near optimal approximation, i.e., (1+1/2((a+1)/(a+2))^(K-1))-approximation where K is the number of iterations in TGB and a(>> 1) is the shape parameters of TGB. The result can approach the true optimum as K increases.

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#### Paper Citation

#### in Harvard Style

Tian W., Huang C. and Wang X. (2017). **A Near Optimal Approach for Symmetric Traveling Salesman Problem in Euclidean Space** . In *Proceedings of the 6th International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,* ISBN 978-989-758-218-9, pages 281-287. DOI: 10.5220/0006125202810287

#### in Bibtex Style

@conference{icores17,

author={Wenhong Tian and Chaojie Huang and Xinyang Wang},

title={A Near Optimal Approach for Symmetric Traveling Salesman Problem in Euclidean Space},

booktitle={Proceedings of the 6th International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,},

year={2017},

pages={281-287},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0006125202810287},

isbn={978-989-758-218-9},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 6th International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,

TI - A Near Optimal Approach for Symmetric Traveling Salesman Problem in Euclidean Space

SN - 978-989-758-218-9

AU - Tian W.

AU - Huang C.

AU - Wang X.

PY - 2017

SP - 281

EP - 287

DO - 10.5220/0006125202810287