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An iterative algorithm to eliminate edges for traveling salesman problem based on a new binomial distribution

Eliminating edges for TSP

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Abstract

Traveling salesman problem (TSP) is one of the extensively studied NP-hard problems. The recent research showed that the TSP on sparse graphs could be resolved in the relatively shorter computation time than that on the complete graph \(K_{n}\). This paper updates a previous probability model for the optimal Hamiltonian cycle edges according to the frequency quadrilaterals in \(K_{n}\). A new binomial distribution for TSP is rebuilt to show the probability that an edge e has the frequency 5 in a frequency quadrilateral. Based on the binomial distribution, an iterative algorithm is designed to compute the sparse graphs for TSP. There are two steps at each computation cycle. Firstly, N frequency quadrilaterals containing an edge e in the input graph is chosen to compute the average frequency \(\bar {f}(e)\) with the frequency quadrilaterals where e has the frequency 5. Secondly, half edges with the small values \(\bar {f}(e)\) are eliminated. The two steps are repeated until a sparse graph is computed. The computation time of the algorithm is \(O(Nn^{2})\). For the TSP instances in the TSPLIB, the experimental results illustrated that the sparse graphs with the \(O(n\log _{2} n)\) edges are computed and the original optimal solution is preserved. The experiments means the optimal Hamiltonian cycle edges have the bigger average frequency \(\bar {f}(e)\) in \(K_{n}\) and the subgraphs of \(K_{n}\) so they are preserved in the computation process.

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Acknowledgements

We acknowledge W. Cook, H. Mittelmann who created the Concorde and G. Reinelt, et al. who provided the TSP data to TSPLIB. The authors also thank the anonymous referees for their suggestions for improvements to presentation of the paper. The authors acknowledge the funds supported by the Fundamental Research Funds for the Central Universities (No.2018MS039 and No.2018ZD09) and NSFC (No.51205129).

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Correspondence to Yong Wang.

Appendix

Appendix

Theorem 1

Given the \(p_{5}(e)=\frac {1}{4}+ \epsilon _{n}\) is the minimum probability that the OHC edge e has the frequency 5 in a frequency quadrilateral in \(K_{n}\), the probability \(p_{5}(e)=\frac {1}{4} + \epsilon _{n}(or\ \epsilon _{n})\) tends to 1 (or 3/4) as n is big enough.

Proof

In the following, we prove that the \(p_{5}(e)\to 1\) holds for \(e\in OHC\) as n is big enough. We assume that the bigger graph \(K_{n + 1}\) originates from a smaller graph \(K_{n}\). Some notations are given first for proof.

Based on the axiom of choice, we will construct such two graphs \(K_{n}\) and \(K_{n + 1}\) which have the two probabilities \({p_{5}^{n}}(e)\) and \(p_{5}^{n + 1}(e^{\prime })\).

\({p_{5}^{n}}(e)=\frac {1}{4} + \epsilon _{n}\)::

The minimum probability that \(e\in OHC\) has the frequency \(f = 5\) in a frequency quadrilateral in \(K_{n}\) where \(\epsilon _{n}\to 0\).

\(p_{5}^{n + 1}(e^{\prime })=\frac {1}{3} + \epsilon _{n + 1}\)::

The minimum probability that \(e^{\prime }\in OHC\) has the frequency \(f = 5\) in a frequency quadrilateral in \(K_{n + 1}\) where \(\epsilon _{n + 1}\to 0\).

\(M_{n}={p_{5}^{n}}(e){{n-2}\choose {2}}\)::

The minimum number of the frequency quadrilaterals where \(e\in OHC\) has the frequency \(f = 5\) in \(K_{n}\).

\(M_{n + 1}=p_{5}^{n + 1}(e^{\prime }){{n-1}\choose {2}}\)::

The minimum number of the frequency quadrilaterals where \(e^{\prime }\in OHC\) has the frequency \(f = 5\) in \(K_{n + 1}\).

Since \(\epsilon _{n + 1}\to 0\) and \(\epsilon _{n}\to 0\), we have the limitations \(p_{5}^{n + 1}(e^{\prime }) \to \frac {1}{4}\) and \({p_{5}^{n}}(e) \to \frac {1}{4}\). Because \(M_{n + 1} \geq M_{n}\), we derive the inequality

$$\frac{p_{5}^{n + 1}(e^{\prime})}{{p_{5}^{n}}(e)} \geq \frac{{{n-2}\choose{2}}}{{{n-1}\choose{2}}} = 1-\frac{2}{n-1}.$$

We assume a big number M to ensure the limitation \(\lim _{n\to M}\frac {2}{n-1} = 0\). For any \(n > M\), we have \(p_{5}^{n + 1}(e^{\prime }) \geq {p_{5}^{n}}(e)\). It means the minimum probability \(p_{5}(e)\) that the OHC edge e has the frequency \(f = 5\) in a frequency quadrilateral in \(K_{n}\) increases according to n until it reaches the maximum value 1. Thus, the OHC edges have the big probability \(p_{5}(e)\) in any subgraphs containing e of \(K_{n}\) as n is big enough. □

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Wang, Y., Remmel, J.B. An iterative algorithm to eliminate edges for traveling salesman problem based on a new binomial distribution. Appl Intell 48, 4470–4484 (2018). https://doi.org/10.1007/s10489-018-1222-2

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