A Method to Compute the Sparse Graphs for Traveling Salesman Problem Based on Frequency Quadrilaterals

In this paper, an iterative algorithm is designed to compute the sparse graphs for traveling salesman problem (TSP) according to the frequency quadrilaterals so that the computation time of the algorithms for TSP will be lowered. At each computation cycle, the algorithm first computes the average fr...

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Bibliographic Details
Published inFrontiers in Algorithmics Vol. 10823; pp. 286 - 299
Main Authors Wang, Yong, Remmel, Jeffrey
Format Book Chapter
LanguageEnglish
Published Switzerland Springer International Publishing AG 2018
Springer International Publishing
SeriesLecture Notes in Computer Science
Subjects
Frequency quadrilateral
Iterative algorithm
Probability model
Sparse graph
Traveling salesman problem
Online AccessGet full text
ISBN3319784544
9783319784540
ISSN0302-9743
1611-3349
DOI10.1007/978-3-319-78455-7_22

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Summary:In this paper, an iterative algorithm is designed to compute the sparse graphs for traveling salesman problem (TSP) according to the frequency quadrilaterals so that the computation time of the algorithms for TSP will be lowered. At each computation cycle, the algorithm first computes the average frequency $$ \overline{f} (e) $$ of an edge e with N frequency quadrilaterals containing e in the input graph G(V, E). Then the 1/3|E| edges with low frequency are eliminated to generate the output graph with a smaller number of edges. The algorithm can be iterated several times and the original optimal Hamiltonian cycle is preserved with a high probability. The experiments demonstrate the algorithm computes the sparse graphs with the O(nlog2n) edges containing the original optimal Hamiltonian cycle for most of the TSP instances in the TSPLIB. The computation time of the iterative algorithm is O(Nn2).
Bibliography:Original Abstract: In this paper, an iterative algorithm is designed to compute the sparse graphs for traveling salesman problem (TSP) according to the frequency quadrilaterals so that the computation time of the algorithms for TSP will be lowered. At each computation cycle, the algorithm first computes the average frequency \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{f} (e) $$\end{document} of an edge e with N frequency quadrilaterals containing e in the input graph G(V, E). Then the 1/3|E| edges with low frequency are eliminated to generate the output graph with a smaller number of edges. The algorithm can be iterated several times and the original optimal Hamiltonian cycle is preserved with a high probability. The experiments demonstrate the algorithm computes the sparse graphs with the O(nlog2n) edges containing the original optimal Hamiltonian cycle for most of the TSP instances in the TSPLIB. The computation time of the iterative algorithm is O(Nn2).
ISBN:3319784544
9783319784540
ISSN:0302-9743
1611-3349
DOI:10.1007/978-3-319-78455-7_22