Approximation Schemes for the Generalized Traveling Salesman Problem

被引:9
|
作者
Khachai, M. Yu. [1 ,2 ,3 ]
Neznakhina, E. D. [1 ,2 ]
机构
[1] Russian Acad Sci, Krasovskii Inst Math & Mech, Ural Branch, Ekaterinburg 620990, Russia
[2] Ural Fed Univ, Ekaterinburg 620000, Russia
[3] Omsk State Tech Univ, Omsk 644050, Russia
基金
俄罗斯科学基金会;
关键词
generalized traveling salesman problem; NP-hard problem; polynomial-time approximation scheme; ALGORITHMS;
D O I
10.1134/S0081543817090127
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
The Generalized Traveling Salesman Problem (GTSP) is defined by a weighted graph G = (V,E,w) and a partition of its vertex set into k disjoint clusters V = V (1) a<feminine ordinal indicator>... a<feminine ordinal indicator> V (k). It is required to find a minimum-weight cycle that contains exactly one vertex of each cluster. We consider a geometric setting of the problem (we call it the EGTSP-k-GC), in which the vertices of the graph are points in the plane, the weight function corresponds to the Euclidean distances between the points, and the partition into clusters is specified implicitly by means of a regular integer grid with step 1. In this setting, a cluster is a subset of vertices lying in the same cell of the grid; the arising ambiguity is resolved arbitrarily. Even in this special setting, the GTSP remains intractable, generalizing in a natural way the classical planar Euclidean TSP. Recently, a -approximation algorithm with complexity depending polynomially both on the number of vertices n and on the number of clusters k has been constructed for this problem. We propose three approximation schemes for this problem. For each fixed k, all the schemes are polynomial and the complexity of the first two is linear in the number of nodes. Furthermore, the first two schemes remain polynomial for k = O(log n), whereas the third scheme is polynomial for k = n - O(log n).
引用
收藏
页码:97 / 105
页数:9
相关论文
共 50 条
  • [1] Approximation Schemes for the Generalized Traveling Salesman Problem
    M. Yu. Khachai
    E. D. Neznakhina
    [J]. Proceedings of the Steklov Institute of Mathematics, 2017, 299 : 97 - 105
  • [2] An efficient transformation of the generalized traveling salesman problem into the traveling salesman problem on digraphs
    Dimitrijevic, V
    Saric, Z
    [J]. INFORMATION SCIENCES, 1997, 102 (1-4) : 105 - 110
  • [3] THE REDUCIBILITY OF THE GENERALIZED TRAVELING SALESMAN PROBLEM TO THE TRAVELING SALESMAN PROBLEM OF SMALLER DIMENSION
    RUBINOV, AR
    [J]. DOKLADY AKADEMII NAUK SSSR, 1982, 264 (05): : 1087 - 1090
  • [4] Approximation algorithms for the traveling salesman problem
    Monnot, J
    Paschos, VT
    Toulouse, S
    [J]. MATHEMATICAL METHODS OF OPERATIONS RESEARCH, 2003, 56 (03) : 387 - 405
  • [5] Approximation algorithms for the traveling salesman problem
    Jérôme Monnot
    Vangelis Th. Paschos
    Sophie Toulouse
    [J]. Mathematical Methods of Operations Research, 2003, 56 : 387 - 405
  • [6] TRANSFORMATION OF THE GENERALIZED TRAVELING-SALESMAN PROBLEM INTO THE STANDARD TRAVELING-SALESMAN PROBLEM
    LIEN, YN
    MA, E
    WAH, BWS
    [J]. INFORMATION SCIENCES, 1993, 74 (1-2) : 177 - 189
  • [7] The generalized covering traveling salesman problem
    Shaelaie, Mohammed H.
    Salari, Majid
    Naji-Azimi, Zahra
    [J]. APPLIED SOFT COMPUTING, 2014, 24 : 867 - 878
  • [8] An LP-based approximation algorithm for the generalized traveling salesman path problem
    Sun, Jian
    Gutin, Gregory
    Li, Ping
    Shi, Peihao
    Zhang, Xiaoyan
    [J]. THEORETICAL COMPUTER SCIENCE, 2023, 941 : 180 - 190
  • [9] An approximation algorithm for the traveling salesman problem with backhauls
    Gendreau, M
    Laporte, G
    Hertz, A
    [J]. OPERATIONS RESEARCH, 1997, 45 (04) : 639 - 641
  • [10] AN APPROXIMATION ALGORITHM FOR A HETEROGENEOUS TRAVELING SALESMAN PROBLEM
    Bae, Jungyun
    Rathinam, Sivakumar
    [J]. PROCEEDINGS OF THE ASME DYNAMIC SYSTEMS AND CONTROL CONFERENCE AND BATH/ASME SYMPOSIUM ON FLUID POWER AND MOTION CONTROL (DSCC 2011), VOL 1, 2012, : 637 - 644