Approximation Algorithms for Euclidean Group TSP

In the Euclidean group Traveling Salesman Problem (TSP), we are given a set of points P in the plane and a set of m connected regions, each containing at least one point of P. We want to find a tour of minimum length that visits at least one point in each region. This unifies the TSP with Neighborhoods and the Group Steiner Tree problem. We give a (9.1α+1)-approximation algorithm for the case when the regions are disjoint α-fat objects with possibly varying size. This considerably improves the best results known, in this case, for both the group Steiner tree problem and the TSP with Neighborhoods problem. We also give the first O(1)-approximation algorithm for the problem with intersecting regions.

[1]  Joseph S. B. Mitchell,et al.  Approximation algorithms for TSP with neighborhoods in the plane , 2001, SODA '01.

[2]  Petr Slavik,et al.  The Errand Scheduling Problem , 1997 .

[3]  Esther M. Arkin,et al.  Approximation Algorithms for the Geometric Covering Salesman Problem , 1994, Discret. Appl. Math..

[4]  Gabriele Reich,et al.  Beyond Steiner's Problem: A VLSI Oriented Generalization , 1989, WG.

[5]  Sanjeev Arora,et al.  Nearly linear time approximation schemes for Euclidean TSP and other geometric problems , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[6]  Joachim Gudmundsson,et al.  TSP with neighborhoods of varying size , 2005, J. Algorithms.

[7]  Christos H. Papadimitriou,et al.  The Euclidean Traveling Salesman Problem is NP-Complete , 1977, Theor. Comput. Sci..

[8]  Joseph S. B. Mitchell,et al.  Guillotine Subdivisions Approximate Polygonal Subdivisions: A Simple Polynomial-Time Approximation Scheme for Geometric TSP, k-MST, and Related Problems , 1999, SIAM J. Comput..

[9]  Oded Schwartz,et al.  On the complexity of approximating tsp with neighborhoods and related problems , 2003, computational complexity.

[10]  Peter Slavík.  A Tight Analysis of the Greedy Algorithm for Set Cover , 1997, J. Algorithms.

[11]  Joachim Gudmundsson,et al.  Hardness Result for TSP with Neighborhoods , 2000 .

[12]  Joseph S. B. Mitchell,et al.  Approximation algorithms for geometric tour and network design problems (extended abstract) , 1995, SCG '95.

[13]  Mark H. Overmars,et al.  Motion planning amidst fat obstacles (extended abstract) , 1994, SCG '94.

[14]  A. Frank van der Stappen,et al.  Motion planning amidst fat obstacles , 1993 .

[15]  J. Sack,et al.  Handbook of computational geometry , 2000 .

[16]  R. Ravi,et al.  A polylogarithmic approximation algorithm for the group Steiner tree problem , 2000, SODA '98.

[17]  Sanjeev Arora,et al.  Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems , 1998, JACM.

[18]  Ronald L. Graham,et al.  Some NP-complete geometric problems , 1976, STOC '76.

[19]  Joseph S. B. Mitchell,et al.  Geometric Shortest Paths and Network Optimization , 2000, Handbook of Computational Geometry.