A PTAS for TSP with neighborhoods among fat regions in the plane

The Euclidean TSP with neighborhoods (TSPN) problem seeks a shortest tour that visits a given collection of n regions (neighborhoods). We present the first polynomial-time approximation scheme for TSPN for a set of regions given by arbitrary disjoint fat regions in the plane. This improves substantially upon the known approximation algorithms, and is the first PTAS for TSPN on regions of non-comparable sizes. Our result is based on a novel extension of the m-guillotine method. The result applies to regions that are "fat" in a very weak sense: each region Pi contains a disk of radius Ω(diam(Pi)), but is otherwise arbitrary. Further, the result applies even if the regions intersect arbitrarily, provided that there exists a packing of disjoint disks, of radii Ω(diam(Pi)), contained within their respective regions. Finally, the PTAS result applies also to the case in which the regions are sets of points or polygons, each each lying within one of a given set of disjoint fat regions.

[1]  Sanjeev Arora,et al.  Approximation schemes for NP-hard geometric optimization problems: a survey , 2003, Math. Program..

[2]  Khaled M. Elbassioni,et al.  Approximation Algorithms for Euclidean Group TSP , 2005, ICALP.

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

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

[5]  Moshe Dror,et al.  Combinatorial Optimization with Explicit Delineation of the Ground Set by a Collection of Subsets , 2008, SIAM J. Discret. Math..

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

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

[8]  Joseph S. B. Mitchell,et al.  Shortest Paths and Networks , 2004, Handbook of Discrete and Computational Geometry, 2nd Ed..

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

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

[11]  Alexander Grigoriev,et al.  Approximation schemes for the generalized geometric problems with geographic clustering , 2005, EuroCG.

[12]  Satish Rao,et al.  Approximating geometrical graphs via “spanners” and “banyans” , 1998, STOC '98.

[13]  Joachim Gudmundsson,et al.  A Fast Approximation Algorithm for TSP with Neighborhoods , 1999, Nord. J. Comput..

[14]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[15]  Khaled M. Elbassioni,et al.  On Approximating the TSP with Intersecting Neighborhoods , 2006, ISAAC.

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

[17]  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..

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