Approximating Watchman Routes

Given a connected polygonal domain P, the watchman route problem is to compute a shortest path or tour for a mobile guard (the "watchman") that is required to see every point of P. While the watchman route problem is polynomially solvable in simple polygons, it is known to be NP-hard in polygons with holes. We present the first polynomial-time approximation algorithm for the watchman route problem in polygonal domains. Our algorithm has an approximation factor O(log2 n). Further, we prove that the problem cannot be approximated in polynomial time to within a factor of c log n, for a constant c > 0, assuming that P≠NP.

[1]  Robert Krauthgamer,et al.  Polylogarithmic inapproximability , 2003, STOC '03.

[2]  Andrzej Pelc,et al.  Optimal Exploration of Terrains with Obstacles , 2010, SWAT.

[3]  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..

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

[5]  Bengt J. Nilsson.  Approximating a Shortest Watchman Route , 2001, Fundam. Informaticae.

[6]  Joseph S. B. Mitchell,et al.  Watchman Routes for Lines and Segments , 2012, SWAT.

[7]  Noga Alon,et al.  Algorithmic construction of sets for k-restrictions , 2006, TALG.

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

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

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

[11]  Svante Carlsson,et al.  Finding the Shortest Watchman Route in a Simple Polygon , 1993, ISAAC.

[12]  R. K. Shyamasundar,et al.  Introduction to algorithms , 1996 .

[13]  Csaba D. Tóth,et al.  Watchman tours for polygons with holes , 2012, CCCG.

[14]  Moshe Dror,et al.  Touring a sequence of polygons , 2003, STOC '03.

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

[16]  Simeon C. Ntafos,et al.  Optimum watchman routes , 1986, SCG '86.

[17]  XUEHOU TAN,et al.  Corrigendum to "An Incremental Algorithm for Constructing Shortest Watchman Routes" , 1999, Int. J. Comput. Geom. Appl..

[18]  Wei-Pand Chin,et al.  Shortest watchman routes in simple polygons , 1990, Discret. Comput. Geom..

[19]  Subhash Suri,et al.  An Optimal Algorithm for Euclidean Shortest Paths in the Plane , 1999, SIAM J. Comput..

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

[21]  Proceedings of the 35th Annual ACM Symposium on Theory of Computing, June 9-11, 2003, San Diego, CA, USA , 2003, STOC.

[22]  Joseph S. B. Mitchell,et al.  A PTAS for TSP with neighborhoods among fat regions in the plane , 2007, SODA '07.

[23]  Tomio Hirata,et al.  An incremental algorithm for constructing shortest watchman routes , 1993, Int. J. Comput. Geom. Appl..

[24]  Satish Rao,et al.  A tight bound on approximating arbitrary metrics by tree metrics , 2003, STOC '03.

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

[26]  Xuehou Tan.  Approximation algorithms for the watchman route and zookeeper's problems , 2004, Discret. Appl. Math..

[27]  Gill Barequet.  A Duality between Small-Face Problems in Arrangements of Lines and Heilbronn-Type Problems , 2000, COCOON.

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

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

[30]  Xuehou Tan,et al.  A linear-time 2-approximation algorithm for the watchman route problem for simple polygons , 2007, Theor. Comput. Sci..

[31]  Khaled M. Elbassioni,et al.  A QPTAS for TSP with fat weakly disjoint neighborhoods in doubling metrics , 2010, SODA '10.

[32]  Bengt J. Nilsson,et al.  Concerning the Time Bounds of Existing Shortest Watchman Route Algorithms , 1997, FCT.

[33]  Joseph S. B. Mitchell,et al.  A constant-factor approximation algorithm for TSP with pairwise-disjoint connected neighborhoods in the plane , 2010, SCG.