On The Watchman Route Problem and Its Related Problems Dissertation Proposal

Transcription

1 On The Watchman Route Problem and Its Related Problems Dissertation Proposal Ning Xu The Graduate Center, The City University of New York 1 / 27

2 Outlines Introduction References Introduction The Watchman Route Problem (WRP) The Touring Polygons Problem (TPP) 2 / 27

3 Scenario: Night Watchman Suppose a night watchman (a person or a robot) is required to guard a building. The night watchman has sensors (eyes or cameras). In the ideal case, the watchman can see infinite far away. Figure: A night watchman in a building Can we find a shortest closed path for the watchman to guard everywhere in the building? 3 / 27

4 Visibility Region Introduction References Visibility Let P be a polygon. Two points x, y P are visible from each other if the line segment xy lies within P; i.e., xy P 1. Visibility Region The visibility region of a point x P is the set of points within P visible from x. Figure: Visibility region: the yellow region (image taken from Wikipedia [1]). 1 Here we assume the unlimited visibility 4 / 27

5 Watchman Route Problem (WRP) Watchman route A watchman route T is a closed path within P so that every point in P is visible from some point on T ; i.e., x P, y T : pq P. The watchman route problem (WRP) The watchman route problem asks a shortest watchman route for a known polygon P. Figure: A shortest watchman route (image taken from [5]) 5 / 27

6 Related Problems: The Art Gallery Problem The WRP was first introduced by Chin and Ntafos in 1986 [6, 7], as a variation of the art gallery problem. Art Gallery Problem The art gallery problem asks the minimum number of stationary guards to watch over all paintings in a gallery of n walls. n 3 cameras are sufficient and sometimes neccessary (Chvátal 1975 [8]). Figure: Four cameras cover the gallery (from Wikipedia [1]) 6 / 27

7 Related Problems: The Euclidean TSP with Neighborhoods (TSPN) Let R 1, R 2,..., R k be k geometric objects (maybe overlapped) of n total vertices in a plane. The TSPN asks a shortest closed path visiting all objects. NP-hard (Papadimitriou 1977[19]) APX-hard (Gudmundsson and Levcopoulos 2000 [15]) O(log n) approximation (Mata and Mitchell 1995 [16], Elbassioni et al [13], Gudmundsson and Levcopoulos 1999 [14]) 7 / 27

8 Related Problems: The Euclidean Traveling Salesman Problem (ETSP) Let p 1, p 2,..., p k be k points in a plane. The ETSP asks a shortest closed path visiting all points. NP-hard (Papadimitriou 1977 [19]) PTAS (Arora 1997 [3], Mitchell 1999 [17], Rao and Smith 1998 [20]) 8 / 27

9 Outlines Introduction References Introduction The Watchman Route Problem (WRP) The Touring Polygons Problem (TPP) 9 / 27

10 The WRP: Three Different Settings On simple polygons (polygon without holes) On polygonal domains (polygon with holes) The WRP on arrangement of connected geometric objects. For example, arrangement of line segments, lines, and etc.. 10 / 27

11 The WRP on Simple Polygons Two cases: The fixed WRP, or the anchored WRP The watchman route must pass a given point s, called the anchored point. The floating WRP No such an anchor point is required, i.e., the general case. (a) The fixed WRP s (b) The floating WRP Figure: The WRP on simple polygons: the red paths are the shortest watchman route. 11 / 27

12 Exact Algorithms of WRP on Simple Polygons Fixed WRP: O(n 3 log n) time (Dror et al [10]) Floating WRP: O(n 4 log n) time (Dror et al [10]) An algorithm for the fixed WRP with running time O(T ) admits an algorithm for the floating WRP with running time O(nT ) (Tan 2001 [21]). Problem to Investigate 1 Can the fixed WRP be solved in less than O(n 3 log n) time? 12 / 27

13 Approximation Algorithms of WRP on Simple Polygons Fixed WRP: 2 ratio, O(n) time (Tab 2004 [23]) Floating WRP: 2 ratio, O(n) time (Tan 2007 [25]) Problem to Investigate 2 Does the fixed WRP has an approximation algorithm better than 2 ratio? 13 / 27

14 The WRP on Polygonal Domains NP-hard (Chin and Ntafos 1986 citecn [7], Dumitrescu and Tóth [12]) O(log 2 n) approximation (Mitchell 2013 [18]) Impossible better than O(log n) ratio unless P NP (Mitchell 2013[18]) Problem to Investigate 3 Can the WRP on polygonal domain be approximated better than O(log 2 n) ratio? 14 / 27

15 The WRP on Arrangement of Connected Line Segments Figure: The WRP on arrangement of line segments: the red paths are the shortest watchman route. A special case of the WRP on polygonal domain, where the polygon collapses into an arrangement of line segments. NP-hard (Xu 2012 [27]) O(log 3 n) approximation (Dumitrescu, Mitchell and Zylinski [11]) 15 / 27

16 The WRP on Arrangement of Connected Line Segments Figure: The WRP on simple polygons: the red path shows the shortest watchman route. Problem to Investigate 4 What is the lower bound of the approximation ratio for the WRP on arrangement of line segments? Problem to Investigate 5 Does the WRP on arrangement of axis-parallel line segments admit a constant factor approximation algorithm, or even a PTAS? 16 / 27

17 The WRP on Arrangement of Lines Can be solved in O(n 8 ) time (Dumitrescu, Mitchell and Zylinski [11]) NP-hard in 3D space (Dumitrescu, Mitchell and Zylinski [11]) Figure: A watchman route (in bold) on arrangement of lines (image taken from [11]). 17 / 27

18 Outlines Introduction References Introduction The Watchman Route Problem (WRP) The Touring Polygons Problem (TPP) 18 / 27

19 The Touring Polygons Problem (TPP) Let P 1,..., P k be k polygons of n totoal vertices in the plane, possibly intersecting. Let s and t be two points in the plane. The TPP asks a shortest path from s to t visiting P 1,..., P k in order, possibly subject to fence constrains Fences Simple polygons F 0,..., F k so that P i P i+1 F i. F1 P2 F2 P3 s F0 P1 F3 t If there is no fence, the TPP is unconstrained. 19 / 27

20 Results on the Unconstrained TPP Convex polygons: O(kn log(n/k)) time (Dror et al [10]) Non-convex polygons: NP-hard (Ahadi, Mozafari, Zarei [2]) Non-convex polygons but the path cannot enter polygons: O(k 2 n log n) time (Ahadi, Mozafari, Zarei [2]) 20 / 27

21 Results on Constrained TPP NP-hard (Dror et al [10]) the part of P i s boundary from which shortest routes may bounce is convex: O(kn 2 log n) time (Dror et al [10]) 21 / 27

22 The Zookeeper s Problem Given a simple polygon P and a set S of convex disjoint convex polygons inside P called cages, each sharing one edge of P, compute a shortest closed path within P that visits every cage but does not enter the interior of any cage. The fixed zookeeper s problem The route are required to pass through a given point s P s (a) The fixed zookeeper s problem (b) The general zookeeper s problem Figure: The zookeeper s problem 22 / 27

23 Exact Algorithms of the Zookeeper s Problem The fixed case: O(n log n) time (Bespamyatnikh 2003 [4]) The general case: O(n 2 ) time (Tan 2001 [22]) Problem to Investigate 6 Can the fixed zookeeper s problem be solved in O(n) time? Problem to Investigate 7 Does the zookeeper s problem admits a sub-quadratic time algorithm, for example, O(n log 2 n) time as conjectured by Tan [22]? 23 / 27

24 Approximation Algorithms of the Zookeeper s Problem The fixed case: 2 ratio (Tan 2004 [23]) The general case: 2 ratio (Tan 2006 [24]) Problem to Investigate 8 Can we find an algorithm for the fixed zookeeper s problem better than 2 ratio? Problem to Investigate 9 Can we find an approximation algorithm for the zookeeper s problem better than 2 ratio? 24 / 27

25 The Safari Problem Introduction References Given a simple polygon P and a set S of convex disjoint convex polygons inside P called cages, each sharing one edge of P, compute a shortest closed path within P that visits every cage. In the safari problem, the route can enter cages The fixed safari problem The route are required to pass through a given point s P. s (a) The fixed safari problem (b) The general safari problem Figure: The safari problem 25 / 27

26 Results on the safari Problem The fixed case: O(kn log n) time (Dror et al [10]) The general case: O(kn 2 log n) time (Dror et al [10]) An algorithm for the fixed safari problem with running time O(T ) admits an algorithm for the general safari problem with running time O(nT ) (Tan and Hirata 2003 [26]). Problem to Investigate 10 Can we improve the current best result, O(kn log n), for the fixed safari problem? 26 / 27

27 The Aquarium-keeper s Problem Given a simple polygon P, find a shortest close path that visits every edge of P. Figure: The aquarium-keeper s problem Optimal algorithm: O(n) time (Czyzowicz et al [9]) 27 / 27

28 Wikipedia, Arash Ahadi, Amirhossein Mozafari, and Alireza Zarei. Touring a sequence of disjoint polygons: Complexity and extension. Theor. Comput. Sci., 556:45 54, Sanjeev Arora. Nearly linear time approximation schemes for euclidean TSP and other geometric problems. In 38th Annual Symposium on Foundations of Computer Science, FOCS 97, Miami Beach, Florida, USA, October 19-22, 1997, pages IEEE Computer Society, Sergei Bespamyatnikh. An o(n log n) algorithm for the zoo-keeper s problem. Computational Geometry: Theory and Applications, 24(2):63 74, Svante Carlsson, Håkan Jonsson, and Bengt J. Nilsson. Finding the shortest watchman route in a simple polygon. Discrete & Computational Geometry, 22(3): , Weipang Chin and Simeon C. Ntafos. Optimum watchman routes. In Proceedings of the 2nd symposium on Computational geometry (SoCG 86), pages 24 33, Weipang Chin and Simeon C. Ntafos. Optimum watchman routes. Information Processing Letters, 28(1):39 44, V Chvátal. A combinatorial theorem in plane geometry. Journal of Combinatorial Theory, Series B, 18(1):39 41, Jurek Czyzowicz, Peter Egyed, Hazel Everett, David Rappaport, Thomas C. Shermer, Diane L. Souvaine, Godfried T. Toussaint, and Jorge Urrutia. 27 / 27

29 The aquarium keeper s problem. In Alok Aggarwal, editor, Proceedings of the Second Annual ACM/SIGACT-SIAM Symposium on Discrete Algorithms, January 1991, San Francisco, California., pages ACM/SIAM, Moshe Dror, Alon Efrat, Anna Lubiw, and Joseph S. B. Mitchell. Touring a sequence of polygons. In Lawrence L. Larmore and Michel X. Goemans, editors, Proceedings of the 35th Symposium on Theory of Computing (STOC 03), pages ACM, Adrian Dumitrescu, Joseph S. B. Mitchell, and Pawel Zylinski. Watchman routes for lines and segments. In Fedor V. Fomin and Petteri Kaski, editors, 13th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 12), volume 7357 of Lecture Notes in Computer Science, pages Springer, Adrian Dumitrescu and Csaba D. Tóth. Watchman tours for polygons with holes. Computional Geometry: Theory and Applications, 45(7): , Khaled M. Elbassioni, Aleksei V. Fishkin, and René Sitters. On approximating the TSP with intersecting neighborhoods. In Tetsuo Asano, editor, Algorithms and Computation, 17th International Symposium, ISAAC 2006, Kolkata, India, December 18-20, 2006, Proceedings, volume 4288 of Lecture Notes in Computer Science, pages Springer, Joachim Gudmundsson and Christos Levcopoulos. A fast approximation algorithm for tsp with neighborhoods. Nord. J. Comput., 6(4): , Joachim Gudmundsson and Christos Levcopoulos. Hardness result for TSP with neighborhoods. In Technical Report LU-CS-TR , Department of Computer Science, Lund University, Sweden., Cristian S. Mata and Joseph S. B. Mitchell. 27 / 27

30 Approximation algorithms for geometric tour and network design problems (extended abstract). In Proceedings of the 11th Symposium on Computational Geometry (SoCG 95), pages , Joseph S. B. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial-time approximation scheme for geometric tsp, k-mst, and related problems. SIAM J. Comput., 28(4): , Joseph S. B. Mitchell. Approximating watchman routes. In Sanjeev Khanna, editor, Proceedings of the 24th Symposium on Discrete Algorithms (SODA 13), pages SIAM, Christos H. Papadimitriou. The euclidean traveling salesman problem is np-complete. Theor. Comput. Sci., 4(3): , Satish Rao and Warren D. Smith. Approximating geometrical graphs via spanners and banyans. In Jeffrey Scott Vitter, editor, Proceedings of the Thirtieth Annual ACM Symposium on the Theory of Computing, Dallas, Texas, USA, May 23-26, 1998, pages ACM, Xuehou Tan. Fast computation of shortest watchman routes in simple polygons. Information Processing Letters, 77(1):27 33, Xuehou Tan. Shortest zookeeper s routes in simple polygons. Inf. Process. Lett., 77(1):23 26, Xuehou Tan. Approximation algorithms for the watchman route and zookeeper s problems. Discrete Applied Mathematics, 136(2-3): , / 27

31 Xuehou Tan. A 2-approximation algorithm for the zookeeper s problem. Inf. Process. Lett., 100(5): , Xuehou Tan. A linear-time 2-approximation algorithm for the watchman route problem for simple polygons. Theoretical Computer Science, 384(1):92 103, Xuehou Tan and Tomio Hirata. Finding shortest safari routes in simple polygons. Inf. Process. Lett., 87(4): , Ning Xu. Complexity of minimum corridor guarding problems. Information Processing Letters, 112(17-18): , / 27