Approximation Algorithms for Maximum Independent Set of Pseudo-Disks

Discrete and Computational Geometry

Volumn 48, Issue 2, 2012, Pages 373-392

  Chan, Timothy M ^a   Har Peled, Sariel ^b  

^a UNIVERSITY OF WATERLOO   (Canada)

^b University of Illinois at Urbana Champaign   (United States)

Author keywords

Approximation algorithms; Fr chet distance; Realistic input models

Indexed keywords

EID: 84864329776     PISSN: 01795376     EISSN: 14320444     Source Type: Journal    
DOI: 10.1007/s00454-012-9417-5     Document Type: Article

References (40)

1. Afshani, P., Chan, T. M.: Dynamic connectivity for axis-parallel rectangles. In: Proc. 14th European Sympos. Algorithms. Lecture Notes Comput. Sci., vol. 4168, pp. 16-27 (2006).
2. Agarwal, P. K., Mustafa, N. H.: Independent set of intersection graphs of convex objects in 2D. Comput. Geom., Theory Appl. 34(2), 83-95 (2006).
3. Agarwal, P. K., Aronov, B., Chan, T. M., Sharir, M.: On levels in arrangements of lines, segments, planes, and triangles. Discrete Comput. Geom. 19, 315-331 (1998).
4. Agarwal, P. K., van Kreveld, M., Suri, S.: Label placement by maximum independent set in rectangles. Comput. Geom., Theory Appl. 11, 209-218 (1998).
5. Agarwal, P. K., Pach, J., Sharir, M.: State of the union-of geometric objects. In: Goodman, J. E., Pach, J., Pollack, R. (eds.) Surveys in Discrete and Computational Geometry Twenty Years Later. Contemporary Mathematics, vol. 453, pp. 9-48. AMS, Providence (2008).
6. Alon, N., Spencer, J. H.: The Probabilistic Method, 2nd edn. Wiley-Interscience, New York (2000).
7. Arora, S.: Polynomial time approximation schemes for Euclidean TSP and other geometric problems. J. Assoc. Comput. Mach. 45(5), 753-782 (1998).
8. Asplund, E., Grübaum, B.: On a coloring problem. Math. Scand. 8, 181-188 (1960).
9. Baker, B. S.: Approximation algorithms for NP-complete problems on planar graphs. J. Assoc. Comput. Mach. 41, 153-180 (1994).
10. Bar-Yehuda, R., Bendel, K., Freund, A., Rawitz, D.: Local ratio: A unified framework for approximation algorithms. In memoriam: Shimon Even 1935-2004. ACM Comput. Surv. 36, 422-463 (2004).
11. Berman, P., Fujito, T.: On approximation properties of the independent set problem for low degree graphs. Theor. Comput. Sci. 32(2), 115-132 (1999).
12. Berman, P., DasGupta, B., Muthukrishnan, S., Ramaswami, S.: Efficient approximation algorithms for tiling and packing problems with rectangles. J. Algorithms 41, 443-470 (2001).
13. Brönnimann, H., Goodrich, M. T.: Almost optimal set covers in finite VC-dimension. Discrete Comput. Geom. 14, 263-279 (1995).
14. Chalermsook, P., Chuzhoy, J.: Maximum independent set of rectangles. In: Proc. 20th ACM-SIAM Sympos. Discrete Algorithms, pp. 892-901 (2009).
15. Chan, T. M.: Polynomial-time approximation schemes for packing and piercing fat objects. J. Algorithms 46(2), 178-189 (2003).
16. Chan, T. M.: A note on maximum independent sets in rectangle intersection graphs. Inf. Process. Lett. 89, 19-23 (2004).
17. Chan, T. M., Har-Peled, S.: Approximation algorithms for maximum independent set of pseudo-disks. In: Proc. 25th Annu. ACM Sympos. Comput. Geom., pp. 333-340 (2009). cs. uiuc. edu/~sariel/papers/08/w_indep.
18. Chekuri, C., Vondrak, J., Zenklusen, R.: Submodular function maximization via the multilinear relaxation and contention resolution schemes. In: Proc. 43rd Annu. ACM Sympos. Theory Comput. (2011, to appear).
19. Clarkson, K. L., Shor, P. W.: Applications of random sampling in computational geometry, II. Discrete Comput. Geom. 4, 387-421 (1989).
20. Clarkson, K. L., Varadarajan, K. R.: Improved approximation algorithms for geometric set cover. Discrete Comput. Geom. 37(1), 43-58 (2007).
21. Efrat, A., Katz, M. J., Nielsen, F., Sharir, M.: Dynamic data structures for fat objects and their applications. Comput. Geom., Theory Appl. 15, 215-227 (2000).
22. Erlebach, T., Jansen, K., Seidel, E.: Polynomial-time approximation schemes for geometric intersection graphs. SIAM J. Comput. 34(6), 1302-1323 (2005).
23. Even, G., Rawitz, D., Shahar, S.: Hitting sets when the VC-dimension is small. Inf. Process. Lett. 95(2), 358-362 (2005).
24. Fox, J., Pach, J.: Computing the independence number of intersection graphs. In: Proc. 22nd ACM-SIAM Sympos. Discrete Algorithms, pp. 1161-1165 (2011).
25. Frederickson, G. N.: Fast algorithms for shortest paths in planar graphs, with applications. SIAM J. Comput. 16(6), 1004-1022 (1987).
26. Halldórsson, M. M.: Approximations of independent sets in graphs. In: The 2nd Intl. Work. Approx. Algs. Combin. Opt. Problems, pp. 1-13 (1998).
27. 1-ε. Acta Math. 182, 105-142 (1996).
28. Kedem, K., Livne, R., Pach, J., Sharir, M.: On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles. Discrete Comput. Geom. 1, 59-71 (1986).
29. Koufogiannakis, C., Young, N. E.: Beating simplex for fractional packing and covering linear programs. In: Proc. 48th Annu. IEEE Sympos. Found. Comput. Sci, pp. 494-506 (2007).
30. Lipton, R. J., Tarjan, R. E.: A separator theorem for planar graphs. SIAM J. Appl. Math. 36, 177-189 (1979).
31. Long, P. M.: Using the pseudo-dimension to analyze approximation algorithms for integer programming. In: Proc. 7th Workshop Algorithms Data Struct. Lecture Notes Comput. Sci., vol. 2125, pp. 26-37 (2001).
32. Marathe, M. V., Breu, H., Hunt, H. B. III, Ravi, S. S., Rosenkrantz, D. J.: Simple heuristics for unit disk graphs. Networks 25, 59-68 (1995).
33. Matoušek, J.: Reporting points in halfspaces. Comput. Geom., Theory Appl. 2(3), 169-186 (1992).
34. Motwani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press, New York (1995).
35. Mustafa, N. H., Ray, S.: PTAS for geometric hitting set problems via local search. In: Proc. 25th Annu. ACM Sympos. Comput. Geom., pp. 17-22 (2009).
36. Nieberg, T., Hurink, J., Kern, W.: A robust PTAS for maximum weight independent set in unit disk graphs. In: Proc. 30th Int. Workshop Graph-Theoretic Concepts in Comput. Sci. Lecture Notes Comput. Sci., vol. 3353, pp. 214-221 (2005).
37. Pyrga, E., Ray, S.: New existence proofs for ε-nets. In: Proc. 24th ACM Sympos. Comput. Geom., pp. 199-207 (2008).
38. Smith, W. D., Wormald, N. C.: Geometric separator theorems and applications. In: Proc. 39th Annu. IEEE Sympos. Found. Comput. Sci., pp. 232-243 (1998).
39. Varadarajan, K.: Weighted geometric set cover via quasi-uniform sampling. In: Proc. 42nd Annu. ACM Sympos. Theory Comput., pp. 641-648 (2010).
40. Whitesides, S., Zhao, R.: k-admissible collections of Jordan curves and offsets of circular arc figures. Technical Report SOCS 90. 08, McGill Univ., Montreal, Quebec (1990).