Abstract
Geometric Covering and Packing problems have been extensively studied in the last few decades and have applications in diverse areas. Several variants and generalizations of these problems have been studied recently. In this paper, we look at the following covering variants where we require that each point is “uniquely” covered, i.e., it is covered by exactly one object: Unique Coverage problem, where we want to maximize the number of uniquely covered points and Exact Cover problem, where we want to uniquely cover every point and minimize the number of objects used for covering. We also look at the following generalizations: Multi-Cover problem, a generalization of Set Cover, where we want to select the minimum subset of objects with the constraint that each input point is covered by at least k objects in the solution. And Shallow Packing problem, a generalization of Packing problem, where we want to select the maximum subset of objects with the constraint that any point in the plane is contained in at most k objects in the solution. The above problems are NP-hard even for unit squares in the plane. Thus, the focus has been on obtaining good approximation algorithms.
Local Search have been quite successful in the recent past in obtaining good approximation algorithms for a wide variety of problems. We consider the Unique Coverage and Multi-Cover problems on non-piercing objects, which is a broad class that includes squares, disks, pseudo-disks, etc. and show that the local search algorithm yields a PTAS approximation under the assumption that the depth of the input points is at most a constant. For the Shallow Packing problem, we show that the local search algorithm yields a PTAS approximation for objects with sub-quadratic union complexity, which is a very broad class of objects that even includes non-piercing objects. For the Exact Cover problem, we show that finding a feasible solution is NP-hard even for unit squares in the plane, thus negating the existence of polynomial time approximation algorithms.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
A set of regions is said to be non-piercing if for any pair of regions A and B, \(A\setminus B\) and \(B\setminus A\) are connected.
- 2.
Union complexity of a set of objects is the description complexity of the boundary of the union of the objects.
- 3.
The depth of a point with respect to a set of objects is the number of objects in the set containing that point.
- 4.
Actually, Lemma 1 proves for the discrete intersection which is a super graph of the bipartite version.
References
Agarwal, P.K., Pach, J., Sharir, M.: State of the union (of geometric objects): a review (2007)
Aschner, R., Katz, M.J., Morgenstern, G., Yuditsky, Y.: Approximation schemes for covering and packing. In: Ghosh, S.K., Tokuyama, T. (eds.) WALCOM 2013. LNCS, vol. 7748, pp. 89–100. Springer, Heidelberg (2013). doi:10.1007/978-3-642-36065-7_10
Ashok, P., Kolay, S., Misra, N., Saurabh, S.: Unique covering problems with geometric sets. In: Xu, D., Du, D., Du, D. (eds.) COCOON 2015. LNCS, vol. 9198, pp. 548–558. Springer, Cham (2015). doi:10.1007/978-3-319-21398-9_43
Bandyapadhyay, S., Basu Roy, A.: Effectiveness of local search for art gallery problems. In: WADS (2017)
Bansal, N., Pruhs, K.: Weighted geometric set multi-cover via quasi-uniform sampling. JoCG 7(1), 221–236 (2016)
Chan, T.M., Har-Peled, S.: Approximation algorithms for maximum independent set of pseudo-disks. Discrete Comput. Geom. 48(2), 373–392 (2012)
Chekuri, C., Clarkson, K.L., Har-Peled, S.: On the set multicover problem in geometric settings. ACM Trans. Algorithms (TALG) 9(1), 9 (2012)
Cohen-Addad, V., Klein, P.N., Mathieu, C.: Local search yields approximation schemes for k-means and k-median in euclidean and minor-free metrics. In: FOCS, pp. 353–364 (2016)
Cohen-Addad, V., Mathieu, C.: Effectiveness of local search for geometric optimization. In: SoCG, pp. 329–343 (2015)
Dahllöf, V., Jonsson, P., Beigel, R.: Algorithms for four variants of the exact satisfiability problem. Theor. Comput. Sci. 320(2–3), 373–394 (2004)
Demaine, E.D., Feige, U., Hajiaghayi, M., Salavatipour, M.R.: Combination can be hard: approximability of the unique coverage problem. SIAM J. Comput. 38(4), 1464–1483 (2008)
Ene, A., Har-Peled, S., Raichel, B.: Geometric packing under non-uniform constraints. In: SoCG, pp. 11–20 (2012)
Erlebach, T., Van Leeuwen, E.J.: Approximating geometric coverage problems. In: Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1267–1276. Society for Industrial and Applied Mathematics (2008)
Fowler, R.J., Paterson, M.S., Tanimoto, S.L.: Optimal packing and covering in the plane are np-complete. Inf. Process. Lett. 12(3), 133–137 (1981)
Frederickson, G.N.: Fast algorithms for shortest paths in planar graphs, with applications. SIAM J. Comput. 16(6), 1004–1022 (1987)
Friggstad, Z., Rezapour, M., Salavatipour, M.R.: Local search yields a PTAS for k-means in doubling metrics. In: FOCS, pp. 365–374 (2016)
Garey, M.R., Johnson, D.S.: Computers and intractability (1979)
Govindarajan, S., Raman, R., Ray, S., Basu Roy, A.: Packing and covering with non-piercing regions. In: ESA, pp. 47:1–47:17 (2016)
Har-Peled, S.: Quasi-polynomial time approximation scheme for sparse subsets of polygons. In: SoCG, pp. 120:120–120:129 (2014)
Ito, T., Nakano, S., Okamoto, Y., Otachi, Y., Uehara, R., Uno, T., Uno, Y.: A 4.31-approximation for the geometric unique coverage problem on unit disks. Theor. Comput. Sci. 544, 14–31 (2014)
Ito, T., Nakano, S.-I., Okamoto, Y., Otachi, Y., Uehara, R., Uno, T., Uno, Y.: A polynomial-time approximation scheme for the geometric unique coverage problem on unit squares. In: Fomin, F.V., Kaski, P. (eds.) SWAT 2012. LNCS, vol. 7357, pp. 24–35. Springer, Heidelberg (2012). doi:10.1007/978-3-642-31155-0_3
Krohn, E., Gibson, M., Kanade, G., Varadarajan, K.: Guarding terrains via local search. J. Comput. Geom. 5(1), 168–178 (2014)
Matoušek, J.: Lectures on Discrete Geometry. Springer, New York (2002)
Misra, N., Moser, H., Raman, V., Saurabh, S., Sikdar, S.: The parameterized complexity of unique coverage and its variants. Algorithmica 65(3), 517–544 (2013)
Mustafa, N.H., Ray, S.: Improved results on geometric hitting set problems. Discrete Comput. Geom. 44(4), 883–895 (2010)
Pach, J., Walczak, B.: Decomposition of multiple packings with subquadratic union complexity. Comb. Probab. Comput. 25(1), 145–153 (2016)
Pyrga, E., Ray, S.: New existence proofs for \(\epsilon \)-nets. In: SoCG, pp. 199–207 (2008)
Schaefer, T.J.: The complexity of satisfiability problems. In: Proceedings of the Tenth Annual ACM Symposium on Theory of Computing, pp. 216–226. ACM (1978)
Whitesides, S., Zhao, R.: K-admissible collections of jordan curves and offsets of circular arc figures. Technical report, McGill University, School of Computer Science (1990)
Author information
Authors and Affiliations
Corresponding authors
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Ashok, P., Basu Roy, A., Govindarajan, S. (2017). Local Search Strikes Again: PTAS for Variants of Geometric Covering and Packing. In: Cao, Y., Chen, J. (eds) Computing and Combinatorics. COCOON 2017. Lecture Notes in Computer Science(), vol 10392. Springer, Cham. https://doi.org/10.1007/978-3-319-62389-4_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-62389-4_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-62388-7
Online ISBN: 978-3-319-62389-4
eBook Packages: Computer ScienceComputer Science (R0)