Document

# How Does Object Fatness Impact the Complexity of Packing in d Dimensions?

## File

LIPIcs.ISAAC.2019.36.pdf
• Filesize: 3.65 MB
• 18 pages

## Cite As

Sándor Kisfaludi-Bak, Dániel Marx, and Tom C. van der Zanden. How Does Object Fatness Impact the Complexity of Packing in d Dimensions?. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 36:1-36:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ISAAC.2019.36

## Abstract

Packing is a classical problem where one is given a set of subsets of Euclidean space called objects, and the goal is to find a maximum size subset of objects that are pairwise non-intersecting. The problem is also known as the Independent Set problem on the intersection graph defined by the objects. Although the problem is NP-complete, there are several subexponential algorithms in the literature. One of the key assumptions of such algorithms has been that the objects are fat, with a few exceptions in two dimensions; for example, the packing problem of a set of polygons in the plane surprisingly admits a subexponential algorithm. In this paper we give tight running time bounds for packing similarly-sized non-fat objects in higher dimensions. We propose an alternative and very weak measure of fatness called the stabbing number, and show that the packing problem in Euclidean space of constant dimension d >=slant 3 for a family of similarly sized objects with stabbing number alpha can be solved in 2^O(n^(1-1/d) alpha) time. We prove that even in the case of axis-parallel boxes of fixed shape, there is no 2^o(n^(1-1/d) alpha) algorithm under ETH. This result smoothly bridges the whole range of having constant-fat objects on one extreme (alpha=1) and a subexponential algorithm of the usual running time, and having very "skinny" objects on the other extreme (alpha=n^(1/d)), where we cannot hope to improve upon the brute force running time of 2^O(n), and thereby characterizes the impact of fatness on the complexity of packing in case of similarly sized objects. We also study the same problem when parameterized by the solution size k, and give a n^O(k^(1-1/d) alpha) algorithm, with an almost matching lower bound: there is no algorithm with running time of the form f(k) n^o(k^(1-1/d) alpha/log k) under ETH. One of our main tools in these reductions is a new wiring theorem that may be of independent interest.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Computational geometry
##### Keywords
• Geometric intersection graph
• Independent Set
• Object fatness

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. Miklós Abért. Symmetric groups as products of abelian subgroups. Bulletin of the London Mathematical Society, 34(4):451-456, 2002.
2. Yohji Akama and Kei Irie. VC dimension of ellipsoids. CoRR, abs/1109.4347, 2011. URL: http://arxiv.org/abs/1109.4347.
3. Jochen Alber and Jirí Fiala. Geometric separation and exact solutions for the parameterized independent set problem on disk graphs. Journal of Algorithms, 52(2):134-151, 2004. URL: https://doi.org/10.1016/j.jalgor.2003.10.001.
4. Julien Baste and Dimitrios M. Thilikos. Contraction-Bidimensionality of Geometric Intersection Graphs. In IPEC 2017, volume 89 of LIPIcs, pages 5:1-5:13, 2018. URL: https://doi.org/10.4230/LIPIcs.IPEC.2017.5.
5. Csaba Biró, Édouard Bonnet, Dániel Marx, Tillmann Miltzow, and Paweł Rzążewski. Fine-grained complexity of coloring unit disks and balls. JoCG, 9(2):47-80, 2018. URL: https://doi.org/10.20382/jocg.v9i2a4.
6. Timothy M. Chan. Polynomial-time approximation schemes for packing and piercing fat objects. J. Algorithms, 46(2):178-189, 2003. URL: https://doi.org/10.1016/S0196-6774(02)00294-8.
7. Miroslav Chlebík and Janka Chlebíková. Approximation hardness of optimization problems in intersection graphs of d-dimensional boxes. In Proceedings of SODA 2005, pages 267-276. SIAM, 2005. URL: http://dl.acm.org/citation.cfm?id=1070432.1070470.
8. Marek Cygan, Fedor V Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015.
9. Mark de Berg, Hans L. Bodlaender, Sándor Kisfaludi-Bak, Dániel Marx, and Tom C. van der Zanden. A framework for ETH-tight algorithms and lower bounds in geometric intersection graphs. In Proceedings of STOC 2018, pages 574-586, 2018. URL: https://doi.org/10.1145/3188745.3188854.
10. Mark de Berg, Hans L. Bodlaender, Sándor Kisfaludi-Bak, Dániel Marx, and Tom C. van der Zanden. A Framework for ETH-Tight Algorithms and Lower Bounds in Geometric Intersection Graphs. CoRR, abs/1803.10633, 2018. URL: http://arxiv.org/abs/1803.10633.
11. Erik D. Demaine, Fedor V. Fomin, Mohammad Taghi Hajiaghayi, and Dimitrios M. Thilikos. Fixed-parameter algorithms for (k,r)-Center in planar graphs and map graphs. ACM Transactions on Algorithms, 1(1):33-47, 2005.
12. Erik D. Demaine, Fedor V. Fomin, Mohammad Taghi Hajiaghayi, and Dimitrios M. Thilikos. Subexponential parameterized algorithms on bounded-genus graphs and H-minor-free graphs. Journal of the ACM, 52(6):866-893, 2005. URL: https://doi.org/10.1145/1101821.1101823.
13. Frederic Dorn, Fedor V. Fomin, and Dimitrios M. Thilikos. Subexponential parameterized algorithms. Computer Science Review, 2(1):29-39, 2008.
14. Frederic Dorn, Fedor V. Fomin, and Dimitrios M. Thilikos. Catalan structures and dynamic programming in H-minor-free graphs. J. Comput. Syst. Sci., 78(5):1606-1622, 2012.
15. Frederic Dorn, Eelko Penninkx, Hans L. Bodlaender, and Fedor V. Fomin. Efficient Exact Algorithms on Planar Graphs: Exploiting Sphere Cut Decompositions. Algorithmica, 58(3):790-810, 2010.
16. Fedor V. Fomin, Daniel Lokshtanov, Venkatesh Raman, and Saket Saurabh. Subexponential algorithms for partial cover problems. Inf. Process. Lett., 111(16):814-818, 2011.
17. Fedor V. Fomin, Daniel Lokshtanov, and Saket Saurabh. Bidimensionality and geometric graphs. In Proceedings of SODA 2012, pages 1563-1575. SIAM, 2012. URL: http://portal.acm.org/citation.cfm?id=2095240&CFID=63838676&CFTOKEN=79617016.
18. Fedor V. Fomin and Dimitrios M. Thilikos. Dominating Sets in Planar Graphs: Branch-Width and Exponential Speed-Up. SIAM J. Comput., 36(2):281-309, 2006.
19. Sariel Har-Peled and Kent Quanrud. Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs. SIAM J. Comput., 46(6):1712-1744, 2017. URL: https://doi.org/10.1137/16M1079336.
20. David Haussler and Emo Welzl. ε-nets and simplex range queries. Discrete & Computational Geometry, 2(2):127-151, June 1987. URL: https://doi.org/10.1007/BF02187876.
21. Russell Impagliazzo and Ramamohan Paturi. On the Complexity of k-SAT. Journal of Computer and System Sciences, 62(2):367-375, 2001. URL: https://doi.org/10.1006/jcss.2000.1727.
22. Fritz John. Extremum Problems with Inequalities as Subsidiary Conditions, pages 197-215. Springer Basel, Basel, 2014. URL: https://doi.org/10.1007/978-3-0348-0439-4_9.
23. Sándor Kisfaludi-Bak, Dániel Marx, and Tom C. van der Zanden. How does object fatness impact the complexity of packing in d dimensions? CoRR, abs/1909.12044, September 2019. URL: http://arxiv.org/abs/1909.12044.
24. Philip N. Klein and Dániel Marx. A subexponential parameterized algorithm for Subset TSP on planar graphs. In SODA 2014 Proc., pages 1812-1830, 2014.
25. Dániel Marx and Michal Pilipczuk. Optimal Parameterized Algorithms for Planar Facility Location Problems Using Voronoi Diagrams. In Proceedings of ESA 2015, volume 9294 of LNCS, pages 865-877. Springer, 2015. URL: https://doi.org/10.1007/978-3-662-48350-3_72.
26. Dániel Marx and Anastasios Sidiropoulos. The limited blessing of low dimensionality: when 1-1/d is the best possible exponent for d-dimensional geometric problems. In Proceedings of SoCG 2014, pages 67-76. ACM, 2014. URL: https://doi.org/10.1145/2582112.2582124.
27. Gary L. Miller, Shang-Hua Teng, William P. Thurston, and Stephen A. Vavasis. Separators for sphere-packings and nearest neighbor graphs. J. ACM, 44(1):1-29, 1997. URL: https://doi.org/10.1145/256292.256294.
28. Marcin Pilipczuk, Michał Pilipczuk, Piotr Sankowski, and Erik Jan van Leeuwen. Subexponential-Time Parameterized Algorithm for Steiner Tree on Planar Graphs. In STACS 2013 Proc., pages 353-364, 2013.
29. Warren D. Smith and Nicholas C. Wormald. Geometric Separator Theorems & Applications. In Proceedings of the 39th Annual Symposium on Foundations of Computer Science, FOCS 1998, pages 232-243. IEEE Computer Society, 1998. URL: https://doi.org/10.1109/SFCS.1998.743449.
30. Dimitrios M. Thilikos. Fast Sub-exponential Algorithms and Compactness in Planar Graphs. In ESA 2011 Proc., pages 358-369, 2011.
31. A. Frank van der Stappen, Dan Halperin, and Mark H. Overmars. The Complexity of the Free Space for a Robot Moving Amidst Fat Obstacles. Comput. Geom., 3:353-373, 1993. URL: https://doi.org/10.1016/0925-7721(93)90007-S.