Skip to main content
SpringerLink
Log in

Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs

  • Conference paper
  • First Online:
Algorithms - ESA 2015

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9294))

Abstract

We investigate the family of intersection graphs of low density objects in low dimensional Euclidean space. This family is quite general, and includes planar graphs. This family of graphs has some interesting properties, and in particular, it is a subset of the family of graphs that have polynomial expansion.

We present efficient (1 + ε)-approximation algorithms for polynomial expansion graphs, for Independent Set, Set Cover, and Dominating Set problems, among others, and these results seem to be new. Naturally, PTAS ’s for these problems are known for subclasses of this graph family.

These results have immediate interesting applications in the geometric domain. For example, the new algorithms yield the only PTAS known for covering points by fat triangles (that are shallow).

We also prove corresponding hardness of approximation for some of these optimization problems, characterizing their intractability with respect to density. For example, we show that there is no PTAS for covering points by fat triangles if they are not shallow, thus matching our PTAS for this problem with respect to depth.

Work on this paper was partially supported by a NSF AF awards CCF-1421231, and CCF-1217462.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Adamaszek, A., Wiese, A.: Approximation schemes for maximum weight independent set of rectangles. In: Proc. 54th Annu. IEEE Sympos. Found. Comput. Sci. (FOCS), pp. 400–409 (2013)

    Google Scholar 

  2. Adamaszek, A., Wiese, A.: A QPTAS for maximum weight independent set of polygons with polylogarithmic many vertices. In: Proc. 25th ACM-SIAM Sympos. Discrete Algs. (SODA), pp. 400–409 (2014)

    Google Scholar 

  3. 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. Amer. Math. Soc. (2008)

    Google Scholar 

  4. Andreev, E.M.: On convex polyhedra in lobachevsky spaces. Sbornik: Mathematics 10, 413–440 (1970)

    Article  Google Scholar 

  5. Aronov, B., de Berg, M., Ezra, E., Sharir, M.: Improved bounds for the union of locally fat objects in the plane. SIAM J. Comput. 43(2), 543–572 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Aronov, B., Ezra, E., Sharir, M.: Small-size ε-nets for axis-parallel rectangles and boxes. SIAM J. Comput. 39(7), 3248–3282 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Baker, B.S.: Approximation algorithms for NP-complete problems on planar graphs. J. Assoc. Comput. Mach. 41, 153–180 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cabello, S., Gajser, D.: Simple ptas’s for families of graphs excluding a minor. CoRR, abs/1410.5778 (2014)

    Google Scholar 

  9. Chalopin, J., Gonçalves, D.: Every planar graph is the intersection graph of segments in the plane: extended abstract. In: Proc. 41st Annu. ACM Sympos. Theory Comput. (STOC), pp. 631–638 (2009)

    Google Scholar 

  10. Chan, T.M.: Polynomial-time approximation schemes for packing and piercing fat objects. J. Algorithms 46(2), 178–189 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chan, T.M., Grant, E.: Exact algorithms and APX-hardness results for geometric set cover. In: Proc. 23rd Canad. Conf. Comput. Geom., CCCG (2011)

    Google Scholar 

  12. Chan, T.M., Har-Peled, S.: Approximation algorithms for maximum independent set of pseudo-disks. Discrete Comput. Geom. 48, 373–392 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Clarkson, K.L., Varadarajan, K.R.: Improved approximation algorithms for geometric set cover. Discrete Comput. Geom. 37(1), 43–58 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Eppstein, D.: Diameter and treewidth in minor-closed graph families. Algorithmica 27, 275–291 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Feder, T., Greene, D.H.: Optimal algorithms for approximate clustering. In: Proc. 20th Annu. ACM Sympos. Theory Comput., STOC, pp. 434–444 (1988)

    Google Scholar 

  16. Frederickson, G.N.: Fast algorithms for shortest paths in planar graphs, with applications. SIAM J. Comput. 16(6), 1004–1022 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  17. Grohe, M.: Local tree-width, excluded minors, and approximation algorithms. Combinatorica 23(4), 613–632 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Grohe, M., Kreutzer, S., Siebertz, S.: Deciding first-order properties of nowhere dense graphs. In: Proc. 46th Annu. ACM Sympos. Theory Comput., STOC, pp. 89–98 (2014)

    Google Scholar 

  19. Har-Peled, S.: Being fat and friendly is not enough. CoRR, abs/0908.2369 (2009)

    Google Scholar 

  20. Har-Peled, S.: Quasi-polynomial time approximation scheme for sparse subsets of polygons. In: Proc. 30th Annu. Sympos. Comput. Geom., SoCG, pp. 120–129 (2014)

    Google Scholar 

  21. Har-Peled, S., Quanrud, K.: Approximation algorithms for low-density graphs. CoRR, abs/1501.00721 (2015), http://arxiv.org/abs/1501.00721

  22. Hastad, J.: Clique is hard to approximate within n 1 − ε. In: Proc. 37th Annu. IEEE Sympos. Found. Comput. Sci., FOCS, pp. 627–636 (1996)

    Google Scholar 

  23. Henzinger, M.R., Klein, P., Rao, S., Subramanian, S.: Faster shortest-path algorithms for planar graphs. J. Comput. Sys. Sci. 55, 3–23 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  24. Karp, R.M.: Reducibility among combinatorial problems. In: Complexity of Computer Computations, pp. 85–103 (1972)

    Google Scholar 

  25. Koebe, P.: Kontaktprobleme der konformen Abbildung. Ber. Verh. Sächs. Akademie der Wissenschaften Leipzig, Math.-Phys. Klasse 88, 141–164 (1936)

    Google Scholar 

  26. Lipton, R.J., Tarjan, R.E.: A separator theorem for planar graphs. SIAM J. Appl. Math. 36, 177–189 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lipton, R.J., Tarjan, R.E.: Applications of a planar separator theorem. SIAM J. Comput. 9(3), 615–627 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  28. Matoušek, J.: Near-optimal separators in string graphs. Combin., Prob. & Comput. 23(1), 135–139 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Miller, G.L., Teng, S.H., Thurston, W.P., Vavasis, S.A.: Separators for sphere-packings and nearest neighbor graphs. J. Assoc. Comput. Mach. 44(1), 1–29 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  30. Mustafa, N.H., Raman, R., Ray, S.: QPTAS for geometric set-cover problems via optimal separators. ArXiv e-prints (2014)

    Google Scholar 

  31. Mustafa, N.H., Raman, R., Ray, S.: Settling the APX-hardness status for geometric set cover. In: Proc. 55th Annu. IEEE Sympos. Found. Comput. Sci., FOCS (2014) (page to appear)

    Google Scholar 

  32. Mustafa, N.H., Ray, S.: Improved results on geometric hitting set problems. Discrete Comput. Geom. 44(4), 883–895 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  33. Nesetril, J., Ossona de Mendez, P.: Grad and classes with bounded expansion I. Decompositions. Eur. J. Comb. 29(3), 760–776 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  34. Nesetril, J., Ossona de Mendez, P.: Grad and classes with bounded expansion II. Algorithmic Aspects 29(3), 777–791 (2008)

    MathSciNet  MATH  Google Scholar 

  35. Nesetril, J., Ossona de Mendez, P.: Sparsity. Alg. Combin., vol. 28. Springer (2012)

    Google Scholar 

  36. Pach, J., Agarwal, P.K.: Combinatorial Geometry. John Wiley & Sons (1995)

    Google Scholar 

  37. Raz, R., Safra, S.: A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP. In: Proc. 29th Annu. ACM Sympos. Theory Comput., STOC, pp. 475–484 (1997)

    Google Scholar 

  38. Schwartz, J.T., Sharir, M.: Efficient motion planning algorithms in environments of bounded local complexity. Report 164, Dept, Math. Sci., New York Univ., New York (1985)

    Google Scholar 

  39. van der Stappen, A.F.: Motion Planning Amidst Fat Obstacles. PhD thesis, Utrecht University, Netherlands (1992)

    Google Scholar 

  40. van der Stappen, A.F., Overmars, M.H., de Berg, M., Vleugels, J.: Motion planning in environments with low obstacle density. Discrete Comput. Geom. 20(4), 561–587 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  41. Verger-Gaugry, J.-L.: Covering a ball with smaller equal balls in ℝn. Discrete Comput. Geom. 33(1), 143–155 (2005)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Illinois, 201 N. Goodwin Avenue, Urbana, IL, 61801, USA

    Sariel Har-Peled & Kent Quanrud

Authors
  1. Sariel Har-Peled
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kent Quanrud
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sariel Har-Peled .

Editor information

Editors and Affiliations

  1. University of Technology, Eindhoven, The Netherlands

    Nikhil Bansal

  2. Sapienza University of Rome, Rome, Italy

    Irene Finocchi

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Har-Peled, S., Quanrud, K. (2015). Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs. In: Bansal, N., Finocchi, I. (eds) Algorithms - ESA 2015. Lecture Notes in Computer Science(), vol 9294. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48350-3_60

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-48350-3_60

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-48349-7

  • Online ISBN: 978-3-662-48350-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation