Optimality of Geometric Local Search

Authors Bruno Jartoux, Nabil H. Mustafa



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Bruno Jartoux
Nabil H. Mustafa

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Bruno Jartoux and Nabil H. Mustafa. Optimality of Geometric Local Search. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 48:1-48:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.SoCG.2018.48

Abstract

Up until a decade ago, the algorithmic status of several basic çlass{NP}-complete problems in geometric combinatorial optimisation was unresolved. This included the existence of polynomial-time approximation schemes (PTASs) for hitting set, set cover, dominating set, independent set, and other problems for some basic geometric objects. These past nine years have seen the resolution of all these problems--interestingly, with the same algorithm: local search. In fact, it was shown that for many of these problems, local search with radius lambda gives a (1+O(lambda^{-1/2}))-approximation with running time n^{O(lambda)}. Setting lambda = Theta(epsilon^{-2}) yields a PTAS with a running time of n^{O(epsilon^{-2})}. On the other hand, hardness results suggest that there do not exist PTASs for these problems with running time poly(n) * f(epsilon) for any arbitrary f. Thus the main question left open in previous work is in improving the exponent of n to o(epsilon^{-2}). We show that in fact the approximation guarantee of local search cannot be improved for any of these problems. The key ingredient, of independent interest, is a new lower bound on locally expanding planar graphs, which is then used to show the impossibility results. Our construction extends to other graph families with small separators.
Keywords
  • local search
  • expansion
  • matchings
  • Hall's marriage theorem

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References

  1. Pankaj K. Agarwal and Nabil H. Mustafa. Independent set of intersection graphs of convex objects in 2D. Computational Geometry, 34(2):83-95, 2006. URL: http://dx.doi.org/10.1016/j.comgeo.2005.12.001.
  2. Noga Alon, Paul Seymour, and Robin Thomas. A separator theorem for nonplanar graphs. Journal of the American Mathematical Society, 3(4), 10 1990. URL: http://dx.doi.org/10.1090/S0894-0347-1990-1065053-0.
  3. Daniel Antunes, Claire Mathieu, and Nabil H. Mustafa. Combinatorics of local search: An optimal 4-local hall’s theorem for planar graphs. In Kirk Pruhs and Christian Sohler, editors, 25th Annual European Symposium on Algorithms (ESA 2017), volume 87 of Leibniz International Proceedings in Informatics (LIPIcs), pages 8:1-8:13, Dagstuhl, Germany, 2017. Schloss Dagstuhl-Leibniz-Zentrum für Informatik. URL: http://dx.doi.org/10.4230/LIPIcs.ESA.2017.8.
  4. Vijay Arya, Naveen Garg, Rohit Khandekar, Adam Meyerson, Kamesh Munagala, and Vinayaka Pandit. Local search heuristics for k-median and facility location problems. SIAM Journal on Computing, 33(3):544-562, 2004. URL: http://dx.doi.org/10.1137/S0097539702416402.
  5. Rom Aschner, Matthew J. Katz, Gila Morgenstern, and Yelena Yuditsky. Approximation schemes for covering and packing. In Subir Kumar Ghosh and Takeshi Tokuyama, editors, WALCOM: Algorithms and Computation: 7th International Workshop, WALCOM 2013, Kharagpur, India, February 14-16, 2013. Proceedings, pages 89-100, Berlin, Heidelberg, 2013. Springer Berlin Heidelberg. URL: http://dx.doi.org/10.1007/978-3-642-36065-7_10.
  6. Norbert Bus, Shashwat Garg, Nabil H. Mustafa, and Saurabh Ray. Limits of local search: Quality and efficiency. Discrete & Computational Geometry, 57(3):607-624, 4 2017. URL: http://dx.doi.org/10.1007/s00454-016-9819-x.
  7. Sergio Cabello and David Gajser. Simple PTAS’s for families of graphs excluding a minor. Discrete Applied Mathematics, 189:41-48, 2015. URL: http://dx.doi.org/10.1016/j.dam.2015.03.004.
  8. Timothy M. Chan and Elyot Grant. Exact algorithms and APX-hardness results for geometric packing and covering problems. Comput. Geom., 47(2, Part A):112-124, 2014. Special Issue: 23rd Canadian Conference on Computational Geometry (CCCG11). URL: http://dx.doi.org/10.1016/j.comgeo.2012.04.001.
  9. Timothy M. Chan and Sariel Har-Peled. Approximation algorithms for maximum independent set of pseudo-disks. Discrete & Computational Geometry, 48(2):373-392, 09 2012. URL: http://dx.doi.org/10.1007/s00454-012-9417-5.
  10. Brent N. Clark, Charles J. Colbourn, and David S. Johnson. Unit disk graphs. Discrete Mathematics, 86(1):165-177, 1990. URL: http://dx.doi.org/10.1016/0012-365X(90)90358-O.
  11. John Conway and Neil J. A. Sloane. Sphere Packings, Lattices and Groups. Grundlehren der mathematischen Wissenschaften. Springer-Verlag New York, New York, NY, USA, 1999. URL: http://dx.doi.org/10.1007/978-1-4757-6568-7.
  12. Alina Ene, Sariel Har-Peled, and Benjamin Raichel. Geometric packing under non-uniform constraints. In Symposium on Computational Geometry 2012, SoCG '12, Chapel Hill, NC, USA, June 17-20, 2012, pages 11-20, 2012. URL: http://dx.doi.org/10.1145/2261250.2261253.
  13. Matt Gibson, Gaurav Kanade, Erik Krohn, and Kasturi Varadarajan. An approximation scheme for terrain guarding. In Irit Dinur, Klaus Jansen, Joseph Naor, and José Rolim, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques: 12th International Workshop, APPROX 2009, and 13th International Workshop, RANDOM 2009, Berkeley, CA, USA, August 21-23, 2009. Proceedings, pages 140-148, Berlin, Heidelberg, 2009. Springer Berlin Heidelberg. URL: http://dx.doi.org/10.1007/978-3-642-03685-9_11.
  14. Matt Gibson and Imran A. Pirwani. Algorithms for dominating set in disk graphs: Breaking the log n barrier. In Mark de Berg and Ulrich Meyer, editors, Proceedings of the 18th Annual European Symposium on Algorithms (ESA), pages 243-254, Berlin, Heidelberg, 2010. Springer Berlin Heidelberg. URL: http://dx.doi.org/10.1007/978-3-642-15775-2_21.
  15. Sathish Govindarajan, Rajiv Raman, Saurabh Ray, and Aniket Basu Roy. Packing and covering with non-piercing regions. In Piotr Sankowski and Christos Zaroliagis, editors, Proceedings of the 22nd Annual European Symposium on Algorithms (ESA), volume 57 of Leibniz International Proceedings in Informatics (LIPIcs), pages 47:1-47:17, Dagstuhl, Germany, 2016. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: http://dx.doi.org/10.4230/LIPIcs.ESA.2016.47.
  16. Sariel Har-Peled and Kent Quanrud. Approximation algorithms for polynomial-expansion and low-density graphs. In Proceedings of the 23rd Annual European Symposium on Algorithms (ESA), pages 717-728, 2015. URL: http://dx.doi.org/10.1007/978-3-662-48350-3_60.
  17. Sariel Har-Peled and Kent Quanrud. Notes on approximation algorithms for polynomial-expansion and low-density graphs, 2016. URL: http://arxiv.org/abs/1603.03098.
  18. Dorit S. Hochbaum and Wolfgang Maass. Fast approximation algorithms for a nonconvex covering problem. Journal of Algorithms, 8(3):305-323, 1987. URL: http://dx.doi.org/10.1016/0196-6774(87)90012-5.
  19. Lynn H. Loomis and Hassler Whitney. An inequality related to the isoperimetric inequality. Bulletin of the American Mathematical Society, 55(10):961-962, 1949. URL: http://dx.doi.org/10.1090/S0002-9904-1949-09320-5.
  20. Dániel Marx. Efficient approximation schemes for geometric problems? In Gerth Stølting Brodal and Stefano Leonardi, editors, Algorithms - ESA 2005: 13th Annual European Symposium, Palma de Mallorca, Spain, October 3-6, 2005. Proceedings, pages 448-459, Berlin, Heidelberg, 2005. Springer Berlin Heidelberg. URL: http://dx.doi.org/10.1007/11561071_41.
  21. Dániel Marx. Parameterized complexity of independence and domination on geometric graphs. In Hans L. Bodlaender and Michael A. Langston, editors, Parameterized and Exact Computation: Second International Workshop, IWPEC 2006, Zürich, Switzerland, September 13-15, 2006. Proceedings, pages 154-165, Berlin, Heidelberg, 2006. Springer Berlin Heidelberg. URL: http://dx.doi.org/10.1007/11847250_14.
  22. Gary L. Miller, Shang-Hua Teng, William Thurston, and Stephen A. Vavasis. Separators for sphere-packings and nearest neighbor graphs. J. ACM, 44(1):1-29, 1997. URL: http://dx.doi.org/10.1145/256292.256294.
  23. Nabil H. Mustafa and Saurabh Ray. Improved results on geometric hitting set problems. Discrete & Computational Geometry, 44(4):883-895, 12 2010. URL: http://dx.doi.org/10.1007/s00454-010-9285-9.
  24. Jaroslav Nešetřil and Patrice Ossona de Mendez. Sparsity, volume 28 of Algorithms and Combinatorics. Springer-Verlag Berlin Heidelberg, 2012. URL: http://dx.doi.org/10.1007/978-3-642-27875-4.
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