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Euclidean Prize-Collecting Steiner Forest

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Abstract

In this paper, we consider Steiner forest and its generalizations, prize-collecting Steiner forest and k-Steiner forest, when the vertices of the input graph are points in the Euclidean plane and the lengths are Euclidean distances. First, we present a simpler analysis of the polynomial-time approximation scheme (PTAS) of Borradaile et al. (Proceedings of the 49th annual IEEE symposium on foundations of computer science, FOCS, pp. 115–124, 2008) for the Euclidean Steiner forest problem. This is done by proving a new structural property and modifying the dynamic programming by adding a new piece of information to each dynamic programming state. Next we develop a PTAS for a well-motivated case, i.e., the multiplicative case, of prize-collecting and budgeted Steiner forest. The ideas used in the algorithm may have applications in design of a broad class of bicriteria PTASs; see Sect. 1.3. At the end, we demonstrate why PTASs for these problems can be hard in the general Euclidean case (and thus for PTASs we cannot go beyond the multiplicative case). In fact, this conjecture was later proved by Bateni, Hajiaghayi and Marx (1006.4339 [abs], 2009).

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References

  1. Archer, A., Bateni, M., Hajiaghayi, M., Karloff, H.: Improved approximation algorithms for prize-collecting steiner tree and TSP. In: Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science, FOCS, pp. 427–436 (2009)

    Chapter  Google Scholar 

  2. Arora, S.: Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. J. ACM 45, 753–782 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  3. Balas, E.: The prize collecting traveling salesman problem. Networks 19, 621–636 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bateni, M., Hajiaghayi, M.: Assignment problem in content distribution networks: unsplittable hard-capacitated facility location. In: Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pp. 805–814 (2009)

    Google Scholar 

  5. Bateni, M., Hajiaghayi, M.: Euclidean prize-collecting Steiner forest. In: Proceedings of the 9th Latin American Theoretical Informatics Symposium, LATIN, pp. 503–514 (2010)

    Google Scholar 

  6. Bateni, M., Hajiaghayi, M., Marx, D.: Approximation schemes for Steiner forest on planar graphs and graphs of bounded treewidth, CoRR. 0911.5143 [abs] (2009)

  7. Bateni, M., Hajiaghayi, M., Marx, D.: Prize-collecting network design on planar graphs, CoRR. 1006.4339 [abs] (2010)

  8. Bern, M., Plassmann, P.: The Steiner problem with edge lengths 1 and 2. Inf. Process. Lett. 32, 171–176 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  9. Bhaskara, A., Charikar, M., Chlamtac, E., Feige, U., Vijayaraghavan, A.: Detecting high log-densities: an O(n 1/4) approximation for densest k-subgraph. In: Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC, pp. 201–210 (2010)

    Chapter  Google Scholar 

  10. Bienstock, D., Goemans, M.X., Simchi-Levi, D., Williamson, D.: A note on the prize collecting traveling salesman problem. Math. Program. 59, 413–420 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  11. Blum, A., Ravi, R., Vempala, S.: A constant-factor approximation algorithm for the k-MST problem. J. Comput. Syst. Sci. 58, 101–108 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  12. Bonsma, P.: Sparsest cuts and concurrent flows in product graphs. Discrete Appl. Math. 136, 173–182 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  13. Borradaile, G., Mathieu, C., Klein, P.N.: A polynomial-time approximation scheme for Steiner tree in planar graphs. In: Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pp. 1285–1294 (2007)

    Google Scholar 

  14. Borradaile, G., Klein, P.N., Mathieu, C.: A polynomial-time approximation scheme for Euclidean Steiner forest. In: Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS, pp. 115–124 (2008). See the updated version available at http://www.math.uwaterloo.ca/glencora/downloads/Steiner-forest-FOCS-update.pdf

    Chapter  Google Scholar 

  15. Chekuri, C., Khanna, S., Shepherd, F.B.: Edge-disjoint paths in planar graphs. In: Proceedings of the 45th Symposium on Foundations of Computer Science, FOCS, pp. 71–80 (2004)

    Chapter  Google Scholar 

  16. Chekuri, C., Khanna, S., Shepherd, F.B.: Multicommodity flow, well-linked terminals, and routing problems. In: Proceedings of the 37th Annual ACM Symposium on Theory of Computing, STOC, pp. 183–192 (2005)

    Google Scholar 

  17. Chudak, F.A., Roughgarden, T., Williamson, D.P.: Approximate k-MSTs and k-Steiner trees via the primal-dual method and Lagrangean relaxation. In: Proceedings of the 8th International Conference on Integer Programming and Combinatorial Optimization, IPCO, pp. 60–70 (2001)

    Google Scholar 

  18. Garg, N.: Saving an epsilon: a 2-approximation for the k-MST problem in graphs. In: Proceedings of the 37th Annual ACM Symposium on Theory of Computing, STOC, pp. 396–402 (2005)

    Google Scholar 

  19. Gassner, E.: The Steiner subgraph problem revisited. Tech. Rep. 2008-17, Graz University of Technology, September 2008

  20. Goemans, M.X., Williamson, D.P.: A general approximation technique for constrained forest problems. SIAM J. Comput. 24, 296–317 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  21. Gupta, A.: Embedding tree metrics into low dimensional Euclidean spaces. Discrete Comput. Geom. 24, 105–116 (2000)

    MATH  MathSciNet  Google Scholar 

  22. Gupta, A., Hajiaghayi, M., Nagarajan, V., Ravi, R.: Dial a ride from k-forest. In: Proceedings of the 15th Annual European Symposium on Algorithms, ESA, pp. 241–252 (2007)

    Google Scholar 

  23. Hajiaghayi, M., Jain, K.: The prize-collecting generalized Steiner tree problem via a new approach of primal-dual schema. In: Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pp. 631–640 (2006)

    Chapter  Google Scholar 

  24. Hajiaghayi, M., Nasri, A.: Prize-collecting Steiner networks via iterative rounding. In: Proceedings of the 9th Latin American Theoretical Informatics Symposium, LATIN, pp. 515–526 (2010)

    Google Scholar 

  25. Jain, K., Vazirani, V.V.: Approximation algorithms for metric facility location and k-median problems using the primal-dual schema and Lagrangian relaxation. J. ACM 48, 274–296 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  26. Johnson, D.S., Minkoff, M., Phillips, S.: The prize collecting Steiner tree problem: theory and practice. In: Proceedings of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pp. 760–769 (2000)

    Google Scholar 

  27. Kolman, P., Scheideler, C.: Improved bounds for the unsplittable flow problem. In: Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pp. 184–193 (2002)

    Google Scholar 

  28. Leighton, T., Rao, S.: Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms. J. ACM 46, 787–832 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  29. Madan, R., Shah, D., Leveque, O.: Product multicommodity flow in wireless networks. IEEE Trans. Inf. Theory 54, 1460–1476 (2008)

    Article  MathSciNet  Google Scholar 

  30. Mitchell, J.C.: Guillotine subdivisions approximate polygonal subdivisions: a simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems. SIAM J. Comput. 28, 1298–1309 (1995)

    Article  Google Scholar 

  31. Ravi, R., Sundaram, R., Marathe, M.V., Rosenkrantz, D.J., Ravi, S.S.: Spanning trees—short or small. SIAM J. Discrete Math. 9, 178–200 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  32. Robins, G., Zelikovsky, A.: Tighter bounds for graph Steiner tree approximation. SIAM J. Discrete Math. 19, 122–134 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  33. Salman, F.S., Cheriyan, J., Ravi, R., Subramanian, S.: Approximating the single-sink link-installation problem in network design. SIAM J. Optim. 11, 595–610 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  34. Sharma, Y., Swamy, C., Williamson, D.P.: Approximation algorithms for prize collecting forest problems with submodular penalty functions. In: Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pp. 1275–1284 (2007)

    Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, Princeton University, Princeton, NJ, 08540, USA

    MohammadHossein Bateni

  2. Center for Computational Intractability, Princeton, NJ, USA

    MohammadHossein Bateni

  3. Computer Science Department, University of Maryland, College Park, MD, 20742, USA

    MohammadTaghi Hajiaghayi

  4. AT&T Labs—Research, Florham Park, NJ, 07932, USA

    MohammadTaghi Hajiaghayi

Authors
  1. MohammadHossein Bateni
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  2. MohammadTaghi Hajiaghayi
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Correspondence to MohammadHossein Bateni.

Additional information

A short version of this paper appears in Proceedings of LATIN 2010 [5].

The first author was supported by a Gordon Wu fellowship, a Charlotte Elizabeth Procter fellowship as well as NSF ITR grants CCF-0205594, CCF-0426582 and NSF CCF 0832797, NSF CAREER award CCF-0237113, MSPA-MCS award 0528414, NSF expeditions award 0832797.

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Bateni, M., Hajiaghayi, M. Euclidean Prize-Collecting Steiner Forest. Algorithmica 62, 906–929 (2012). https://doi.org/10.1007/s00453-011-9491-8

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