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An FPT Algorithm for Minimum Additive Spanner Problem

Author Yusuke Kobayashi

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Author Details

Yusuke Kobayashi
  • Research Institute for Mathematical Sciences, Kyoto University, Japan

Funding

  • Kobayashi, Yusuke: Supported by JST ACT-I Grant Number JPMJPR17UB, and JSPS KAKENHI Grant Numbers JP16K16010, 16H03118, JP18H05291, and JP19H05485, Japan.

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Yusuke Kobayashi. An FPT Algorithm for Minimum Additive Spanner Problem. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 11:1-11:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.STACS.2020.11

Abstract

For a positive integer t and a graph G, an additive t-spanner of G is a spanning subgraph in which the distance between every pair of vertices is at most the original distance plus t. The Minimum Additive t-Spanner Problem is to find an additive t-spanner with the minimum number of edges in a given graph, which is known to be NP-hard. Since we need to care about global properties of graphs when we deal with additive t-spanners, the Minimum Additive t-Spanner Problem is hard to handle and hence only few results are known for it. In this paper, we study the Minimum Additive t-Spanner Problem from the viewpoint of parameterized complexity. We formulate a parameterized version of the problem in which the number of removed edges is regarded as a parameter, and give a fixed-parameter algorithm for it. We also extend our result to the case with both a multiplicative approximation factor α and an additive approximation parameter β, which we call (α, β)-spanners.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
Keywords
  • Graph algorithms
  • Fixed-parameter tractability
  • Graph spanner

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References

  1. Amir Abboud and Greg Bodwin. The 4/3 additive spanner exponent is tight. J. ACM, 64(4):28:1-28:20, 2017. URL: https://doi.org/10.1145/3088511.
  2. D. Aingworth, C. Chekuri, P. Indyk, and R. Motwani. Fast estimation of diameter and shortest paths (without matrix multiplication). SIAM Journal on Computing, 28(4):1167-1181, 1999. URL: https://doi.org/10.1137/S0097539796303421.
  3. Ingo Althöfer, Gautam Das, David Dobkin, Deborah Joseph, and José Soares. On sparse spanners of weighted graphs. Discrete & Computational Geometry, 9:81-100, 1993. URL: https://doi.org/10.1007/BF02189308.
  4. Baruch Awerbuch. Complexity of network synchronization. J. ACM, 32(4):804-823, 1985. URL: https://doi.org/10.1145/4221.4227.
  5. Jørgen Bang-Jensen, Manu Basavaraju, Kristine Vitting Klinkby, Pranabendu Misra, M. S. Ramanujan, Saket Saurabh, and Meirav Zehavi. Parameterized algorithms for survivable network design with uniform demands. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'18), pages 2838-2850, 2018. URL: https://doi.org/10.1137/1.9781611975031.180.
  6. Surender Baswana and Telikepalli Kavitha. Faster algorithms for approximate distance oracles and all-pairs small stretch paths. In Proceedings of the Forty-seventh Annual IEEE Symposium on Foundations of Computer Science (FOCS'06), pages 591-602, 2006. URL: https://doi.org/10.1109/FOCS.2006.29.
  7. Surender Baswana, Telikepalli Kavitha, Kurt Mehlhorn, and Seth Pettie. Additive spanners and (α, β)-spanners. ACM Trans. Algorithms, 7(1):5:1-5:26, 2010. URL: https://doi.org/10.1145/1868237.1868242.
  8. Piotr Berman, Arnab Bhattacharyya, Konstantin Makarychev, Sofya Raskhodnikova, and Grigory Yaroslavtsev. Approximation algorithms for spanner problems and directed steiner forest. Information and Computation, 222:93-107, 2013. URL: https://doi.org/10.1016/j.ic.2012.10.007.
  9. Béla Bollobás, Don Coppersmith, and Michael Elkin. Sparse distance preservers and additive spanners. SIAM J. Discret. Math., 19(4):1029-1055, 2005. URL: https://doi.org/10.1137/S0895480103431046.
  10. Ulrik Brandes and Dagmar Handke. NP-completeness results for minimum planar spanners. In Proceedings of the 23rd International Workshop Graph-Theoretic Concepts in Computer Science (WG'97), pages 85-99, 1997. URL: https://doi.org/10.1007/BFb0024490.
  11. Leizhen Cai. NP-completeness of minimum spanner problems. Discrete Applied Mathematics, 48(2):187-194, 1994. URL: https://doi.org/10.1016/0166-218X(94)90073-6.
  12. Leizhen Cai and Derek G. Corneil. Tree spanners. SIAM J. Discrete Math., 8:359-387, 1995. URL: https://doi.org/10.1137/S0895480192237403.
  13. Leizhen Cai and Mark Keil. Spanners in graphs of bounded degree. Networks, 24(4):233-249, 1994. URL: https://doi.org/10.1002/net.3230240406.
  14. Shiri Chechik. New additive spanners. In Proceedings of the Twenty-fourth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'13), pages 498-512, 2013. URL: https://doi.org/10.1137/1.9781611973105.36.
  15. Victor D. Chepoi, Feodor F. Dragan, and Chenyu Yan. Additive sparse spanners for graphs with bounded length of largest induced cycle. Theoretical Computer Science, 347(1):54-75, 2005. URL: https://doi.org/10.1016/j.tcs.2005.05.017.
  16. Eden Chlamtáč, Michael Dinitz, Guy Kortsarz, and Bundit Laekhanukit. Approximating spanners and directed steiner forest: Upper and lower bounds. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'17), pages 534-553, 2017. URL: https://doi.org/10.1137/1.9781611974782.34.
  17. Edith Cohen. Fast algorithms for constructing t-spanners and paths with stretch t. SIAM Journal on Computing, 28(1):210-236, 1998. URL: https://doi.org/10.1137/S0097539794261295.
  18. Edith Cohen. Polylog-time and near-linear work approximation scheme for undirected shortest paths. J. ACM, 47(1):132-166, 2000. URL: https://doi.org/10.1145/331605.331610.
  19. Lenore J. Cowen and Christopher G. Wagner. Compact roundtrip routing in directed networks. Journal of Algorithms, 50(1):79-95, 2004. URL: https://doi.org/10.1016/j.jalgor.2003.08.001.
  20. M. Cygan, F.V. Fomin, L. Kowalik, D. Lokshtanov, D. Marx, M. Pilipczuk, M. Pilipczuk, and S. Saurabh. Parameterized Algorithms. Springer, 2015. Google Scholar
  21. Michael Dinitz, Guy Kortsarz, and Ran Raz. Label cover instances with large girth and the hardness of approximating basick-spanner. ACM Transactions on Algorithms, 12(2):1-16, 2016. URL: https://doi.org/10.1145/2818375.
  22. Michael Dinitz and Zeyu Zhang. Approximating low-stretch spanners. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'16), pages 821-840, 2016. URL: https://doi.org/10.1137/1.9781611974331.ch59.
  23. Feodor F. Dragan, Fedor V. Fomin, and Petr A. Golovach. Approximation of minimum weight spanners for sparse graphs. Theoretical Computer Science, 412(8):846-852, 2011. URL: https://doi.org/10.1016/j.tcs.2010.11.034.
  24. William Duckworth, Nicholas C Wormald, and Michele Zito. A PTAS for the sparsest 2-spanner of 4-connected planar triangulations. Journal of Discrete Algorithms, 1(1):67-76, 2003. URL: https://doi.org/10.1016/S1570-8667(03)00007-8.
  25. M. Elkin and D. Peleg. (1 + ε, β)-spanner constructions for general graphs. SIAM Journal on Computing, 33(3):608-631, 2004. URL: https://doi.org/10.1137/S0097539701393384.
  26. Michael Elkin. Computing almost shortest paths. ACM Trans. Algorithms, 1(2):283-323, 2005. URL: https://doi.org/10.1145/1103963.1103968.
  27. Michael Elkin and Ofer Neiman. Efficient algorithms for constructing very sparse spanners and emulators. ACM Trans. Algorithms, 15(1):4:1-4:29, 2018. URL: https://doi.org/10.1145/3274651.
  28. Michael Elkin and David Peleg. Approximating k-spanner problems for k>2. Theoretical Computer Science, 337(1):249-277, 2005. URL: https://doi.org/10.1016/j.tcs.2004.11.022.
  29. Michael Elkin and David Peleg. The hardness of approximating spanner problems. Theory of Computing Systems, 41(4):691-729, 2007. URL: https://doi.org/10.1007/s00224-006-1266-2.
  30. P. Erdős. Extremal problems in graph theory. In Theory of Graphs and Its Applications, pages 29-36, 1964. Google Scholar
  31. Jörg Flum and Martin Grohe. Parameterized Complexity Theory. Springer, 2006. Google Scholar
  32. Mathias Bæk Tejs Knudsen. Additive spanners: A simple construction. In Proceedings of the 14th Scandinavian Workshop on Algorithm Theory (SWAT'14), pages 277-281, 2014. URL: https://doi.org/10.1007/978-3-319-08404-6_24.
  33. Yusuke Kobayashi. NP-hardness and fixed-parameter tractability of the minimum spanner problem. Theoretical Computer Science, 746:88-97, 2018. URL: https://doi.org/10.1016/j.tcs.2018.06.031.
  34. Guy Kortsarz and David Peleg. Generating sparse 2-spanners. Journal of Algorithms, 17(2):222-236, 1994. URL: https://doi.org/10.1006/jagm.1994.1032.
  35. Dieter Kratsch, Hoàng-Oanh Le, Haiko Müller, Erich Prisner, and Dorothea Wagner. Additive tree spanners. SIAM Journal on Discrete Mathematics, 17(2):332-340, 2003. URL: https://doi.org/10.1137/s0895480195295471.
  36. Arthur Liestman and Thomas Shermer. Additive spanners for hypercubes. Parallel Processing Letters, 1:35-42, September 1991. URL: https://doi.org/10.1142/S0129626491000197.
  37. Arthur L. Liestman and Thomas C. Shermer. Additive graph spanners. Networks, 23:343-363, 1993. URL: https://doi.org/10.1002/net.3230230417.
  38. M.S. Madanlal, G. Venkatesan, and C. Pandu Rangan. Tree 3-spanners on interval, permutation and regular bipartite graphs. Information Processing Letters, 59(2):97-102, 1996. URL: https://doi.org/10.1016/0020-0190(96)00078-6.
  39. R. Niedermeier. Invitation to Fixed Parameter Algorithms. Oxford University Press, 2006. Google Scholar
  40. David Peleg and Alejandro A. Schäffer. Graph spanners. J. Graph Theory, 13(1):99-116, 1989. URL: https://doi.org/10.1002/jgt.3190130114.
  41. David Peleg and Jeffrey D. Ullman. An optimal synchronizer for the hypercube. SIAM J. Computing, 18(4):740-747, 1989. URL: https://doi.org/10.1137/0218050.
  42. David Peleg and Eli Upfal. A trade-off between space and efficiency for routing tables. J. ACM, 36(3):510-530, 1989. URL: https://doi.org/10.1145/65950.65953.
  43. Seth Pettie. Low distortion spanners. ACM Trans. Algorithms, 6(1):7:1-7:22, 2009. URL: https://doi.org/10.1145/1644015.1644022.
  44. Liam Roditty, Mikkel Thorup, and Uri Zwick. Deterministic constructions of approximate distance oracles and spanners. In Proceedings of the 32nd International Conference on Automata, Languages and Programming (ICALP'05), pages 261-272, 2005. URL: https://doi.org/10.1007/11523468_22.
  45. Mikkel Thorup and Uri Zwick. Approximate distance oracles. J. ACM, 52(1):1-24, 2005. URL: https://doi.org/10.1145/1044731.1044732.
  46. Mikkel Thorup and Uri Zwick. Spanners and emulators with sublinear distance errors. In Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithm (SODA'06), pages 802-809, 2006. URL: https://doi.org/10.1145/1109557.1109645.
  47. G. Venkatesan, U. Rotics, M.S. Madanlal, J.A. Makowsky, and C.Pandu Rangan. Restrictions of minimum spanner problems. Information and Computation, 136(2):143-164, 1997. URL: https://doi.org/10.1006/inco.1997.2641.
  48. David P. Woodruff. Lower bounds for additive spanners, emulators, and more. In Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06), pages 389-398, 2006. URL: https://doi.org/10.1109/FOCS.2006.45.
  49. David P. Woodruff. Additive spanners in nearly quadratic time. In Proceedings of the 37th International Colloquium Conference on Automata, Languages and Programming (ICALP'10), pages 463-474, 2010. URL: https://doi.org/10.1007/978-3-642-14165-2_40.
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