Advertisement

Theory of Computing Systems

, Volume 49, Issue 4, pp 672–697 | Cite as

Local Approximability of Max-Min and Min-Max Linear Programs

  • Patrik Floréen
  • Marja Hassinen
  • Joel Kaasinen
  • Petteri Kaski
  • Topi Musto
  • Jukka Suomela
Article

Abstract

In a max-min LP, the objective is to maximise ω subject to A x1, C xω 1, and x0. In a min-max LP, the objective is to minimise ρ subject to A xρ 1, C x1, and x0. The matrices A and C are nonnegative and sparse: each row a i of A has at most Δ I positive elements, and each row c k of C has at most Δ K positive elements.

We study the approximability of max-min LPs and min-max LPs in a distributed setting; in particular, we focus on local algorithms (constant-time distributed algorithms). We show that for any Δ I ≥2, Δ K ≥2, and ε>0 there exists a local algorithm that achieves the approximation ratio Δ I (1−1/Δ K )+ε. We also show that this result is the best possible: no local algorithm can achieve the approximation ratio Δ I (1−1/Δ K ) for any Δ I ≥2 and Δ K ≥2.

Keywords

Approximation algorithms Distributed algorithms Linear programs Local algorithms 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Angluin, D.: Local and global properties in networks of processors. In: Proc. 12th Annual ACM Symposium on Theory of Computing (STOC), Los Angeles, CA, USA, April 1980, pp. 82–93. ACM, New York (1980) Google Scholar
  2. 2.
    Åstrand, M., Floréen, P., Polishchuk, V., Rybicki, J., Suomela, J., Uitto, J.: A local 2-approximation algorithm for the vertex cover problem. In: Proc. 23rd International Symposium on Distributed Computing (DISC), Elche, Spain, September 2009. Lecture Notes in Computer Science, vol. 5805, pp. 191–205. Springer, Berlin (2009) Google Scholar
  3. 3.
    Åstrand, M., Suomela, J.: Fast distributed approximation algorithms for vertex cover and set cover in anonymous networks. In: Proc. 22nd Annual ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), Santorini, Greece, June 2010, pp. 294–302. ACM, New York (2010) Google Scholar
  4. 4.
    Awerbuch, B., Sipser, M.: Dynamic networks are as fast as static networks. In: Proc. 29th Annual Symposium on Foundations of Computer Science (FOCS), White Plains, NY, USA, October 1988, pp. 206–219. IEEE, Piscataway (1988). Google Scholar
  5. 5.
    Awerbuch, B., Varghese, G.: Distributed program checking: a paradigm for building self-stabilizing distributed protocols. In: Proc. 32nd Annual Symposium on Foundations of Computer Science (FOCS), San Juan, Puerto Rico, October 1988, pp. 258–267. IEEE, Piscataway (1991). CrossRefGoogle Scholar
  6. 6.
    Baker, B.S.: Approximation algorithms for NP-complete problems on planar graphs. J. ACM 41(1), 153–180 (1994) zbMATHCrossRefGoogle Scholar
  7. 7.
    Czygrinow, A., Hańćkowiak, M., Wawrzyniak, W.: Fast distributed approximations in planar graphs. In: Proc. 22nd International Symposium on Distributed Computing (DISC), Arcachon, France, September 2008. Lecture Notes in Computer Science, vol. 5218, pp. 78–92. Springer, Berlin (2008) Google Scholar
  8. 8.
    Floréen, P., Hassinen, M., Kaski, P., Suomela, J.: Tight local approximation results for max-min linear programs. In: Proc. 4th International Workshop on Algorithmic Aspects of Wireless Sensor Networks (Algosensors), Reykjavík, Iceland, July 2008. Lecture Notes in Computer Science, vol. 5389, pp. 2–17. Springer, Berlin (2008) CrossRefGoogle Scholar
  9. 9.
    Floréen, P., Kaasinen, J., Kaski, P., Suomela, J.: An optimal local approximation algorithm for max-min linear programs. In: Proc. 21st Annual ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), Calgary, Canada, August 2009, pp. 260–269. ACM, New York (2009) Google Scholar
  10. 10.
    Floréen, P., Kaski, P., Musto, T., Suomela, J.: Approximating max-min linear programs with local algorithms. In: Proc. 22nd IEEE International Parallel and Distributed Processing Symposium (IPDPS), Miami, FL, USA, April 2008. IEEE, Piscataway (2008) Google Scholar
  11. 11.
    Floréen, P., Kaski, P., Polishchuk, V., Suomela, J.: Almost stable matchings by truncating the Gale–Shapley algorithm. Algorithmica 58(1), 102–118 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Hańćkowiak, M., Karoński, M., Panconesi, A.: On the distributed complexity of computing maximal matchings. In: Proc. 9th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), San Francisco, CA, USA, January 1998, pp. 219–225. Society for Industrial and Applied Mathematics, Philadelphia (1998) Google Scholar
  13. 13.
    Hochbaum, D.S., Maass, W.: Approximation schemes for covering and packing problems in image processing and VLSI. J. ACM 32(1), 130–136 (1985) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Hoory, S.: On graphs of high girth. PhD thesis, Hebrew University, Jerusalem (March 2002) Google Scholar
  15. 15.
    Kuhn, F.: The price of locality: exploring the complexity of distributed coordination primitives. PhD thesis, ETH Zürich (2005) Google Scholar
  16. 16.
    Kuhn, F., Moscibroda, T., Wattenhofer, R.: Fault-tolerant clustering in ad hoc and sensor networks. In: Proc. 26th IEEE International Conference on Distributed Computing Systems (ICDCS), Lisboa, Portugal, July 2006. IEEE Computer Society Press, Los Alamitos (2006) Google Scholar
  17. 17.
    Kuhn, F., Moscibroda, T., Wattenhofer, R.: The price of being near-sighted. In: Proc. 17th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), Miami, FL, USA, January 2006, pp. 980–989. ACM, New York (2006) CrossRefGoogle Scholar
  18. 18.
    Lenzen, C., Oswald, Y.A., Wattenhofer, R.: What can be approximated locally? In: Proc. 20th Annual ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), Munich, Germany, June 2008, pp. 46–54. ACM, New York (2008) Google Scholar
  19. 19.
    Lenzen, C., Suomela, J., Wattenhofer, R.: Local algorithms: self-stabilization on speed. In: Proc. 11th International Symposium on Stabilization, Safety, and Security of Distributed Systems (SSS), Lyon, France, November 2009. Lecture Notes in Computer Science, vol. 5873, pp. 17–34. Springer, Berlin (2009) CrossRefGoogle Scholar
  20. 20.
    Lenzen, C., Wattenhofer, R.: Leveraging Linial’s locality limit. In: Proc. 22nd International Symposium on Distributed Computing (DISC), Arcachon, France, September 2008. Lecture Notes in Computer Science, vol. 5218, pp. 394–407. Springer, Berlin (2008) Google Scholar
  21. 21.
    Linial, N.: Locality in distributed graph algorithms. SIAM J. Comput. 21(1), 193–201 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Mayer, A., Naor, M., Stockmeyer, L.: Local computations on static and dynamic graphs. In: Proc. 3rd Israel Symposium on the Theory of Computing and Systems (ISTCS), Tel Aviv, Israel, January 1995, pp. 268–278. IEEE, Piscataway (1995). CrossRefGoogle Scholar
  23. 23.
    Moscibroda, T.: Locality, scheduling, and selfishness: algorithmic foundations of highly decentralized networks. PhD thesis, ETH Zürich (2006) Google Scholar
  24. 24.
    Naor, M., Stockmeyer, L.: What can be computed locally? SIAM J. Comput. 24(6), 1259–1277 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Papadimitriou, C.H., Yannakakis, M.: Linear programming without the matrix. In: Proc. 25th Annual ACM Symposium on Theory of Computing (STOC), San Diego, CA, USA, May 1993, pp. 121–129. ACM, New York (1993) Google Scholar
  26. 26.
    Polishchuk, V., Suomela, J.: A simple local 3-approximation algorithm for vertex cover. Inf. Process. Lett. 109(12), 642–645 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Suomela, J.: Optimisation problems in wireless sensor networks: local algorithms and local graphs. PhD thesis, University of Helsinki, Department of Computer Science, Helsinki, Finland (May 2009) Google Scholar
  28. 28.
    Suomela, J.: Survey of local algorithms. http://www.iki.fi/jukka.suomela/local-survey (2009). Manuscript submitted for publication
  29. 29.
    Young, N.E.: Sequential and parallel algorithms for mixed packing and covering. In: Proc. 42nd Annual Symposium on Foundations of Computer Science (FOCS), Las Vegas, NV, USA, October 2001, pp. 538–546. IEEE Computer Society Press, Los Alamitos (2001) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Patrik Floréen
    • 1
  • Marja Hassinen
    • 1
  • Joel Kaasinen
    • 1
  • Petteri Kaski
    • 1
  • Topi Musto
    • 1
  • Jukka Suomela
    • 1
  1. 1.Helsinki Institute for Information Technology HIITUniversity of HelsinkiHelsinkiFinland

Personalised recommendations