Distributed Approximation of k-Service Assignment

Authors Magnús M. Halldórsson, Sven Köhler, Dror Rawitz



PDF
Thumbnail PDF

File

LIPIcs.OPODIS.2015.11.pdf
  • Filesize: 0.56 MB
  • 16 pages

Document Identifiers

Author Details

Magnús M. Halldórsson
Sven Köhler
Dror Rawitz

Cite As Get BibTex

Magnús M. Halldórsson, Sven Köhler, and Dror Rawitz. Distributed Approximation of k-Service Assignment. In 19th International Conference on Principles of Distributed Systems (OPODIS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 46, pp. 11:1-11:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.OPODIS.2015.11

Abstract

We consider the k-Service Assignment problem (k-SA), defined as follows. The input consists of a network that contains servers and clients, and an integer k. Each server has a finite capacity, and each client is associated with a demand and a profit. A feasible solution is an assignment of clients to neighboring servers such that (i) the total demand assigned to a server is at most its capacity, and (ii) a client is assigned either to k servers or to none. The profit of an assignment is the total profit of clients that are assigned to k servers, and the goal is to find a maximum profit assignment. In the r-restricted version of k-SA, no client requires more than an r-fraction of the capacity of any adjacent server. The k-SA problem is motivated by backup placement in networks and by resource allocation in 4G cellular networks. It can also be viewed as machine scheduling on related machines with assignment restrictions. 

We present a centralized polynomial time greedy (k+1-r)/(1-r)-approximation algorithm for r-restricted k-SA. We then show that a variant of this algorithm achieves an approximation ratio of k+1 using a resource augmentation factor of 1+r. We use the latter to present a (k+1)^2-approximation algorithm for k-SA. In the distributed setting, we present: (i) a (1+epsilon)*(k +1-r)/(1-r)-approximation algorithm for r-restricted k-SA, (ii) a (1+epsilon)(k+1)-approximation algorithm that uses a resource augmentation factor of 1+r for r-restricted k-SA, both for any constant epsilon>0, and (iii) an O{k^2}-approximation algorithm for k-SA (in expectation). The three distributed algorithms compute a solution with high probability and terminate in O(k^2 *log^3(n)) rounds.

Subject Classification

Keywords
  • approximation algorithms
  • distributed algorithms
  • related machines

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. David Amzallag, Reuven Bar-Yehuda, Danny Raz, and Gabriel Scalosub. Cell selection in 4G cellular networks. IEEE Trans. Mobile Comput., 12(7):1443-1455, 2013. Google Scholar
  2. Baruch Awerbuch. Complexity of network synchronization. J. ACM, 32(4):804-823, 1985. Google Scholar
  3. Amotz Bar-Noy, Reuven Bar-Yehuda, Ari Freund, Joseph Naor, and Baurch Schieber. A unified approach to approximating resource allocation and schedualing. J. ACM, 48(5):1069-1090, 2001. Google Scholar
  4. Reuven Bar-Yehuda, Keren Bendel, Ari Freund, and Dror Rawitz. Local ratio: A unified framework for approximation algorithms. ACM Comput. Surv., 36(4):422-463, 2004. Google Scholar
  5. Reuven Bar-Yehuda and Shimon Even. A local-ratio theorem for approximating the weighted vertex cover problem. Annals of Discrete Mathematics, 25:27-46, 1985. Google Scholar
  6. Chandra Chekuri and Sanjeev Khanna. On multidimensional packing problems. SIAM J. Comput., 33(4):837-851, 2004. Google Scholar
  7. Chandra Chekuri and Sanjeev Khanna. A polynomial time approximation scheme for the multiple knapsack problem. SIAM J. Comput., 35(3):713-728, 2005. Google Scholar
  8. Milind Dawande, Jayant Kalagnanam, Pinar Keskinocak, F. Sibel Salman, and R. Ravi. Approximation algorithms for the multiple knapsack problem with assignment restrictions. Journal of Combinatorial Optimization, 4(2):171-186, 2000. Google Scholar
  9. Yuval Emek, Magnús M. Halldórsson, Yishay Mansour, Boaz Patt-Shamir, Jaikumar Radhakrishnan, and Dror Rawitz. Online set packing. SIAM J. Comput., 41(4):728-746, 2012. Google Scholar
  10. Paul Erdös and András Hajnal. On chromatic number of graphs and set-systems. Acta Mathematica Hungarica, 17(1-2):61-99, 1966. Google Scholar
  11. Lisa Fleischer, Michel X. Goemans, Vahab S. Mirrokni, and Maxim Sviridenko. Tight approximation algorithms for maximum general assignment problems. In 17th SODA, pages 611-620, 2006. Google Scholar
  12. A. M. Frieze and M. R. B. Clarke. Approximation algorithms for the m-dimensional 0-1 knapsack problem: worst-case and probabilistic analyses. European Journal of Operational Research, 15:100-109, 1984. Google Scholar
  13. Magnús M. Halldórsson, Sven Köhler, Boaz Patt-Shamir, and Dror Rawitz. Distributed backup placement in networks. In 27th ACM SPAA, pages 274-283, 2015. Google Scholar
  14. Elad Hazan, Shmuel Safra, and Oded Schwartz. On the complexity of approximating k-set packing. Computational Complexity, 15(1):20-39, 2006. Google Scholar
  15. Oscar H. Ibarra and Chul E. Kim. Fast approximation algorithms for the knapsack and sum of subset problems. J. ACM, 22(4):463-468, 1975. Google Scholar
  16. Michael Luby. A simple parallel algorithm for the maximal independent set problem. SIAM J. Comput., 15(4):1036-1053, 1986. Google Scholar
  17. Michael J. Magazine and Maw-Sheng Chern. A note on approximation schemes for multidimensional knapsack problems. Mathematics of Operations Research, 9(2):244-247, 1984. Google Scholar
  18. Boaz Patt-Shamir, Dror Rawitz, and Gabriel Scalosub. Distributed approximation of cellular coverage. J. Parallel Distrib. Comput., 72(3):402-408, 2012. Google Scholar
  19. David Peleg. Distributed Computing: A Locality-sensitive Approach. SIAM, 2000. Google Scholar
  20. Prabhakar Raghavan and Clark D. Thompson. Randomized rounding: a technique for provably good algorithms and algorithmic proofs. Combinatorica, 7(4):365-374, 1987. Google Scholar
  21. Sartaj Sahni. Approximate algorithms for the 0/1 knapsack problem. J. ACM, 22(1):115-124, 1975. Google Scholar
  22. David B. Shmoys and Éva Tardos. An approximation algorithm for the generalized assignment problem. Mathematical Programming, 62:461-474, 1993. Google Scholar
  23. Aravind Srinivasan. Improved approximation guarantees for packing and covering integer programs. SIAM J. Comput., 29(2):648-670, 1999. Google Scholar
  24. Roger Wattenhofer. Principles of distributed computing: Maximal independent set. http://www.dcg.ethz.ch/lectures/fs15/podc/lecture/chapter7.pdf. Accessed 2015-08-27.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail