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Message Passing Algorithm for the Generalized Assignment Problem

  • Mindi Yuan
  • Chong Jiang
  • Shen Li
  • Wei Shen
  • Yannis Pavlidis
  • Jun Li
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8707)

Abstract

The generalized assignment problem (GAP) is NP-hard. It is even APX-hard to approximate it. The best known approximation algorithm is the LP-rounding algorithm in [1] with a \((1-\frac{1}{e})\) approximation ratio. We investigate the max-product belief propagation algorithm for the GAP, which is suitable for distributed implementation. The basic algorithm passes an exponential number of real-valued messages in each iteration. We show that the algorithm can be simplified so that only a linear number of real-valued messages are passed in each iteration. In particular, the computation of the messages from machines to jobs decomposes into two knapsack problems, which are also present in each iteration of the LP-rounding algorithm. The messages can be computed in parallel at each iteration. We observe that for small instances of GAP where the optimal solution can be computed, the message passing algorithm converges to the optimal solution when it is unique. We then show how to add small deterministic perturbations to ensure the uniqueness of the optimum. Finally, we prove GAP remains strongly NP-hard even if the optimum is unique.

References

  1. 1.
    Fleischer, L., Goemans, M.X., Mirrokni, V.S., Sviridenko, M.: Tight approximation algorithms for maximum separable assignment problems. Math. of Operations Research 36, 416–431 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Wikipedia, Generalized assignment problem, http://en.wikipedia.org/wiki/Generalized_assignment_problem
  3. 3.
    Shmoys, D.B., Tardos, E.: An approximation algorithm for the generalized assignment problem. Math. Program 62, 461–474 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Chekuri, C., Khanna, S.: A PTAS for the multiple knapsack problem. In: Proceedings of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 213–222 (2000)Google Scholar
  5. 5.
    Cohen, R., Katzir, L., Raz, D.: An efficient approximation for the generalized assignment problem. Info. Processing Letters 100, 162–166 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Koller, D., Friedman, N.: Probabilistic Graphical Models: Principles and Techniques. MIT Press, USA (2009)Google Scholar
  7. 7.
    Yuan, M., Li, S., Shen, W., Pavlidis, Y.: Belief propagation for minimax weight matching. University of Illinois, Tech. Rep (2013)Google Scholar
  8. 8.
    Yuan, M., Shen, W., Li, J., Pavlidis, Y., Li, S.: Auction/belief propagation algorithms for constrained assignment problem. Walmart Labs, Tech. Rep. (2013)Google Scholar
  9. 9.
    Bayati, M., Shah, D., Sharma, M.: Max-product for maximum weight matching: convergence, correctness, and LP duality. IEEE Trans. Info. Theory 54, 1241–1251 (2008)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Bayati, M., Borgs, C., Chayes, J., Zecchina, R.: Belief propagation for weighted b-matchings on arbitrary graphs and its relation to linear programs with integer solutions. SIAM J. Discrete Math. 25, 989–1011 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Sanghavi, S.: Equivalence of LP relaxation and max-product for weighted matching in general graphs. In: IEEE Info. Theory Workshop, pp. 242–247 (2007)Google Scholar
  12. 12.
    Wikipedia, Knapsack problem, http://en.wikipedia.org/wiki/Knapsack_problem
  13. 13.
  14. 14.
  15. 15.
    Garey, M.R., Johnson, D.S.: Strong np-completeness results: Motivation, examples, and implications. Journal of the ACM 25, 499–508 (1978)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© IFIP International Federation for Information Processing 2014

Authors and Affiliations

  • Mindi Yuan
    • 1
  • Chong Jiang
    • 1
  • Shen Li
    • 1
  • Wei Shen
    • 1
  • Yannis Pavlidis
    • 1
  • Jun Li
    • 1
  1. 1.Walmart Labs and University of Illinois at Urbana-ChampaignUSA

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