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Faster Algorithms for Semi-matching Problems (Extended Abstract)

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Automata, Languages and Programming (ICALP 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6198))

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Abstract

We consider the problem of finding semi-matching in bipartite graphs which is also extensively studied under various names in the scheduling literature. We give faster algorithms for both weighted and unweighted case.

For the weighted case, we give an O(nmlogn)-time algorithm, where n is the number of vertices and m is the number of edges, by exploiting the geometric structure of the problem. This improves the classical O(n 3) algorithms by Horn [Operations Research 1973] and Bruno, Coffman and Sethi [Communications of the ACM 1974].

For the unweighted case, the bound could be improved even further. We give a simple divide-and-conquer algorithm which runs in \(O(\sqrt{n}m\log n)\) time, improving two previous O(nm)-time algorithms by Abraham [MSc thesis, University of Glasgow 2003] and Harvey, Ladner, Lovász and Tamir [WADS 2003 and Journal of Algorithms 2006]. We also extend this algorithm to solve the Balance Edge Cover problem in \(O(\sqrt{n}m\log n)\) time, improving the previous O(nm)-time algorithm by Harada, Ono, Sadakane and Yamashita [ISAAC 2008].

Full version can be found in [9].

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References

  1. Abraham, D.: Algorithmics of two-sided matching problems. Master’s thesis, Department of Computer Science, University of Glasgow (2003)

    Google Scholar 

  2. Blażewicz, J., Ecker, K., Pesch, E., Schmidt, G., Weglarz, J.: Handbook on scheduling: from theory to applications

    Google Scholar 

  3. Bokal, D., Bresar, B., Jerebic, J.: A generalization of hungarian method and hall’s theorem with applications in wireless sensor networks. arXiv/0911.1269 (2009)

    Google Scholar 

  4. Bruno, J.L., Coffman Jr., E.G., Sethi, R.: Algorithms for minimizing mean flow time. In: IFIP Congress, pp. 504–510 (1974)

    Google Scholar 

  5. Bruno, J.L., Coffman Jr., E.G., Sethi, R.: Scheduling independent tasks to reduce mean finishing time. ACM Commun. 17(7), 382–387 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  6. Dinic, E.A.: Algorithm for solution of a problem of maximum flow in networks with power estimation. Soviet Mathematics Doklady 11, 1277–1280 (1970)

    Google Scholar 

  7. Dinitz, Y.: Dinitz’ algorithm: The original version and even’s version. In: Goldreich, O., Rosenberg, A.L., Selman, A.L. (eds.) Theoretical Computer Science. LNCS, vol. 3895, pp. 218–240. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  8. Edmonds, J., Karp, R.M.: Theoretical improvements in algorithmic efficiency for network flow problems. In: Combinatorical Structures and Their Applications, pp. 93–96. Gordon and Breach, New York (1970)

    Google Scholar 

  9. Fakcharoenphol, J., Lekhanukit, B., Nanongkai, D.: Faster algorithms for semi-matching problems. arXiv/1004.3363 (2010)

    Google Scholar 

  10. Graham, R.L., Lawler, E.L., Lenstra, J.K., Kan, A.H.G.R.: Optimization and approximation in deterministic sequencing and scheduling: a survey. Annals of Discrete Mathematics 5(2), 287–326 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  11. Harada, Y., Ono, H., Sadakane, K., Yamashita, M.: Optimal balanced semi-matchings for weighted bipartite graphs. IPSJ Digital Courier 3, 693–702 (2007)

    Article  Google Scholar 

  12. Harada, Y., Ono, H., Sadakane, K., Yamashita, M.: The balanced edge cover problem. In: Hong, S.-H., Nagamochi, H., Fukunaga, T. (eds.) ISAAC 2008. LNCS, vol. 5369, pp. 246–257. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  13. Harvey, N.J.A.: Semi-matchings for bipartite graphs and load balancing (slide) (July 2003), http://people.csail.mit.edu/nickh/Publications/SemiMatching/Semi-Matching.ppt

  14. Harvey, N.J.A., Ladner, R.E., Lovász, L., Tamir, T.: Semi-matchings for bipartite graphs and load balancing. In: Dehne, F., Sack, J.-R., Smid, M. (eds.) WADS 2003. LNCS, vol. 2748, pp. 53–78. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  15. Horn, W.A.: Minimizing average flow time with parallel machines. Operations Research, 846–847 (1973)

    Google Scholar 

  16. Kao, M.-Y., Lam, T.W., Sung, W.-K., Ting, H.-F.: An even faster and more unifying algorithm for comparing trees via unbalanced bipartite matchings. J. Algorithms 40(2), 212–233 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kothari, A., Suri, S., Tóth, C.D., Zhou, Y.: Congestion games, load balancing, and price of anarchy. In: López-Ortiz, A., Hamel, A.M. (eds.) CAAN 2004. LNCS, vol. 3405, pp. 13–27. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  18. Lee, K., Leung, J.Y.-T., Pinedo, M.L.: Scheduling jobs with equal processing times subject to machine eligibility constraints. Working Paper (2009)

    Google Scholar 

  19. Lee, K., Leung, J.Y.T., Pinedo, M.L.: A note on an approximation algorithm for the load-balanced semi-matching problem in weighted bipartite graphs. Inf. Process. Lett. 109(12), 608–610 (2009)

    Article  MathSciNet  Google Scholar 

  20. Leung, J.Y.-T., Li, C.-L.: Scheduling with processing set restrictions: A survey. International Journal of Production Economics 116(2), 251–262 (2008)

    MathSciNet  Google Scholar 

  21. Li, C.-L.: Scheduling unit-length jobs with machine eligibility restrictions. European Journal of Operational Research 174(2), 1325–1328 (2006)

    Article  MATH  Google Scholar 

  22. Lin, Y., Li, W.: Parallel machine scheduling of machine-dependent jobs with unit-length. European Journal of Operational Research 156(1), 261–266 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  23. Logendran, R., Subur, F.: Unrelated parallel machine scheduling with job splitting. IIE Transactions 36(4), 359–372 (2004)

    Article  Google Scholar 

  24. Lovász, L.: The membership problem in jump systems. J. Comb. Theory, Ser. B 70(1), 45–66 (1997)

    Article  MATH  Google Scholar 

  25. Low, C.P.: An efficient retrieval selection algorithm for video servers with random duplicated assignment storage technique. Inf. Process. Lett. 83(6), 315–321 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  26. Low, C.P.: An approximation algorithm for the load-balanced semi-matching problem in weighted bipartite graphs. Inf. Process. Lett. 100(4), 154–161 (2006) (also appeared in TAMC 2006)

    Article  MATH  MathSciNet  Google Scholar 

  27. Machado, R., Tekinay, S.: A survey of game-theoretic approaches in wireless sensor networks. Computer Networks 52(16), 3047–3061 (2008)

    Article  MATH  Google Scholar 

  28. Pinedo, M.: Scheduling: Theory, Algorithms, and Systems, 2nd edn. Prentice Hall, Englewood Cliffs (2001)

    Google Scholar 

  29. Sadagopan, N., Singh, M., Krishnamachari, B.: Decentralized utility-based sensor network design. Mob. Netw. Appl. 11(3), 341–350 (2006)

    Article  Google Scholar 

  30. Sitters, R.: Two NP-hardness results for preemptive minsum scheduling of unrelated parallel machines. In: Aardal, K., Gerards, B. (eds.) IPCO 2001. LNCS, vol. 2081, pp. 396–405. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  31. Spyropoulos, C.D., Evans, D.J.: Analysis of the q.a.d. algorithm for an homogeneous multiprocessor computing model with independent memories. International Journal of Computer Mathematics, 237–255

    Google Scholar 

  32. Su, L.H.: Scheduling on identical parallel machines to minimize total completion time with deadline and machine eligibility constraints. The International Journal of Advanced Manufacturing Technology 40(5), 572–581 (2009)

    Article  Google Scholar 

  33. Suri, S., Tóth, C.D., Zhou, Y.: Uncoordinated load balancing and congestion games in p2p systems. In: IPTPS, pp. 123–130 (2004)

    Google Scholar 

  34. Suri, S., Tóth, C.D., Zhou, Y.: Selfish load balancing and atomic congestion games. Algorithmica 47(1), 79–96 (2007) (also appeared in SPAA 2004)

    Article  MATH  MathSciNet  Google Scholar 

  35. Tamir, A.: Least majorized elements and generalized polymatroids. Mathematics of Operations Research, 583–589 (1995)

    Google Scholar 

  36. Tamir, T., Vaksendiser, B.: Algorithms for storage allocation based on client preferences. In: International Symposium on Combinatorial Optimization (2008)

    Google Scholar 

  37. Tarjan, R.E.: Data structures and network algorithms. Society for Industrial and Applied Mathematics (1983)

    Google Scholar 

  38. Tomizawa, N.: On some techniques useful for solution of transportation network problems. In: Networks 1, pp. 173–194 (1971)

    Google Scholar 

  39. Vaik, Z.: On scheduling problems with parallel multi-purpose machines. Technical report, Technical Reports, Egervary Research Group on Combinatorial Optimization, Hungary (2005), http://www.cs.elte.hu/egres/tr/egres-05-02.pdf

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Author information

Authors and Affiliations

  1. Kasetsart University, Bangkok, 10900, Thailand

    Jittat Fakcharoenphol

  2. University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Bundit Laekhanukit

  3. Georgia Institute of Technology, Atlanta, GA, 30332

    Danupon Nanongkai

Authors
  1. Jittat Fakcharoenphol
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  2. Bundit Laekhanukit
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  3. Danupon Nanongkai
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Editor information

Editors and Affiliations

  1. Oxford University Computing Laboratory, Wolfson Building, Parks Road, OX1 3QD, Oxford, UK

    Samson Abramsky

  2. Université de Bordeaux (LaBRI) & INRIA, 351, cours de la Libération, 33405, Talence Cedex, France

    Cyril Gavoille

  3. INRIA, Centre de Recherche Bordeaux – Sud-Ouest, 351 cours de la Libération, 33405, Talence Cedex, France

    Claude Kirchner

  4. Heinz Nixdorf Institute, University of Paderborn, Fürstenallee 11, 33102, Paderborn, Germany

    Friedhelm Meyer auf der Heide

  5. University of Patras and RACTI, 26500, Patras, Greece

    Paul G. Spirakis

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Fakcharoenphol, J., Laekhanukit, B., Nanongkai, D. (2010). Faster Algorithms for Semi-matching Problems (Extended Abstract). In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds) Automata, Languages and Programming. ICALP 2010. Lecture Notes in Computer Science, vol 6198. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14165-2_16

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  • DOI: https://doi.org/10.1007/978-3-642-14165-2_16

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-14164-5

  • Online ISBN: 978-3-642-14165-2

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