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Note on time bounds of two-phase algorithms for L-convex function minimization

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Abstract

We analyze minimization algorithms, called the two-phase algorithms, for L\(^{\natural }\)-convex functions in discrete convex analysis and derive tight bounds for the number of iterations.

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Notes

  1. A weaker statement than Theorem 1.1 is given in an unpublished technical report [13].

  2. Due to L\(^{\natural }\)-convexity, a minimal minimizer is uniquely determined if it exists.

  3. While the algorithm TwoPhaseMinMin in [14] is proposed as an algorithm for a specific L\(^{\natural }\)-convex function (i.e., Lyapunov function), the algorithm as well as its analysis can be naturally extended to general L\(^{\natural }\)-convex functions.

References

  1. Andersson, T., Erlanson, A.: Multi-item Vickrey-English-Dutch auctions. Games Econ. Behav. 81, 116–129 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ausubel, L.M.: An efficient dynamic auction for heterogeneous commodities. Amer. Econ. Rev. 96, 602–629 (2006)

    Article  Google Scholar 

  3. Blumrosen, L., Nisan, N.: Combinatorial auction. In: Nisan, N., Roughgarden, T., Tardos, É., Vazirani, V.V. (eds.) Algorithm. Game Theory, pp. 267–299. Cambridge Univ. Press, Cambridge (2007)

    Chapter  Google Scholar 

  4. Cramton, P., Shoham, Y., Steinberg, R.: Combinatorial auctions. MIT Press, Cambridge (2006)

    MATH  Google Scholar 

  5. Demange, G., Gale, D., Sotomayor, M.: Multi-item auctions. J. Polit. Econ. 94, 863–872 (1986)

    Article  Google Scholar 

  6. Gul, F., Stacchetti, E.: The English auction with differentiated commodities. J. Econom. Theory 92, 66–95 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Kelso Jr., A.S., Crawford, V.P.: Job matching, coalition formation, and gross substitutes. Econometrica 50, 1483–1504 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kolmogorov, V., Shioura, A.: New algorithms for convex cost tension problem with application to computer vision. Discrete Optim. 6, 378–393 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Mishra, D., Parkes, D.C.: Multi-item Vickrey-Dutch auctions. Games Econ. Behav. 66, 326–347 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Murota, K.: Discrete Convex Anal. SIAM, Philadelphia (2003)

    Book  MATH  Google Scholar 

  11. Murota, K.: Discrete convex analysis: a tool for economics and game theory. J. Mech. Inst. Design 1, 151–273 (2016)

    Google Scholar 

  12. Murota, K., Shioura, A.: Exact bounds for steepest descent algorithms of L-convex function minimization. Oper. Res. Lett. 42, 361–366 (2014)

    Article  MathSciNet  Google Scholar 

  13. Murota, K., Shioura, A., Yang, Z.: Computing a Walrasian equilibrium in iterative auctions with multiple differentiated items. Extended abstract version In: Cai, L., Cheng, S.-W., Lam, T.W. (eds.) Proceedings of the 24th International Symposium on Algorithms and Computation (ISAAC 2013), LNCS 8283, pp. 468–478. Springer, Berlin (2013); Full paper version in: Technical Report METR 2013-10, University of Tokyo (2013)

  14. Murota, K., Shioura, A., Yang, Z.: Time bounds for iterative auctions: a unified approach by discrete convex analysis. Discrete Optim. 19, 36–62 (2016)

    Article  MathSciNet  Google Scholar 

  15. Shioura, A.: Algorithms for L-convex function minimization: connection between discrete convex analysis and other research fields. J. Oper. Res. Soc. Japan 60 (2017) (to appear)

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Authors and Affiliations

  1. School of Business Administration, Tokyo Metropolitan University, Tokyo, 192-0397, Japan

    Kazuo Murota

  2. Department of Industrial Engineering and Economics, Tokyo Institute of Technology, Tokyo, 152-8550, Japan

    Akiyoshi Shioura

Authors
  1. Kazuo Murota
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  2. Akiyoshi Shioura
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Corresponding author

Correspondence to Akiyoshi Shioura.

Additional information

This work was supported by The Mitsubishi Foundation, CREST, JST, Grant Number JPMJCR14D2, and JSPS KAKENHI Grant Numbers 26280004, 15K00030, 15H00848.

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Murota, K., Shioura, A. Note on time bounds of two-phase algorithms for L-convex function minimization. Japan J. Indust. Appl. Math. 34, 429–440 (2017). https://doi.org/10.1007/s13160-017-0246-z

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  • DOI: https://doi.org/10.1007/s13160-017-0246-z

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