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A large class of facets for the K-median polytope

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Abstract

The polyhedral structure of the K-median problem is examined. We present an extended formulation that is integral but grows exponentially with the number of nodes. Then, some extra variables are projected out. Based on the reduced formulation, we develop two basic properties for facets of K-median problem. By applying the two properties, we generalize two known classes of facets, de Vries facets and de Farias facets. The computational study illustrates that the generalization is significant.

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References

  1. Avella P., Sassano A.: On the p-median polytope. Math. Program. 89, 395–411 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  2. Balas E.: Projection and lifting in combinatorial optimization. In: Jünger, M., Naddef, D. (eds) Computational Combinatorial Optimization, LNCS 2241, pp. 26–56. Springer, Berlin (2001)

    Chapter  Google Scholar 

  3. Bazaraa M.S., Jarvis J.J., Sherali H.D.: Linear Programming and Network Flows. 3rd edn. Wiley, NJ (2005)

    MATH  Google Scholar 

  4. de Farias I.R.: A family of facets for the p-median polytope. Oper. Res. Lett. 28, 161–167 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. de Vries S., Posner M.E., Vohra R.V.: Polyhedral properties of the K-median problem on a tree. Math. Program. 110, 261–285 (2003)

    Article  Google Scholar 

  6. Goemans M.: Notes on the Median Polytope on Trees, Notes. Massachusetts Institute of Technology, Cambridge (1992)

    Google Scholar 

  7. Hakimi S.: Optimum locations of switching centres and the absolute centres and medians on a graph. Oper. Res. 12, 450–459 (1964)

    Article  MATH  Google Scholar 

  8. Hsu W.L.: The distance-domination numbers of trees. Oper. Res. Lett. 1, 96–100 (1982)

    Article  MATH  Google Scholar 

  9. Kariv O., Hakimi S.L.: An algorithmic approach to network location problems II: the p-medians. SIAM J. Appl. Math. 37, 539–560 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  10. Tamir A.: An O(pn 2) algorithm for the p-median and related Problems on tree graphs. Oper. Res. Lett. 19, 59–64 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. Zhao, W., Posner, M.E.: New Classes of Facets for the K-median Polytope, Working Paper, The Ohio State University, Columbus, OH (2007)

  12. Zhao, W., Posner M.E.: Valid Inequalities for the K-median Problem on a Tree, Working Paper, The Ohio State University, Columbus, OH (2007)

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Authors and Affiliations

  1. Olin Business School, Washington University in St. Louis, 1 Brookings Dr, St. Louis, MO, 63130, USA

    Wenhui Zhao

  2. Department of Integrated Systems Engineering, The Ohio State University, 1971 Neil Avenue, Columbus, OH, 43210, USA

    Marc E. Posner

Authors
  1. Wenhui Zhao
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  2. Marc E. Posner
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Correspondence to Wenhui Zhao.

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Zhao, W., Posner, M.E. A large class of facets for the K-median polytope. Math. Program. 128, 171–203 (2011). https://doi.org/10.1007/s10107-009-0301-x

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  • DOI: https://doi.org/10.1007/s10107-009-0301-x

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