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Finding a core of a tree with pos/neg weight

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  • Mehdi Zaferanieh
  • Jafar Fathali

Abstract

Let T = (V, E) be a tree. A core of T is a path P, for which the sum of the weighted distances from all vertices to this path is minimized. In this paper, we consider the semi-obnoxious case in which the vertices have positive or negative weights. We prove that, when the sum of the weights of vertices is negative, the core must be a single vertex and that, when the sum of the vertices’ weights is zero there exists a core that is a vertex. Morgan and Slater (J Algorithms 1:247–258, 1980 ) presented a linear time algorithm to find the core of a tree with only positive weights of vertices. We show that their algorithm also works for semi-obnoxious problems. Copyright Springer-Verlag 2012

Suggested Citation

  • Mehdi Zaferanieh & Jafar Fathali, 2012. "Finding a core of a tree with pos/neg weight," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 76(2), pages 147-160, October.
    • Handle: RePEc:spr:mathme:v:76:y:2012:i:2:p:147-160
    DOI: 10.1007/s00186-012-0394-5
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    References listed on IDEAS

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    1. S. S. Ting, 1984. "A Linear-Time Algorithm for Maxisum Facility Location on Tree Networks," Transportation Science, INFORMS, vol. 18(1), pages 76-84, February.
    2. Rainer Burkard & Jafar Fathali, 2007. "A polynomial method for the pos/neg weighted 3-median problem on a tree," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 65(2), pages 229-238, April.
    3. Richard L. Church & Robert S. Garfinkel, 1978. "Locating an Obnoxious Facility on a Network," Transportation Science, INFORMS, vol. 12(2), pages 107-118, May.
    4. Peter J. Slater, 1982. "Locating Central Paths in a Graph," Transportation Science, INFORMS, vol. 16(1), pages 1-18, February.
    5. S. Mitchell Hedetniemi & E. J. Cockayne & S. T. Hedetniemi, 1981. "Linear Algorithms for Finding the Jordan Center and Path Center of a Tree," Transportation Science, INFORMS, vol. 15(2), pages 98-114, May.
    6. A. J. Goldman, 1971. "Optimal Center Location in Simple Networks," Transportation Science, INFORMS, vol. 5(2), pages 212-221, May.
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