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The inverse 1-maxian problem with edge length modification

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Abstract

We consider the problem of modifying the lengths of the edges of a graph at minimum cost such that a prespecified vertex becomes a 1-maxian with respect to the new edge lengths. The inverse 1-maxian problem with edge length modification is shown to be strongly \(\mathcal{N}\mathcal{P}\) -hard and remains weakly \(\mathcal{NP}\) -hard even on series-parallel graphs. Moreover, a transformation of the inverse 1-maxian problem with edge length modification on a tree to a minimum cost circulation problem is given which solves the original problem in \(\mathcal{O}(n\log n)\) .

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References

  • Ahuja RK, Magnanti TL, Orlin JB (1993) Network flows: theory, algorithms, and applications. Prentice Hall, New Jersey

    Google Scholar 

  • Bein WW, Brucker P, Tamir A (1985) Minimum cost flow algorithms for series-parallel networks. Discret Appl Math 10:117–124

    Article  MATH  MathSciNet  Google Scholar 

  • Berman O, Ingco DI, Odoni A (1992) Improving the location of minisum facilities through network modification. Ann Oper Res 40:1–16

    Article  MATH  MathSciNet  Google Scholar 

  • Berman O, Ingco DI, Odoni A (1994) Improving the location of minimax facilities through network modification. Networks 24:31–41

    Article  MATH  MathSciNet  Google Scholar 

  • Booth H, Tarjan RE (1993) Finding the minimum-cost maximum flow in a series-parallel network. J Algorithms 15:416–446

    Article  MATH  MathSciNet  Google Scholar 

  • Burkard RE, Pleschiutschnig C, Zhang J (2004) Inverse median problems. Discret Optim 1:23–39

    Article  MATH  MathSciNet  Google Scholar 

  • Burkard RE, Gassner E, Hatzl J (2006) A linear time algorithm for the reverse 1-median problem on a cycle. Networks 48:16–23

    Article  MATH  MathSciNet  Google Scholar 

  • Burkard RE, Gassner E, Hatzl J (2007) Reverse 2-median problem on trees, accpeted for publication in Discret Appl Math

  • Garey MR, Johnson DS (1979) Computers and intractability: a guide to the theory of NP-completeness. Freeman, New York

    MATH  Google Scholar 

  • Goldman AJ (1971) Optimum center location in simple networks. Transp Sci 5:212–221

    Article  Google Scholar 

  • Henzinger MR, Klein PN, Rao S, Subramanian S (1997) Faster shortest-path algorithms for planar graphs. J Comput Syst Sci 55:3–23

    Article  MATH  MathSciNet  Google Scholar 

  • Heuberger C (2004) Inverse combinatorial optimization: a survey on problems, methods, and results. J Comb Optim 8:329–361

    Article  MATH  MathSciNet  Google Scholar 

  • Robertson N, Seymour PD (1986) Graph minors II. Algorithmic aspects of tree-width. J Algorithms 7:309–322

    Article  MATH  MathSciNet  Google Scholar 

  • Tamir A (1991) Obnoxious facility location on graphs. SIAM J Discret Math 4:550–567

    Article  MATH  MathSciNet  Google Scholar 

  • Ting SS (1984) A linear-time algorithm for maxisum facility location on tree networks. Transp Sci 18:76–84

    MathSciNet  Google Scholar 

  • Zhang JZ, Yang XG, Cai MC (1999) Reverse center location problem. Lect Notes Comput Sci 1741:279–294

    Article  MathSciNet  Google Scholar 

  • Zhang J, Liu Z, Ma Z (2000) Some reverse location problems. Eur J Oper Res 124:77–88

    Article  MATH  MathSciNet  Google Scholar 

  • Zhang JZ, Yang XG, Cai MC (2004) Inapproximability and a polynomially solvable special case of a network improvement problem. Eur J Oper Res 155:251–257

    Article  MATH  MathSciNet  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Optimization and Discrete Mathematics, Graz University of Technology, Steyrergasse 30, 8010, Graz, Austria

    Elisabeth Gassner

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  1. Elisabeth Gassner
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Correspondence to Elisabeth Gassner.

Additional information

This research has been supported by the Austrian Science Fund (FWF) Project P18918-N18.

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Gassner, E. The inverse 1-maxian problem with edge length modification. J Comb Optim 16, 50–67 (2008). https://doi.org/10.1007/s10878-007-9098-9

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  • DOI: https://doi.org/10.1007/s10878-007-9098-9

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