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Minimum-weight two-connected spanning networks

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Abstract

We consider the problem of constructing a minimum-weight, two-connected network spanning all the points in a setV. We assume a symmetric, nonnegative distance functiond(·) defined onV × V which satisfies the triangle inequality. We obtain a structural characterization of optimal solutions. Specifically, there exists an optimal two-connected solution whose vertices all have degree 2 or 3, and such that the removal of any edge or pair of edges leaves a bridge in the resulting connected components. These are the strongest possible conditions on the structure of an optimal solution since we also show thatany two-connected graph satisfying these conditions is theunique optimal solution for a particular choice of ‘canonical’ distances satisfying the triangle inequality. We use these properties to show that the weight of an optimal traveling salesman cycle is at most 4/3 times the weight of an optimal two-connected solution; examples are provided which approach this bound arbitrarily closely. In addition, we obtain similar results for the variation of this problem where the network need only span a prespecified subset of the points.

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Authors and Affiliations

  1. Bell Communications Research, 07960, Morristown, NJ, USA

    Clyde L. Monma

  2. AT&T Bell Laboratories, 07733, Holmdel, NJ, USA

    Beth Spellman Munson

  3. University of Waterloo, N2L 3G1, Waterloo, Ont., Canada

    William R. Pulleyblank

Authors
  1. Clyde L. Monma
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  2. Beth Spellman Munson
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  3. William R. Pulleyblank
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Monma, C.L., Munson, B.S. & Pulleyblank, W.R. Minimum-weight two-connected spanning networks. Mathematical Programming 46, 153–171 (1990). https://doi.org/10.1007/BF01585735

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  • DOI: https://doi.org/10.1007/BF01585735

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