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A comparison of directed formulations for the capacitated minimal spanning tree problem

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Abstract

In this paper, we compare linear integer programming directed formulations for the capacitated minimal spanning tree (CMST) problem. This problem is directly related to network design and consists of finding the cheapest way to link a set of terminals at different locations with a central node (a computer site, for instance). The present work focuses on a single commodity flow formulation presented in [3,4]. This formulation includes two sets of variables and, hence, it also must include coupling constraints between the two sets of variables. These coupling constraints can be defined in several ways. The main result of this work establishes that when the strongest form of the coupling constraints is used in the flow formulation (as in [4]), the “equivalent” formulation using only theX ij variables satisfies the so-called multistar constraints. These constraints are shown to be stronger than the weakest form of the generalized subtour breaking constraints and, hence, our result improves on the characterization given in [5] for the flow formulation presented in [4]. For certain parameters, the multistar constraints induce facets of the non-directed CMST polytope (see [2]). We also show that it is not possible to compare the linear relaxations of a formulation that involves the multistar constraints with one that involves the generalized subtour breaking constraints.

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  1. DEIO - CEAUL, Faculdade de Ciências da Universidade de Lisboa, Av. 24 de Julho, 134 5o, 1300, Lisboa, Portugal

    Luis Gouveia

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  1. Luis Gouveia
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Gouveia, L. A comparison of directed formulations for the capacitated minimal spanning tree problem. Telecommunication Systems 1, 51–76 (1993). https://doi.org/10.1007/BF02136155

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  • DOI: https://doi.org/10.1007/BF02136155

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