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Polyhedral results and a Branch-and-cut algorithm for the \(k\)-cardinality tree problem

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Abstract

Given an undirected graph \(G\) with vertex and edge weights, the \(k\)-cardinality tree problem asks for a minimum weight tree of \(G\) containing exactly \(k\) edges. In this paper we consider a directed graph reformulation of the problem and carry out a polyhedral investigation of the polytope defined by the convex hull of its feasible integral solutions. Three new families of valid inequalities are identified and two of them are proven to be facet implying for that polytope. Additionally, a Branch-and-cut algorithm that separates the new inequalities is implemented and computationally tested. The results obtained indicate that our algorithm outperforms its counterparts from the literature. Such a performance could be attributed, to a large extent, to the use of the new inequalities and also to some pre-processing tests introduced in this study.

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Acknowledgments

The authors wish to thank Ivana Ljubić and Maria Kandyba-Chimani for sending us detailed computational results of their algorithm.

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Correspondence to Abilio Lucena.

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Luidi Simonetti is partially funded by FAPERJ grant E-26/111.819/2010 and CNPq grants 483.243/2010-8, 304793/2011-6. Alexandre Salles da Cunha is partially funded by CNPq grants 302276/2009-2, 477863/2010-8 and FAPEMIG PRONEX APQ-01201-09. Abilio Lucena is partially funded by CNPq grant 310561/2009-4 and FAPERJ grant E26-110.552/2010.

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Simonetti, L., da Cunha, A.S. & Lucena, A. Polyhedral results and a Branch-and-cut algorithm for the \(k\)-cardinality tree problem. Math. Program. 142, 511–538 (2013). https://doi.org/10.1007/s10107-012-0590-3

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