Binary Steiner trees: Structural results and an exact solution approach
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- 作者：Frauke Liers frauke.liers@math.uni-erlangen.de" class="auth_mail" title="E-mail the corresponding author ; Alexander Martin alexander.martin@math.uni-erlangen.de" class="auth_mail" title="E-mail the corresponding author ; Susanne Pape ; susanne.pape@math.uni-erlangen.de" class="auth_mail" title="E-mail the corresponding author
- 关键词：Steiner tree ; Degree constraint ; Steiner ratio ; Integer programming ; Polytope ; Branch-&-cut
- 刊名：Discrete Optimization
- 出版年：2016
- 出版时间：August 2016
- 年：2016
- 卷：21
- 期：Complete
- 页码：85-117
- 全文大小：1061 K
文摘
In this paper we study the Steiner tree problem with degree constraints. Motivated by an application in computational biology we focus on binary Steiner trees in which all node degrees are required to be at most three. It is shown that finding a binary Steiner tree is NP-complete for arbitrary graphs. We relate the problem to Steiner trees without degree constraints as well as degree-constrained spanning trees by proving approximation ratios. Further, we give integer programming formulations for this problem on undirected and directed graphs and study the associated polytopes for both cases. Some classes of facets are introduced. Based on this study a branch-&-cut approach is developed and evaluated on biological instances coming from the reconstruction of phylogenetic trees. We are able to solve nearly all instances up to 200 nodes to optimality within a limited amount of time. This shows the effectiveness of our approach.