An improved flow-based formulation and reduction principles for the minimum connectivity inference problem

Abstract

The Minimum Connectivity Inference (MCI) problem represents an $\mathcal{N}P$-hard generalization of the well-known minimum spanning tree problem and has been studied in different fields of research independently. Let an undirected complete graph and finitely many subsets (clusters) of its vertex set be given. Then, the MCI problem is to find a minimal subset of edges so that every cluster is connected with respect to this minimal subset. Whereas, in general, existing approaches can only be applied to find approximate solutions or optimal edge sets of rather small instances, concepts to optimally cope with more meaningful problem sizes have not been proposed yet in literature. For this reason, we present a new mixed integer linear programming formulation for the MCI problem, and introduce new instance reduction methods that can be applied to downsize the complexity of a given instance prior to the optimization. Based on theoretical and computational results both contributions are shown to be beneficial for solving larger instances.

Keywords:

• Minimum connectivity inference
• reduction rules
• mixed integer linear programming model
• subset interconnection design

Acknowledgements

The authors would like to thank an anonymous reviewer for his comments, in particular leading to simplified proofs for the correctness of Rules 4 and 5.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported in parts by a scholarship of the Governmental Scholarship Programme Pakistan – DAAD/HEC Overseas and by the German Research Foundation (DFG) in the Collaborative Research Center 912 ‘Highly Adaptive Energy-Efficient Computing (HAEC)’.