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A Computational Study of Reduction Techniques for the Minimum Connectivity Inference Problem

Chapter
Part of the Advances in Mechanics and Mathematics book series (AMMA, volume 41)

Abstract

The minimum connectivity inference (MCI) problem is an NP-hard discrete optimization problem. Its description is based on a simple, undirected, and complete graph given by a vertex set. Moreover, a finite number of clusters (subsets of the vertex set) are given. These clusters may overlap each other. The problem consists in determining an edge set with minimal cardinality so that the vertices in each cluster are connected by edges of this set which have both vertices in the cluster. Research on the MCI problem can be traced back from 1976 to the present and includes complexity results, reduction techniques, heuristic solution approaches, and various applications. Some years ago, the MCI problem has been modeled as a mixed integer linear programming (MILP) problem which enables to solve MCI instances exactly up to a small size. An improved MILP formulation, recently introduced by the authors of the present contribution, allows for successfully tackling moderately sized instances. To further increase the size of MCI instances that can be solved exactly (or to reduce the required computation time), reduction techniques can be very helpful. Such techniques aim at converting a given instance into an instance with fewer clusters or vertices or both. Our contribution will briefly review the improved MILP-based solution approach as well as reduction techniques. Based on this, we will present a computational study on the influence of several reduction techniques on the problem size and the computation time. In addition, we will also discuss the effect of a heuristic reduction technique.

Notes

Acknowledgements

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).”

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institute of Numerical Mathematics, Technische Universität DresdenDresdenGermany

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