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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Agarwal, D., Araujo, J.-C.S., Caillouet, C., Cazals, F., Coudert, D., Pérennes, S.: Connectivity inference in mass spectrometry based structure determination. In: Bodlaender, H.L., Italiano, G.F. (eds.) European Symposium on Algorithms, Lecture Notes in Computer Science vol. 8125, pp. 289–300. Springer, Berlin (2013)
Agarwal, D., Caillouet, C., Coudert, D., Cazals, F.: Unveiling contacts within macromolecular assemblies by solving minimum weight connectivity inference (MWC) problems. Molecular & Cellular Proteomics 14(8), 2274–2284 (2015)
Angluin, D., Aspnes, J., Reyzin, L.: Inferring social networks from outbreaks. In: Hutter, M., Stephan, F., Vovk, V., Zeugmann, T. (eds.) Algorithmic Learning Theory. Lecture Notes in Computer Science vol. 6331, pp. 104–118. Springer, Berlin (2010)
Chen, C., Jacobsen, H.-A., Vitenberg, R.: Algorithms based on divide and conquer for topic-based publish/subscribe overlay design. IEEE/ACM Transactions on Networking 24(1), 422–436 (2016)
Chen, J., Komusiewicz, C., Niedermeier, R., Sorge, M., Suchý, O., Weller, M.: Polynomial-time data reduction for the subset interconnection design problem. SIAM Journal on Discrete Mathematics 29, 1–25 (2015)
Chockler, G., Melamed, R., Tock, Y., Vitenberg, R.: Constructing scalable overlays for pub-sub with many topics. In: Gupta, I. (ed.) Proceedings of the Twenty-Sixth Annual ACM Symposium on Principles of Distributed Computing, pp.109–118. ACM, New York (2007)
Dar, A., Fischer, A., Martinovic, J., Scheithauer, G.: An improved flow-based formulation and reduction principles for the Minimum Connectivity Inference problem. Optimization (2018). https://doi.org/10.1080/02331934.2018.1465944
Du, D.-Z.: An optimization problem on graphs. Discrete Applied Mathematics 14(1), 101–104 (1986)
Du, D.-Z.: Curriculum Vitae of Ding-Zuh Du. http://www.utdallas.edu/~dxd056000/. Accessed 25 July 2017
Du, D.-Z., Chen, Y.-M.: Placement of valves in vacuum systems. Communication on Electric Light Source Technology 4, 22–28 (in Chinese, 1976)
Du, D.-Z., Miller, Z.: Matroids and subset interconnection design. SIAM Journal on Discrete Mathematics 1(4), 416–424 (1988)
Fan, H., Hundt, C., Wu, Y.-L., Ernst, J.: Algorithms and implementation for interconnection graph problem. In: Yang, B., Du, D.-Z., Wang, C. A. (eds.) Combinatorial Optimization and Applications. Lecture Notes in Computer Science vol. 5165, pp. 201–210. Springer, Berlin (2008)
Hosoda, J., Hromkovič, J., Izumi, T., Ono, H., Steinová, M., Wada, K.: On the approximability and hardness of minimum topic connected overlay and its special instances. Theoretical Computer Science 429, 144–154 (2012)
Magnanti, T.L., Wolsey, L.A.: Optimal Trees. In: Ball, M.O., Magnanti, T.L., Monma, B.L., Nemhauser, G. (eds.) Network Routing, Handbooks in Operations Research and Management Science vol. 7, pp. 503–615, Elsevier, Amsterdam (1995)
Prisner, E.: Two algorithms for the subset interconnection design problem. Networks 22(4), 385–395 (1992)
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).”
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Dar, M.A., Fischer, A., Martinovic, J., Scheithauer, G. (2019). A Computational Study of Reduction Techniques for the Minimum Connectivity Inference Problem. In: Singh, V., Gao, D., Fischer, A. (eds) Advances in Mathematical Methods and High Performance Computing. Advances in Mechanics and Mathematics, vol 41. Springer, Cham. https://doi.org/10.1007/978-3-030-02487-1_7
Download citation
DOI: https://doi.org/10.1007/978-3-030-02487-1_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-02486-4
Online ISBN: 978-3-030-02487-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)