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Branch and Cut for Partitioning a Graph into a Cycle of Clusters

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Combinatorial Optimization (ISCO 2024)

Abstract

In this paper we study formulations and algorithms for the cycle clustering problem, a partitioning problem over the vertex set of a directed graph with nonnegative arc weights that is used to identify cyclic behavior in simulation data generated from nonreversible Markov state models. Here, in addition to partitioning the vertices into a set of coherent clusters, the resulting clusters must be ordered into a cycle such as to maximize the total net flow in the forward direction of the cycle. We provide a problem-specific binary programming formulation and compare it to a formulation based on the reformulation-linearization technique (RLT). We present theoretical results on the polytope associated with our custom formulation and develop primal heuristics and separation routines for both formulations. In computational experiments on simulation data from biology we find that branch and cut based on the problem-specific formulation outperforms the one based on RLT.

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Acknowledgments

We wish to thank Konstantin Fackeldey, Andreas Grever, and Marcus Weber for supplying us with simulation data for our experiments. This work has been supported by the Research Campus MODAL funded by the German Federal Ministry of Education and Research (BMBF grants 05M14ZAM, 05M20ZBM).

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Correspondence to Ambros Gleixner .

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Eifler, L., Witzig, J., Gleixner, A. (2024). Branch and Cut for Partitioning a Graph into a Cycle of Clusters. In: Basu, A., Mahjoub, A.R., Salazar González, J.J. (eds) Combinatorial Optimization. ISCO 2024. Lecture Notes in Computer Science, vol 14594. Springer, Cham. https://doi.org/10.1007/978-3-031-60924-4_8

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  • DOI: https://doi.org/10.1007/978-3-031-60924-4_8

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