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General variable neighborhood search approach to group steiner tree problem

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Abstract

In this paper, we consider the Group Steiner Tree (GST) problem that can be stated as follows: For a given non-negative edge weighted graph \(G = (V, E)\), an integer k, and the corresponding family \(g_1, \ldots , g_k\) containing non-empty subsets of V called groups, we need to find a minimum cost tree \(T = (V_T, E_T)\) where \(V_T \subseteq V\) and \(E_T\subseteq E\) that spans at least one vertex from each of the groups. Numerous applications of this NP-hard problem initiated researchers to study it from both theoretical and algorithmic aspects. One of the challenges is to provide a good heuristic solution within the reasonable amount of CPU time. We propose the application of metaheuristic framework based on Variable Neighborhood Search (VNS) and related approaches. One of our main objectives is to find a neighborhood structure that ensures efficient implementation. We develop Variable Neighborhood Descend (VND) algorithm that can be the main ingredient of several local search approaches. Experimental evaluation involves comparison of our heuristic to exact approach based on Integer Linear Programming solvers and other metaheuristic approaches, such as genetic algorithm. The obtained results show that the proposed method always outperforms genetic algorithm. Exact method is outperformed in the case of instances with large number of groups.

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Notes

  1. https://cpu.userbenchmark.com/Compare/AMD-Ryzen-TR-3990X-vs-AMD-Ryzen-7-1800X/m1035665vs3916.

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Authors and Affiliations

  1. Mathematical Institute, Serbian Academy of Science and Arts, Knez-Mihailova 36/III, Belgrade, 11001, Serbia

    Luka Matijević & Tatjana Davidović

  2. Department of Mathematics, Josip Juraj Strossmayer University of Osijek, Trg Svetog Trojstva 3, Osijek, 31000, Croatia

    Slobodan Jelić

Authors
  1. Luka Matijević
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  2. Slobodan Jelić
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  3. Tatjana Davidović
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Correspondence to Luka Matijević.

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Dedicated to the memory of Professor Nenad Mladenović, one of the founders of the VNS method.

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Matijević, L., Jelić, S. & Davidović, T. General variable neighborhood search approach to group steiner tree problem. Optim Lett 17, 2087–2111 (2023). https://doi.org/10.1007/s11590-022-01904-7

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