Abstract
Let G=(V,E) denote a weighted graph of n nodes and m edges, and let G[ V ′ ] denote the subgraph of G induced by a subset of nodes V′ ⊆ V. The radius of G[ V ′ ] is the maximum length of a shortest path in G[ V ′ ] emanating from its center (i.e., a node of G[ V ′ ] of minimum eccentricity). In this paper, we focus on the problem of partitioning the nodes of G into exactly p non-empty subsets, so as to minimize the sum of the induced subgraph radii. We show that this problem – which is of significance in facility location applications – is NP-hard when p is part of the input, but for a fixed constant p > 2 it can be solved in O(n 2p/p!) time. Moreover, for the notable case p=2, we present an efficient O(mn 2+n 3 logn) time algorithm.
Work partially supported by the Research Project GRID.IT, funded by the Italian Ministry of Education, University and Research, by the European Union under COST 295 (DYNAMO), and by the Swiss SBF under grant no. C05.0047. Part of this work has been developed while the first author was visiting ETH Zürich.
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Proietti, G., Widmayer, P. (2006). Partitioning the Nodes of a Graph to Minimize the Sum of Subgraph Radii. In: Asano, T. (eds) Algorithms and Computation. ISAAC 2006. Lecture Notes in Computer Science, vol 4288. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11940128_58
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DOI: https://doi.org/10.1007/11940128_58
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