A linear time algorithm for finding tree-decompositions of small treewidth
STOC '93 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing
Clustering to minimize the sum of cluster diameters
Journal of Computer and System Sciences - STOC 2001
Minimum-cost coverage of point sets by disks
Proceedings of the twenty-second annual symposium on Computational geometry
Polynomial time approximation schemes for base station coverage with minimum total radii
Computer Networks: The International Journal of Computer and Telecommunications Networking
Geometric clustering to minimize the sum of cluster sizes
ESA'05 Proceedings of the 13th annual European conference on Algorithms
Partitioning the nodes of a graph to minimize the sum of subgraph radii
ISAAC'06 Proceedings of the 17th international conference on Algorithms and Computation
An O(pn2) algorithm for the p -median and related problems on tree graphs
Operations Research Letters
The 2-radius and 2-radiian problems on trees
Theoretical Computer Science
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Let G = (V,E) denote an undirected weighted graph of n nodes and m edges, and let U ⊆ V. The relative eccentricity of a node v ∈ U is the maximum distance in G between v and any other node of U, while the radius of U in G is the minimum relative eccentricity of all the nodes in U. Several facility location problems ask for partitioning the nodes of G so as to minimize some global optimization function of the radii of the subsets of the partition. Here, we focus on the problem of partitioning the nodes of G into exactly p ≥ 2 non-empty subsets, so as to minimize the sum of the subset radii, called the total radius of the partition. This problem can be easily seen to be NP-hard when p is part of the input, but when p is fixed it can be solved in polynomial time by reducing it to a similar partitioning problem. In this paper, we first present an efficient O(n3) time algorithm for the notable case p = 2, which improves the O(mn2 + n3 log n) running time obtainable by applying the aforementioned reduction. Then, in an effort of characterizing meaningful polynomial-time solvable instances of the problem when p is part of the input, we show that (i) when G is a tree, then the problem can be solved in O(n3p3) time, and (ii) when G has bounded treewidth h, then the problem can be solved in O(n4h+4p3) time.