Finding subsets maximizing minimum structures
Magnús M. Halldórsson,Kazuo Iwano,Naoki Katoh,Takeshi Tokuyama +3 more
- 22 Jan 1995
- pp 150-159
TL;DR: In this paper, the authors consider the problem of finding a set of k vertices in a graph that are in some sense remote, where the structure to be minimized is a spanning tree, Steiner tree, or traveling salesperson tour.
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Abstract: We consider the problem of nding a set of k vertices in a graph that are in some sense remote. Stated more formally, given a graph G and an integer k, nd a set P of k vertices for which the total weight of a minimum structure on P is maximized. In particular, we are interested in three problems of this type, where the structure to be minimized is a spanning tree (Remote-MST), Steiner tree, or traveling salesperson tour. We study a natural greedy algorithm that simultaneously approximates all three problems on metric graphs. For instance, its performance ratio for Remote-MST is exactly 4, while this problem is NP-hard to approximate within a factor of less than 2. We also give a better approximation for graphs induced by Euclidean points in the plane, present an exact algorithm for graphs whose distances correspond to shortest-path distances in a tree, and prove hardness and approximability results for general graphs.
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Citations
An O(pn2) algorithm for the p -median and related problems on tree graphs
TL;DR: This work improves the complexity bound of the p-median problem on trees by showing that the total running time of the ''leaves to root'' dynamic programming algorithm is O(pn^2).
263
Max-Sum diversification, monotone submodular functions and dynamic updates
Allan Borodin,Hyun Chul Lee,Yuli Ye +2 more
- 21 May 2012
TL;DR: This paper considers the setting where the authors are given a set of elements in a metric space and a set valuation function f defined on every subset and shows that a natural single swap local search algorithm provides a 2-approximation in this more general setting.
165
•Posted Content
Max-Sum Diversification, Monotone Submodular Functions and Dynamic Updates
TL;DR: In this article, a generalization of the max sum diversification problem is studied, where the constraint is given by independence in a matroid, where quality is determined by a monotone submodular function.
90
An annotated bibliography of combinatorial optimization problems with fixed cardinality constraints
Maurizio Bruglieri,Matthias Ehrgott,Horst W. Hamacher,Francesco Maffioli +3 more
- 01 Jun 2006
TL;DR: This paper formally defines the problem, mentions some examples and summarizes general results, and provides an annotated bibliography of combinatorial optimization problems of which versions with cardinality constraint have been considered in the literature.
74
Max-Sum Diversification, Monotone Submodular Functions, and Dynamic Updates
TL;DR: A greedy algorithm for a cardinality constraint and a local search algorithm for an arbitrary matroid constraint are proposed and it is proved that both algorithms achieve constant approximation ratios.
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