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Integrality ratio for group steiner trees and directed steiner trees
 In 14th Annual ACMSIAM Symposium on Discrete Algorithms
, 2003
"... The natural relaxation for the Group Steiner Tree problem, as well as for its generalization, the Directed Steiner Tree problem, is a flowbased linear programming relaxation. We prove new lower bounds on the integrality ratio of this relaxation. For the Group Steiner Tree problem, we show the integ ..."
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Cited by 29 (6 self)
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The natural relaxation for the Group Steiner Tree problem, as well as for its generalization, the Directed Steiner Tree problem, is a flowbased linear programming relaxation. We prove new lower bounds on the integrality ratio of this relaxation. For the Group Steiner Tree problem, we show the integrality ratio is Ω(log 2 k), where k denotes the number of groups; this holds even for input graphs that are Hierarchically WellSeparated Trees, introduced by Bartal [Symp. Foundations of Computer Science, pp. 184–193, 1996], in which case this lower bound is tight. This also applies for the Directed Steiner Tree problem. In terms of the number n of vertices, our results for the Directed Steiner problem log imply an Ω( 2 n (log log n) 2) integrality ratio. For both problems, these are the first lower bounds on the integrality ratio that are superlogarithmic in the input size. This exhibits, for the first time, a relaxation of a natural optimization problem whose integrality ratio is known to be superlogarithmic but subpolynomial. Our results and techniques have been used by Halperin and Krauthgamer [Symp. on Theory of Computing, pp. 585–594, 2003] to show comparable inapproximability results, assuming that NP has no quasipolynomial LasVegas algorithms. We also show algorithmically that the integrality ratio for Group Steiner Tree is much better for certain families of instances, which helps pinpoint the types of instances (parametrized by optimal solutions to their flowbased relaxations) that appear to be most difficult to approximate.
Aspects of Network Design
, 2007
"... In this dissertation we study two problems from the area of network design. The first part discusses the multicommodity buyatbulk network design problem, a problem that occurs naturally in the design of telecommunication and transportation networks. We are given an underlying graph and associated ..."
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Cited by 1 (1 self)
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In this dissertation we study two problems from the area of network design. The first part discusses the multicommodity buyatbulk network design problem, a problem that occurs naturally in the design of telecommunication and transportation networks. We are given an underlying graph and associated with each edge of the graph, a cost function that represents the price of routing demand along this edge. We are also given a set of demands between pairs of vertices each of which needs to be satisfied by paying for sufficient capacity along a path connecting the vertices of the pair. In the multicommodity network design problem the objective is to minimize the cost of satisfying all demands. There are often situations where there is an initial fixed cost of utilizing an edge, or there is discounting or economies of scale that give rise to concave cost functions. We have an instance of the buyatbulk network design problem when the cost functions along all edges are concave. Unlike the case of linear cost functions, for which polynomial time algorithms exist, the buyatbulk network design problem is NPhard. We give the first nontrivial approx
Minimum Latency Submodular Cover∗
, 2013
"... We study the Minimum Latency Submodular Cover problem (MLSC), which consists of a metric (V, d) with source r ∈ V and m monotone submodular functions f1, f2,..., fm: 2V → [0, 1]. The goal is to find a path originating at r that minimizes the total cover time of all functions. This generalizes wells ..."
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We study the Minimum Latency Submodular Cover problem (MLSC), which consists of a metric (V, d) with source r ∈ V and m monotone submodular functions f1, f2,..., fm: 2V → [0, 1]. The goal is to find a path originating at r that minimizes the total cover time of all functions. This generalizes wellstudied problems, such as Submodular Ranking [1] and Group Steiner Tree [16]. We give a polynomial time O(log 1 · log2+δ V )approximation algorithm for MLSC, where > 0 is the smallest nonzero marginal increase of any {fi}mi=1 and δ> 0 is any constant. We also consider the Latency Covering Steiner Tree problem (LCST), which is the special case of MLSC where the fis are multicoverage functions. This is a common generalization of the Latency Group Steiner Tree [20, 8] and Generalized Minsum Set Cover [2, 3] problems. We obtain an O(log2 V )approximation algorithm for LCST. Finally we study a natural stochastic extension of the Submodular Ranking problem, and obtain an adaptive algorithm with an O(log 1/) approximation ratio, which is best possible. This result also generalizes some previously studied stochastic optimization problems, such as Stochastic Set Cover [17] and Shared Filter Evaluation [27, 26]. 1