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An Approximate MaxSteinerTreePacking MinSteinerCut Theorem
"... Given an undirected multigraph G and a subset of vertices S ` V (G), the STEINER TREE PACKING problem is to find a largest collection of edgedisjoint trees that each connects S. This problem and its generalizations have attracted considerable attention from researchers in different areas because of ..."
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Cited by 32 (4 self)
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Given an undirected multigraph G and a subset of vertices S ` V (G), the STEINER TREE PACKING problem is to find a largest collection of edgedisjoint trees that each connects S. This problem and its generalizations have attracted considerable attention from researchers in different areas because of their wide applicability. This problem was shown to be APXhard (no polynomial time approximation scheme unless P=NP). In fact, prior to this paper, not evenan approximation algorithm with asymptotic ratio o(n) wasknown despite several attempts. In this work, we close this huge gap by presenting the first polynomial time constant factor approximation algorithm for the STEINER TREE PACKING problem. The maintheorem is an approximate minmax relation between the maximum number of edgedisjoint trees that each connects S (i.e. Strees) and the minimum size of an edgecut thatdisconnects some pair of vertices in S (i.e. Scut). Specifically, we prove that if the minimum Scut in G has 26k edges, then G has at least k edgedisjoint Strees; this answers Kriesell's conjecture affirmatively up to a constant multiple. The techniques that we use are purely combinatorial, where matroid theory is the underlying ground work.
Design of a Multisender 3D Videoconferencing Application over an End System Multicast Protocol
 Proceedings of the eleventh ACM international conference on Multimedia
, 2003
"... Videoconferencing in the context of 3D virtual environments promises better spatial consistency and mutual awareness for its participants. However, in the absence of IP Multicast and limited upload bandwidth of today's DSL connections, the feasibility of such systems in supporting even a small ..."
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Cited by 17 (1 self)
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Videoconferencing in the context of 3D virtual environments promises better spatial consistency and mutual awareness for its participants. However, in the absence of IP Multicast and limited upload bandwidth of today's DSL connections, the feasibility of such systems in supporting even a small group of users is in question. This paper presents the design and implementation of an awareness driven 3D videoconferencing application that runs on a peertopeer architecture and our own End System Multicast protocol. The paper highlights the unique requirements of multiparty videoconferencing applications and presents a solution that can support 410 bandwidthlimited users without the need for IP Multicast capability.
A graph reduction step preserving elementconnectivity and applications
 IN INTERNATIONAL COLLOQUIUM ON AUTOMATA, LANGUAGES AND PROGRAMMING
, 2009
"... Given an undirected graph G = (V, E) and subset of terminals T ⊆ V, the elementconnectivity κ ′ G (u, v) of two terminals u, v ∈ T is the maximum number of uv paths that are pairwise disjoint in both edges and nonterminals V \ T (the paths need not be disjoint in terminals). Elementconnectivity ..."
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Cited by 11 (2 self)
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Given an undirected graph G = (V, E) and subset of terminals T ⊆ V, the elementconnectivity κ ′ G (u, v) of two terminals u, v ∈ T is the maximum number of uv paths that are pairwise disjoint in both edges and nonterminals V \ T (the paths need not be disjoint in terminals). Elementconnectivity is more general than edgeconnectivity and less general than vertexconnectivity. Hind and Oellermann [21] gave a graph reduction step that preserves the global elementconnectivity of the graph. We show that this step also preserves local connectivity, that is, all the pairwise elementconnectivities of the terminals. We give two applications of this reduction step to connectivity and network design problems. • Given a graph G and disjoint terminal sets T1, T2,..., Tm, we seek a maximum number of elementdisjoint Steiner forests where each forest connects each Ti. We prove that if each Ti is k element k connected then there exist Ω( log hlog m) elementdisjoint Steiner forests, where h =  i Ti. If G is planar (or more generally, has fixed genus), we show that there exist Ω(k) Steiner forests. Our proofs are constructive, giving polytime algorithms to find these forests; these are the first nontrivial algorithms for packing elementdisjoint Steiner Forests. • We give a very short and intuitive proof of a spiderdecomposition theorem of Chuzhoy and Khanna [12] in the context of the singlesink kvertexconnectivity problem; this yields a simple and alternative analysis of an O(k log n) approximation. Our results highlight the effectiveness of the elementconnectivity reduction step; we believe it will find more applications in the future.
On approximate minmax theorems of graph connectivity problems
, 2006
"... Given an undirected graph G and a subset of vertices S ` V (G), we call the vertices in S the terminal vertices and the vertices in V (G) S the Steiner vertices. In this thesis, we study two problems whose goals are to achieve high "connectivity " among the terminal vertices. The ..."
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Given an undirected graph G and a subset of vertices S ` V (G), we call the vertices in S the terminal vertices and the vertices in V (G) S the Steiner vertices. In this thesis, we study two problems whose goals are to achieve high &quot;connectivity &quot; among the terminal vertices. The first problem is the Steiner Tree Packing problem, where a Steiner tree is a tree that connects the terminal vertices (Steiner vertices are optional). The goal of this problem is to find a largest collection of edgedisjoint Steiner trees. The second problem is the Steiner RootedOrientation problem. In this problem, there is a root vertex r among the terminal vertices. The goal is to find an orientation of all the edges in G so that the Steiner rootedconnectivity is maximized in the resulting directed graph D. Here, the Steiner rootedconnectivity is defined to be the maximum k so that the root vertex has k arcdisjoint paths to each terminal vertex in D. Both problems are generalizations of two classical graph theoretical problems: the edgedisjoint s, tpaths problem and the edgedisjoint spanning trees problem. Polynomial time algorithms and exact minmax relations are known for the classical problems. However, both problems that we study are NPcomplete, and thus exact minmax relations are not expected. In the following, we say S is ledgeconnected in G if we need to remove at least l edges in order to disconnect two vertices in S. Clearly, the maximum iii l for which S is ledgeconnected in G is an upper bound on the optimal value for both problems that we study (i.e. the number of edgedisjoint Steiner trees, and the Steiner rootedconnectivity in an orientation). The main result of the Steiner Tree Packing problem is the following approximate minmax relation:
Approximation Algorithms and Hardness Results for Packing ElementDisjoint Steiner Trees
, 2008
"... We study the problem of packing elementdisjoint Steiner trees in graphs. We are given a graph and a designated subset of terminal nodes, and the goal is to find a maximum cardinality set of elementdisjoint trees such that each tree contains every terminal node. An element means a nonterminal node ..."
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We study the problem of packing elementdisjoint Steiner trees in graphs. We are given a graph and a designated subset of terminal nodes, and the goal is to find a maximum cardinality set of elementdisjoint trees such that each tree contains every terminal node. An element means a nonterminal node or an edge. (Thus, each nonterminal node and each edge must be in at most one of the trees.) We show that the problem is APXhard when there are only three terminal nodes, thus answering an open question. Our main focus is on the special case when the graph is planar. We show that the problem of finding two elementdisjoint Steiner trees in a planar graph is NPhard. Similarly, the problem of finding two edgedisjoint Steiner trees in a planar graph is NPhard. We design an algorithm for planar graphs that achieves an approximation guarantee close to 2. In fact, given a planar graph that is k elementconnected on the terminals (k is an upper bound on the number of elementdisjoint Steiner trees), the algorithm returns ⌊ ⌋ k 2 − 1 elementdisjoint Steiner trees. Using this algorithm, we get an approximation algorithm for the edgedisjoint version of the problem on planar graphs that improves on the previous approximation guarantees. We also show that the natural LP relaxation of the planar problem has an integrality ratio approaching 2.
Models for the Steiner Tree Packing Problem
, 2013
"... The Steiner tree packing problem is a long studied problem in combinatorial optimization. In contrast to many other problems, where an enormous progress has been made in the practical problem solving, the Steiner tree packing problem remains very difficult. Most heuristics schemes are ineffective ..."
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The Steiner tree packing problem is a long studied problem in combinatorial optimization. In contrast to many other problems, where an enormous progress has been made in the practical problem solving, the Steiner tree packing problem remains very difficult. Most heuristics schemes are ineffective and even finding feasible solutions is already NPhard. What makes this problem special, is that in order to reach an overall optimal solution nonoptimal solutions to the underlying NPhard Steiner tree problems must be used. Any nonglobal approach to the Steiner tree packing problem is likely to fail. Integer programming is currently the best approach for computing optimal solutions. The goal of this master thesis is to give a survey of models relating to the Steiner tree packing problem from the literature. In addition, a closer look