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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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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.
Approximation Algorithms for NETWORK DESIGN AND ORIENTEERING
, 2010
"... This thesis presents approximation algorithms for some N PHard combinatorial optimization problems on graphs and networks; in particular, we study problems related to Network Design. Under the widelybelieved complexitytheoretic assumption that P ̸ = N P, there are no efficient (i.e., polynomialt ..."
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This thesis presents approximation algorithms for some N PHard combinatorial optimization problems on graphs and networks; in particular, we study problems related to Network Design. Under the widelybelieved complexitytheoretic assumption that P ̸ = N P, there are no efficient (i.e., polynomialtime) algorithms that solve these problems exactly. Hence, if one desires efficient algorithms for such problems, it is necessary to consider approximate solutions: An approximation algorithm for an N PHard problem is a polynomial time algorithm which, for any instance of the problem, finds a solution whose value is guaranteed to be within a multiplicative factor ρ of the value of an optimal solution to that instance. We attempt to design algorithms for which this factor ρ, referred to as the approximation ratio of the algorithm, is as small as possible. The field of Network Design comprises a large class of problems that deal with constructing networks of low cost and/or high capacity, routing data through existing networks, and many related issues. In this thesis, we focus chiefly on designing faulttolerant networks. Two vertices u, v in a network are said to be kedgeconnected if deleting any set of k − 1 edges leaves u and v connected; similarly, they are kvertex connected if deleting any set of k − 1 other vertices or edges leaves u and v connected. We focus on building networks that are highly connected, meaning