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Polynomialtime approximation schemes for subsetconnectivity problems in boundedgenus graphs
, 2009
"... We present the first polynomialtime approximation schemes (PTASes) for the following subsetconnectivity problems in edgeweighted graphs of bounded genus: Steiner tree, lowconnectivity survivablenetwork design, and subset TSP. The schemes run in O(n log n) time for graphs embedded on both orien ..."
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We present the first polynomialtime approximation schemes (PTASes) for the following subsetconnectivity problems in edgeweighted graphs of bounded genus: Steiner tree, lowconnectivity survivablenetwork design, and subset TSP. The schemes run in O(n log n) time for graphs embedded on both orientable and nonorientable surfaces. This work generalizes the PTAS frameworks of Borradaile, Klein, and Mathieu [BMK07, Kle06] from planar graphs to boundedgenus graphs: any future problems shown to admit the required structure theorem for planar graphs will similarly extend to boundedgenus graphs.
Secluded connectivity problems
 in Proc. 21st ESA
, 2013
"... Consider a setting where possibly sensitive information sent over a path in a network is visible to every neighbor of the path, i.e., every neighbor of some node on the path, thus including the nodes on the path itself. The exposure of a path P can be measured as the number of nodes adjacent to it, ..."
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Consider a setting where possibly sensitive information sent over a path in a network is visible to every neighbor of the path, i.e., every neighbor of some node on the path, thus including the nodes on the path itself. The exposure of a path P can be measured as the number of nodes adjacent to it, denoted by N [P]. A path is said to be secluded if its exposure is small. A similar measure can be applied to other connected subgraphs, such as Steiner trees connecting a given set of terminals. Such subgraphs may be relevant due to considerations of privacy, security or revenue maximization. This paper considers problems related to minimum exposure connectivity structures such as paths and Steiner trees. It is shown that on unweighted undirected nnode graphs, the problem of finding the minimum exposure path connecting a given pair of vertices is strongly inapproximable, i.e., hard to approximate within a factor of O(2log 1− n) for any > 0 (under an appropriate complexity assumption), but is approximable with ratio ∆+3, where ∆ is the maximum degree in the graph. One of our main results concerns the class of boundeddegree graphs, which is shown to exhibit the following interesting dichotomy. On the one hand, the minimum exposure path problem is NPhard on nodeweighted or directed boundeddegree graphs (even when the maximum degree is 4). On the other hand, we present
Subexponential Parameterized Odd Cycle Transversal on Planar Graphs
"... In the Odd Cycle Transversal (OCT) problem we are given a graph G on n vertices and an integer k, and the objective is to determine whether there exists a vertex set O in G of size at most k such that G \ O is bipartite. Reed, Smith, and Vetta [Oper. Res. Lett., 2004] gave an algorithm for OCT with ..."
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In the Odd Cycle Transversal (OCT) problem we are given a graph G on n vertices and an integer k, and the objective is to determine whether there exists a vertex set O in G of size at most k such that G \ O is bipartite. Reed, Smith, and Vetta [Oper. Res. Lett., 2004] gave an algorithm for OCT with running time 3 k n O(1). Assuming the exponential time hypothesis of Impagliazzo, Paturi and Zane, the running time cannot be improved to 2 o(k) n O(1). We show that OCT admits a randomized algorithm running in O(n O(1) + 2 O( √ k log k) n) time when the input graph is planar. As a byproduct we also obtain a linear time algorithm for OCT on planar graphs with running time O(2 O(k log k) n) time. This improves over an algorithm of Fiorini et
Speedingup Dynamic Programming with Representative Sets ⋆ An Experimental Evaluation of Algorithms for Steiner Tree on Tree Decompositions
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Tree Decomposition based Steiner Tree Computation over Large Graphs Fang WeiKleinera
"... In this paper, we present an exact algorithm for the Steiner tree problem. The algorithm is based on certain precomputed index structures. Our algorithm offers a practical solution for the Steiner tree problems on graphs of large size and bounded number of terminals. ..."
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In this paper, we present an exact algorithm for the Steiner tree problem. The algorithm is based on certain precomputed index structures. Our algorithm offers a practical solution for the Steiner tree problems on graphs of large size and bounded number of terminals.
Strong Steiner Tree Approximations in Practice?
"... Abstract. In this experimental study we consider Steiner tree approximations that guarantee a constant approximation of ratio less than 2. The considered greedy algorithms and approaches based on linear programming involve the incorporation of krestricted full components for some k ≥ 3. For most ..."
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Abstract. In this experimental study we consider Steiner tree approximations that guarantee a constant approximation of ratio less than 2. The considered greedy algorithms and approaches based on linear programming involve the incorporation of krestricted full components for some k ≥ 3. For most of the algorithms, their strongest theoretical approximation bounds are only achieved for k →∞. However, the running time is also exponentially dependent on k, so only small k are tractable in practice. We investigate different implementation aspects and parameter choices that finally allow us to construct algorithms (somewhat) feasible for practical use. We compare the algorithms against each other, to an exact LPbased algorithm, and to fast and simple 2approximations. 1