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Improved Approximation Algorithms for (Budgeted) Nodeweighted Steiner Problems
"... Abstract. Moss and Rabani [12] study constrained nodeweighted Steiner tree problems with two independent weight values associated with each node, namely, cost and prize (or penalty). They give an O(logn)approximation algorithm for the prizecollecting nodeweighted Steiner tree problem (PCST)—wher ..."
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Abstract. Moss and Rabani [12] study constrained nodeweighted Steiner tree problems with two independent weight values associated with each node, namely, cost and prize (or penalty). They give an O(logn)approximation algorithm for the prizecollecting nodeweighted Steiner tree problem (PCST)—where the goal is to minimize the cost of a tree plus the penalty of vertices not covered by the tree. They use the algorithm for PCST to obtain a bicriteria (2, O(logn))approximation algorithm for the Budgeted nodeweighted Steiner tree problem— where the goal is to maximize the prize of a tree with a given budget for its cost. Their solution may cost up to twice the budget, but collects a factor Ω ( 1 logn) of the optimal prize. We improve these results from at least two aspects. Our first main result is a primaldual O(log h)approximation algorithm for a more general problem, prizecollecting nodeweighted Steiner forest (PCSF), where we have h demands each requesting the connectivity of a pair of vertices. Our approximation guarantee is tight within constant factors. Our algo
In Survivable Network problems (a.k.a. Survivable Network Design Problem – SNDP) we are
"... given a graph with costs/weights on edges and/or nodes and prescribed connectivity requirements/demands. Among the subgraphs of G that satisfy the requirements, we seek to find one of minimum cost. Formally, the problem is defined as follows. Given a graph G = (V,E) and Q ⊆ V, the Qconnectivity λQ ..."
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given a graph with costs/weights on edges and/or nodes and prescribed connectivity requirements/demands. Among the subgraphs of G that satisfy the requirements, we seek to find one of minimum cost. Formally, the problem is defined as follows. Given a graph G = (V,E) and Q ⊆ V, the Qconnectivity λQG(uv) of uv in G is the maximum number of edgedisjoint uvpaths such that no two of them have a node in Q − {u, v} in common. The case S = ∅ is just the edgeconnectivity when the paths should be edgedisjoint, and the case S = V is just the nodeconnectivity when the paths should be internally nodedisjoint. Survivable Network Instance: A (possibly directed) graph G = (V,E) with edge/nodecosts, a node subset Q ⊆ V, and a nonnegative integer requirements {ruv: uv ∈ D} on a set D of demand pairs on a set S ⊆ V of terminals. Objective: Find a minimum cost subgraph G ′ of G such that λQG′(uv)> ruv for all uv ∈ D. Extensively studied particular choices of Q are edgeconnectivity (Q = ∅), nodeconnectivity (Q = V), and elementconnectivity (ruv = 0 whenever u ∈ Q or v ∈ Q). Given an instance of Survivable Network let k = maxuv∈D ruv denote the maximum con
NodeWeighted Prize Collecting Steiner Tree and Applications
"... I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii The Steiner Tree problem has appeared in the Karp’s li ..."
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I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii The Steiner Tree problem has appeared in the Karp’s list of the first 21 NPhard problems and is well known as one of the most fundamental problems in Network Design area. We study the NodeWeighted version of the Prize Collecting Steiner Tree problem. In this problem, we are given a simple graph with a cost and penalty value associated with each node. Our goal is to find a subtree T of the graph minimizing the cost of the nodes in T plus penalty of the nodes not in T. By a reduction from set cover problem it can be easily shown that the problem cannot be approximated in polynomial time within factor of (1 − o(1)) lnn unless NP has quasipolynomial time algorithms, where n is the number of vertices of the graph. Moss and Rabani claimed an O(log n)approximation algorithm for the problem using