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Prizecollecting Survivable Network Design in Nodeweighted Graphs
"... We consider nodeweighted network design problems, in particular the survivable network design problem (SNDP) and its prizecollecting version (PCSNDP). The input consists of a nodeweighted undirected graph G = (V, E) and integral connectivity requirements r(st) for each pair of nodes st. The go ..."
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We consider nodeweighted network design problems, in particular the survivable network design problem (SNDP) and its prizecollecting version (PCSNDP). The input consists of a nodeweighted undirected graph G = (V, E) and integral connectivity requirements r(st) for each pair of nodes st. The goal is to find a minimum nodeweighted subgraph H of G such that, for each pair st, H contains r(st) edgedisjoint paths between s and t. PCSNDP is a generalization in which the input also includes a penalty π(st) for each pair, and the goal is to find a subgraph H to minimize the sum of the weight of H and the sum of the penalties for all pairs whose connectivity requirements are not fully satisfied by H. Let k = maxst r(st) be the maximum requirement. There has been no nontrivial approximation for nodeweighted PCSNDP for k> 1, the main reason being the lack of an LP relaxation based approach for nodeweighted SNDP. In this paper we describe multirouteflow based relaxations for the two problems and obtain approximation algorithms for PCSNDP through them. The approximation ratios we obtain for PCSNDP are similar to those that were previously known for SNDP via combinatorial algorithms. Specifically, we obtain an O(k 2 log n)approximation in general graphs and an O(k 2)approximation in graphs that exclude a fixed minor. The approximation ratios can be improved by a factor of k but the running times of the algorithms depend polynomially on n k.
APPROXIMATION ALGORITHMS FOR SUBMODULAR OPTIMIZATION AND GRAPH PROBLEMS
, 2013
"... In this thesis, we consider combinatorial optimization problems involving submodular functions and graphs. The problems we study are NPhard and therefore, assuming that P 6 = NP, there do not exist polynomialtime algorithms that always output an optimal solution. In order to cope with the intracta ..."
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In this thesis, we consider combinatorial optimization problems involving submodular functions and graphs. The problems we study are NPhard and therefore, assuming that P 6 = NP, there do not exist polynomialtime algorithms that always output an optimal solution. In order to cope with the intractability of these problems, we focus on algorithms that construct approximate solutions: An approximation algorithm is a polynomialtime algorithm that, for any instance of the problem, it outputs a solution whose value is within a multiplicative factor ρ of the value of the optimal solution for the instance. The quantity ρ is the approximation ratio of the algorithm and we aim to achieve the smallest ratio possible. Our focus in this thesis is on designing approximation algorithms for several combinatorial optimization problems. In the first part of this thesis, we study a class of constrained submodular minimization problems. We introduce a model that captures allocation problems with submodular costs and we give a generic approach for designing approximation algorithms for problems in this model. Our model captures several problems of interest, such as nonmetric facility location, multiway cut problems in graphs and hypergraphs, uniform metric labeling and its generalization
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