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International Workshop on Approximation Algorithms for Combinatorial Optimization

International Workshop on Randomization and Approximation Techniques in Computer Science

APPROX 2012, RANDOM 2012: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques pp 98–109Cite as

Prize-Collecting Survivable Network Design in Node-Weighted Graphs

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7408))

Abstract

We consider node-weighted network design problems, in particular the survivable network design problem (SNDP) and its prize-collecting version (PC-SNDP). The input consists of a node-weighted undirected graph G = (V,E) and integral connectivity requirements r(st) for each pair of nodes st. The goal is to find a minimum node-weighted subgraph H of G such that, for each pair st, H contains r(st) edge-disjoint paths between s and t. PC-SNDP 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 =  max st r(st) be the maximum requirement. There has been no non-trivial approximation for node-weighted PC-SNDP for k > 1, the main reason being the lack of an LP relaxation based approach for node-weighted SNDP. In this paper we describe multiroute-flow based relaxations for the two problems and obtain approximation algorithms for PC-SNDP through them. The approximation ratios we obtain for PC-SNDP are similar to those that were previously known for SNDP via combinatorial algorithms. Specifically, we obtain an O(k 2 logn)-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.

Partially supported by NSF grant CCF-1016684. A longer version containing the omitted proofs will be made available on the authors’ webpages.

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Authors and Affiliations

  1. Dept. of Computer Science, University of Illinois, Urbana, IL, 61801, USA

    Chandra Chekuri, Alina Ene & Ali Vakilian

Authors
  1. Chandra Chekuri
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  2. Alina Ene
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  3. Ali Vakilian
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Editor information

Editors and Affiliations

  1. Department of Computer Science, Carnegie Mellon University, Gates Building 7203, 15213, Pittsburgh, PA, USA

    Anupam Gupta

  2. Department of Computer Science, University Kiel, Olshausenstraße 40, 24098, Kiel, Germany

    Klaus Jansen

  3. Centre Universitaire d’Informatique, University of Geneva, Battelle A, 7 route de Drize, 1227, Carouge, Switzerland

    José Rolim

  4. Department of Computer Science, Foundation School of Engineering and Applied Science, Columbia University, Amsterdam Avenue 1214, 10027-7003, New York, NY, USA

    Rocco Servedio

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Chekuri, C., Ene, A., Vakilian, A. (2012). Prize-Collecting Survivable Network Design in Node-Weighted Graphs. In: Gupta, A., Jansen, K., Rolim, J., Servedio, R. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2012 2012. Lecture Notes in Computer Science, vol 7408. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32512-0_9

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  • DOI: https://doi.org/10.1007/978-3-642-32512-0_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-32511-3

  • Online ISBN: 978-3-642-32512-0

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