Non-Uniform Robust Network Design in Planar Graphs
Robust optimization is concerned with constructing solutions that remain feasible also when a limited number of resources is removed from the solution. Most studies of robust combinatorial optimization to date made the assumption that every resource is equally vulnerable, and that the set of scenarios is implicitly given by a single budget constraint. This paper studies a robustness model of a different kind. We focus on Bulk-Robustness, a model recently introduced (Adjiashvili, Stiller, Zenklusen 2015) for addressing the need to model non-uniform failure patterns in systems.
We significantly extend the techniques used by Adjiashvili et al. to design approximation algorithm for bulk-robust network design problems in planar graphs. Our techniques use an augmentation framework, combined with linear programming (LP) rounding that depends on a planar embedding of the input graph. A connection to cut covering problems and the dominating set problem in circle graphs is established. Our methods use few of the specifics of bulk-robust optimization, hence it is conceivable that they can be adapted to solve other robust network design problems.
Robust optimization
Network design
Planar graph
Approximation algorithm
LP rounding
61-77
Regular Paper
David
Adjiashvili
David Adjiashvili
10.4230/LIPIcs.APPROX-RANDOM.2015.61
D. Adjiashvili. Structural Robustness in Combinatorial Optimization. PhD thesis, ETH Zürich, Zürich, Switzerland, 2012.
D. Adjiashvili, G. Oriolo, and M. Senatore. The online replacement path problem. In Proceedings of 18th Annual European Symposium on Algorithms (ESA), pages 1-12. Springer, 2013.
D. Adjiashvili, S. Stiller, and R. Zenklusen. Bulk-robust combinatorial optimization. Mathematical Programming, 149(1-2):361-390, 2014.
D. Adjiashvili and R. Zenklusen. An s-t connection problem with adaptability. Discrete Applied Mathematics, 159:695-705, 2011.
H. Aissi, C. Bazgan, and D. Vanderpooten. Min–max and min–max regret versions of combinatorial optimization problems: A survey. European Journal of Operational Research, 197(2):427-438, 2009.
N. Bansal and K. Pruhs. The geometry of scheduling. In 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 407-414, 2010.
D. Bertsimas, D. B. Brown, and C. Caramanis. Theory and applications of robust optimization. SIAM review, 53:464-501, 2011.
C. Chekuri, J. Vondrák, and R. Zenklusen. Multi-budgeted matchings and matroid intersection via dependent rounding. In Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1080-1097, 2011.
J. Cheriyan and R. Thurimella. Approximating Minimum-Size k-Connected Spanning Subgraphs via Matching. SIAM J. Comput, 30:292-301, 2000.
M. Damian-Iordache and S. V. Pemmaraju. A (2+ ε)-approximation scheme for minimum domination on circle graphs. Journal of Algorithms, 42(2):255-276, 2002.
K. Dhamdhere, V. Goyal, R. Ravi, and M. Singh. How to pay, come what may: approximation algorithms for demand-robust covering problems. In 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 367-376, October 2005.
U. Feige, K. Jain, M. Mahdian, and V. Mirrokni. Robust combinatorial optimization with exponential scenarios. In Matteo Fischetti and David Williamson, editors, IPCO 2007, volume 4513 of Lecture Notes in Computer Science, pages 439-453. Springer Berlin / Heidelberg, 2007.
H. N. Gabow, M. X. Goemans, É. Tardos, and D. P. Williamson. Approximating the smallest k-edge connected spanning subgraph by lp-rounding. Networks, 53(4):345-357, 2009.
H. N. Gabow, M.X. Goemans, and D.P. Williamson. An efficient approximation algorithm for the survivable network design problem. Mathematical Programming, Series B, 82(1-2):13-40, 1998.
D. Golovin, V. Goyal, and R. Ravi. Pay today for a rainy day: Improved approximation algorithms for demand-robust min-cut and shortest path problems. In Bruno Durand and Wolfgang Thomas, editors, STACS 2006, volume 3884 of Lecture Notes in Computer Science, pages 206-217. Springer Berlin / Heidelberg, 2006.
F. Grandoni, R. Ravi, M. Singh, and R. Zenklusen. New approaches to multi-objective optimization. Mathematical Programming, 146(1-2):525-554, 2014.
V. Guruswami, S. Sachdeva, and R. Saket. Inapproximability of minimum vertex cover on k-uniform k-partite hypergraphs. SIAM Journal on Discrete Mathematics, 29(1):36-58, 2015.
E. Israeli and R. K. Wood. Shortest-path network interdiction. Networks, 40:97-111, 2002.
K. Jain. A factor 2 approximation algorithm for the generalized steiner network problem. Combinatorica, 21:39-60, 2001.
H. Kerivin and A.R. Mahjoub. Design of survivable networks: A survey. Networks, 46(1):1-21, 2005.
R. Khandekar, G. Kortsarz, V. Mirrokni, and M. Salavatipour. Two-stage robust network design with exponential scenarios. In Dan Halperin and Kurt Mehlhorn, editors, ESA 2008, volume 5193 of Lecture Notes in Computer Science, pages 589-600. Springer Berlin / Heidelberg, 2008.
P. Kouvelis and G. Yu. Robust Discrete Optimization and Its Applications. Kluwer Academic Publishers, Boston., 1997.
N.-K. Olver. Robust network design. PhD thesis, McGill University, Montreal, Quebec, Canada, 2010.
C. H. Papadimitriou and M. Yannakakis. On the approximability of trade-offs and optimal access of web sources. In 41st Annual Symposium on Foundations of Computer Science (FOCS), pages 86-92, 2000.
R. Ravi, M. V. Marathe, Ravi S. S., Rosenkrantz D. J., and Hunt H. B. Many birds with one stone: Multi-objective approximation algorithms. In 25th annual ACM Symposium on the Theory of Computing (STOC), pages 438-447, 1993.
D.P. Williamson, M.X. Goemans, M. Mihail, and V.V. Vazirani. A primal-dual approximation algorithm for generalized steiner network problems. Combinatorica, 15(3):435-454, 1995.
G. Yu and J. Yang. On the robust shortest path problem. Computers & Operations Research, 25(6):457-468, 1998.
R. Zenklusen. Matching interdiction. Discrete Applied Mathematics, 158(15):1676-1690, 2010.
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode