Abstract
We discuss extensions of Jain’s framework for network design [8] that go beyond undirected graphs. The main problem is approximating a minimum cost set of directed edges that covers a crossing supermodular function. We show that iterated rounding gives a factor 3 approximation, where factor 4 was previously known and factor 2 was conjectured. Our bound is tight for the simplest interpretation of iterated rounding. We also show that (the simplest version of) iterated rounding has unbounded approximation ratio when the problem is extended to mixed graphs.
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References
Cheriyan, J., Vempala, S.: Edge covers of setpairs and the iterative rounding method. In:Proc. 8th International Integer Programming and Combinatorial Optimization Conf., pp. 30–44 (2001)
Cheriyan, J., Vempala, S., Vetta, A.: Approximation algorithms for minimum-cost k-vertex connected subgraphs. In:Proc. 34th Annual ACM Symp. on Theory of Comput., pp. 306–312 (2002)
Frank A. (1979). Kernel systems of directed graphs. Acta Sci. Math., Szeged, Hungary 41: 63–76
Frank A. (1993). Submodular functions in graph theory. Discrete Math. 111: 231–243
Frederickson G.N. and Ja’Ja’ J. (1981). Approximation algorithms for several graph augmentation problems. SIAM J. Comput. 10: 270–283
Fleischer, L., Jain, K., Williamson, D.P.: Iterative rounding 2-approximation algorithms for minimum-cost vertex connectivity problems, submitted for publication; union of An iterative rounding 2-approximation algorithm for the element connectivity problem. In: Proc. 42nd Annual IEEE Symp. on Foundations of Comp. Sci., pp. 339–347 (2001); A 2-approximation for minimum cost {0,1,2} vertex connectivity. In: Proc. 8th International Integer Programming and Combinatorial Optimization Conf., pp. 115–129 (2001)
Gabow, H.N., Goemans, M. X., Tardos, E., Williamson, D.P.: Approximating the smallest k-edge connected spanning subgraph by LP-rounding. In: Proc. 16th Annual ACM-SIAM Symp. on Disc. Algorithms, pp. 562–571 (2005)
Jain K. (2001). A factor 2 approximation algorithm for the generalized Steiner network problem. Combinatorica 21: 39–60
Khuller S. and Vishkin U. (1994). Biconnectivity approximations and graph carvings. J. ACM 41: 214–235
Kortsarz, G., Nutov, Z.: Approximation algorithm for k-node connected subgraphs via critical graphs. In:Proc. 36th Annual ACM Symp. on Theory of Comput., pp. 138–145 (2004)
Melkonian V. and Tardos E. (2004). Algorithms for a network design problem with crossing supermodular demands. Networks 43: 256–265
Schrijver A. (2003). Combinatorial Optimization: Polyhedra and Efficiency. Springer, Berlin Heidelberg New York
Vazirani V.V. (2001). Approximation Algorithms. Springer, Berlin Heidelberg New York
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Gabow, H.N. On the L∞-norm of extreme points for crossing supermodular directed network LPs. Math. Program. 110, 111–144 (2007). https://doi.org/10.1007/s10107-006-0061-9
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DOI: https://doi.org/10.1007/s10107-006-0061-9
Keywords
- Approximation algorithms
- Network design
- Iterated rounding
- Linear programming
- Extreme point
- Basic feasible solution
- L∞-norm
- Directed networks
- Crossing supermodular function