Towards Duality of Multicommodity Multiroute Cuts and Flows: Multilevel Ball-Growing
An elementary h-route flow, for an integer h≥1, is a set of h edge-disjoint paths between a source and a sink, each path carrying a unit of flow, and an h-route flow is a non-negative linear combination of elementary h-route flows. An h-route cut is a set of edges whose removal decreases the maximum h-route flow between a given source-sink pair (or between every source-sink pair in the multicommodity setting) to zero. The main result of this paper is an approximate duality theorem for multicommodity h-route cuts and flows, for h≤3: The size of a minimum h-route cut is at least f/h and at most O(log4 k⋅f) where f is the size of the maximum h-route flow and k is the number of commodities. The main step towards the proof of this duality is the design and analysis of a polynomial-time approximation algorithm for the minimum h-route cut problem for h=3 that has an approximation ratio of O(log4 k). Previously, polylogarithmic approximation was known only for h-route cuts for h≤2. A key ingredient of our algorithm is a novel rounding technique that we call multilevel ball-growing. Though the proof of the duality relies on this algorithm, it is not a straightforward corollary of it as in the case of classical multicommodity flows and cuts. Similar results are shown also for the sparsest multiroute cut problem.
KeywordsMulticommodity flow Approximation algorithms Duality
The first author would like to thank Jiří Sgall and Thomas Erlebach for stimulating discussions.
- 3.Bagchi, A., Chaudhary, A., Kolman, P., Sgall, J.: A simple combinatorial proof for the duality of multiroute flows and cuts. Technical Report 2004-662, Charles University, Prague (2004) Google Scholar
- 5.Barman, S., Chawla, S.: Region growing for multi-route cuts. In: Proceedings of the 21th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) (2010) Google Scholar
- 6.Bruhn, H., Černý, J., Hall, A., Kolman, P., Sgall, J.: Single source multiroute flows and cuts on uniform capacity networks. Theory Comput. 4(1), 1–20 (2008). Preliminary version in Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) (2007) MathSciNetCrossRefGoogle Scholar
- 7.Chekuri, C., Khanna, S.: Algorithms for 2-route cut problems. In: Proceedings of the 35th International Colloquium on Automata (ICALP). Lecture Notes in Computer Science, vol. 5125, pp. 472–484 (2008) Google Scholar
- 11.Kahale, N.: On reducing the cut ratio to the multicut problem. DIMACS Technical report 93-78 (1993) Google Scholar
- 12.Kishimoto, W., Takeuchi, M.: On m-route flows in a network. IEICE Trans. J-76-A(8), 1185–1200 (1993) (in Japanese) Google Scholar
- 13.Kolman, P., Scheideler, C.: Towards duality of multicommodity multiroute cuts and flows: multilevel ball-growing. In: Proceedings of the 28th International Symposium on Theoretical Aspects of Computer Science (STACS), Leibniz International Proceedings in Informatics (LIPIcs) (2011) Google Scholar
- 14.Kolman, P., Scheideler, C.: Approximate duality of multicommodity multiroute flows and cuts: single source case. In: Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 800–810 (2012) Google Scholar
- 15.Leighton, T., Rao, S.: Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms. J. ACM 46(6), 787–832 (1999). Preliminary version in Proceedings of the 29th Annual IEEE Symposium on Foundations of Computer Science (FOCS) (1988) MathSciNetzbMATHCrossRefGoogle Scholar
- 17.Shmoys, D.B.: Cut problems and their application to divide-and-conquer. In: Hochbaum, D.S. (ed.) Approximation Algorithms for NP-hard Problems, pp. 192–235. PWS, Boston (1997) Google Scholar
- 18.Vazirani, V.V.: Approximation Algorithms. Springer, Berlin (2001) Google Scholar