# Towards Duality of Multicommodity Multiroute Cuts and Flows: Multilevel Ball-Growing

- 155 Downloads
- 1 Citations

## Abstract

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*(log^{4} *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*(log^{4} *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.

## Keywords

Multicommodity flow Approximation algorithms Duality## Notes

### Acknowledgements

The first author would like to thank Jiří Sgall and Thomas Erlebach for stimulating discussions.

## References

- 1.Aggarwal, C.C., Orlin, J.B.: On multiroute maximum flows in networks. Networks
**39**, 43–52 (2002) MathSciNetMATHCrossRefGoogle Scholar - 2.Aumann, Y., Rabani, Y.: An
*O*(log*k*) approximate min-cut max-flow theorem and approximation algorithm. SIAM J. Comput.**27**(1), 291–301 (1998) MathSciNetMATHCrossRefGoogle Scholar - 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
- 4.Baier, G., Erlebach, T., Hall, A., Köhler, E., Kolman, P., Pangrác, O., Schilling, H., Skutella, M.: Length-bounded cuts and flows. ACM Trans. Algorithms
**7**(1), 4–27 (2010) MathSciNetCrossRefGoogle 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
- 8.Dahlhaus, E., Johnson, D.S., Papadimitriou, C.H., Seymour, P.D., Yannakakis, M.: The complexity of multiterminal cuts. SIAM J. Comput.
**23**(4), 864–894 (1994). Preliminary version in Proceedings of the 24th ACM Symposium on Theory of Computing (STOC) (1992) MathSciNetMATHCrossRefGoogle Scholar - 9.Ford, L.R., Fulkerson, D.R.: Maximum flow through a network. Can. J. Math.
**8**, 399–404 (1956) MathSciNetMATHCrossRefGoogle Scholar - 10.Garg, N., Vazirani, V.V., Yannakakis, M.: Approximate max-flow min-cut theorems and their applications. SIAM J. Comput.
**25**(2), 235–251 (1996) MathSciNetMATHCrossRefGoogle 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) MathSciNetMATHCrossRefGoogle Scholar - 16.Linial, N., London, E., Rabinovich, Y.: The geometry of graphs and some of its algorithmic applications. Combinatorica
**15**, 215–245 (1995) MathSciNetMATHCrossRefGoogle 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
- 19.Williamson, D.P., Shmoys, D.B.: The Design of Approximation Algorithms. Cambridge University Press, Cambridge (2011) MATHCrossRefGoogle Scholar