Abstract
Given an unweighted tree \(T=(V,E)\) with terminals \(K \subset V\), we show how to obtain a 2-quality vertex flow and cut sparsifier H with \(V_H = K\). We prove that our result is essentially tight by providing a \(2-o(1)\) lower-bound on the quality of any cut sparsifier for stars.
In addition we give improved results for quasi-bipartite graphs. First, we show how to obtain a 2-quality flow sparsifier with \(V_H = K\) for such graphs. We then consider the other extreme and construct exact sparsifiers of size \(O(2^{k})\), when the input graph is unweighted.
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Notes
- 1.
Alternatively, one can view this step as contracting an arbitrary child-edge of v.
- 2.
Note that the dual requires that \(\delta _{st}\) is at most the length of the shortest s-t path. In our scenario this is always a 2-hop path. Hence, the above formulation is correct.
References
Alon, N., Schieber, B.: Optimal preprocessing for answering on-line product queries. Technical report, Tel Aviv University (1987)
Andersen, R., Feige, U.: Interchanging distance and capacity in probabilistic mappings. CoRR, arXiv:abs/0907.3631 (2009)
Andoni, A., Gupta, A., Krauthgamer, R.: Towards (1+ \(\varepsilon \))-approximate flow sparsifiers. In: Proceedings of the 25th SODA, pp. 279–293 (2014)
Benczúr, A.A., Karger, D.R.: Approximating s-t minimum cuts in Õ(n\({}^{\text{2}}\)) time. In: Proceedings of the 28th STOC, pp. 47–55 (1996)
Chekuri, C., Shepherd, F.B., Oriolo, G., Scutellà, M.G.: Hardness of robust network design. Networks 50(1), 50–54 (2007)
Cheung, Y.K., Goranci, G., Henzinger, M.: Graph minors for preserving terminal distances approximately - lower and upper bounds. In: Proceedings of the 43rd ICALP, pp. 131:1–131:14 (2016)
Chuzhoy, J.: On vertex sparsifiers with steiner nodes. In: Proceedings of the 44th STOC, pp. 673–688 (2012)
Gupta, A.: Steiner points in tree metrics don’t (really) help. In: Proceedings of the 12th SODA, pp. 220–227 (2001)
Hagerup, T., Katajainen, J., Nishimura, N., Ragde, P.: Characterizing multiterminal flow networks and computing flows in networks of small treewidth. J. Comput. Syst. Sci. 57(3), 366–375 (1998)
Kamma, L., Krauthgamer, R., Nguyen, H.L.: Cutting corners cheaply, or how to remove steiner points. SIAM J. Comput. 44(4), 975–995 (2015)
Khan, A., Raghavendra, P.: On mimicking networks representing minimum terminal cuts. Inf. Process. Lett. 114(7), 365–371 (2014)
Krauthgamer, R., Nguyen, H.L., Zondiner, T.: Preserving terminal distances using minors. SIAM J. Discrete Math. 28(1), 127–141 (2014)
Krauthgamer, R., Rika, I.: Mimicking networks and succinct representations of terminal cuts. In: Proceedings of the 24th SODA, pp. 1789–1799 (2013)
Leighton, F.T., Moitra, A.: Extensions and limits to vertex sparsification. In: Proceedings of the 42nd STOC, pp. 47–56 (2010)
Makarychev, K., Makarychev, Y.: Metric extension operators, vertex sparsifiers and lipschitz extendability. In: Proceedings of the 51th FOCS, pp. 255–264 (2010)
Moitra, A.: Approximation algorithms for multicommodity-type problems with guarantees independent of the graph size. In: Proceedings of the 50th FOCS (2009)
Räcke, H.: Optimal hierarchical decompositions for congestion minimization in networks. In: Proceedings of the 40th STOC, pp. 255–264 (2008)
Räcke, H., Shah, C., Täubig, H.: Computing cut-based hierarchical decompositions in almost linear time. In: Proceedings of the 25th SODA, pp. 227–238 (2014)
Spielman, D.A., Teng, S.: Spectral sparsification of graphs. SIAM J. Comput. 40(4), 981–1025 (2011)
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Goranci, G., Räcke, H. (2017). Vertex Sparsification in Trees. In: Jansen, K., Mastrolilli, M. (eds) Approximation and Online Algorithms. WAOA 2016. Lecture Notes in Computer Science(), vol 10138. Springer, Cham. https://doi.org/10.1007/978-3-319-51741-4_9
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