Abstract
We introduce the Minimum-size bounded-capacity cut (MinSBCC) problem, in which we are given a graph with an identified source and seek to find a cut minimizing the number of nodes on the source side, subject to the constraint that its capacity not exceed a prescribed bound B. Besides being of interest in the study of graph cuts, this problem arises in many practical settings, such as in epidemiology, disaster control, military containment, as well as finding dense subgraphs and communities in graphs.
In general, the MinSBCC problem is NP-complete. We present an efficient \((\frac{1}{{\rm \lambda}},\frac{1}{1-{\rm \lambda}})\)-bicriteria approximation algorithm for any 0 < λ < 1; that is, the algorithm finds a cut of capacity at most \(\frac{1}{{\rm \lambda}}B\), leaving at most \(\frac{1}{1-{\rm \lambda}}\) times more vertices on the source side than the optimal solution with capacity B. In fact, the algorithm’s solution either violates the budget constraint, or exceeds the optimal number of source-side nodes, but not both. For graphs of bounded treewidth, we show that the problem with unit weight nodes can be solved optimally in polynomial time, and when the nodes have weights, approximated arbitrarily well by a PTAS.
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References
Ahuja, R., Magnanti, T., Orlin, J.: Network Flows. Prentice-Hall, Englewood Cliffs (1993)
Arora, S., Rao, S., Vazirani, U.: Expander flows, geometric embeddings and graph partitioning. In: STOC (2004)
Asahiro, Y., Iwama, K., Tamaki, H., Tokuyama, T.: Greedily finding a dense subgraph. Journal of Algorithms, 34 (2000)
Bailey, N.: The Mathematical Theory of Infectious Diseases and its Applications. Hafner Press, New York (1975)
Bodlaender, H.L.: A linear time algorithm for finding tree-decompositions of small treewidth. SIAM J. on Computing 25, 1305–1317 (1996)
Develin, M., Hartke, S.G.: Fire containment in grids of dimension three and higher (2004) (Submitted)
Eubank, S., Guclu, H., Kumar, V.S.A., Marathe, M.V., Srinivasan, A., Toroczkai, Z., Wang, N.: Modelling disease outbreaks in realistic urban social networks. Nature 429, 180–184 (2004)
Eubank, S., Kumar, V.S.A., Marathe, M.V., Srinivasan, A., Wang, N.: Structure of social contact networks and their impact on epidemics. AMS-DIMACS Special Volume on Epidemiology
Eubank, S., Kumar, V.S.A., Marathe, M.V., Srinivasan, A., Wang, N.: Structural and algorithmic aspects of massive social networks. In: SODA (2004)
Feige, U., Kortsarz, G., Peleg, D.: The dense k-subgraph problem. In: STOC (1993)
Feige, U., Krauthgamer, R.: A polylogarithmic approximation of the minimum bisection. SIAM J. on Computing 31, 1090–1118 (2002)
Feige, U., Krauthgamer, R., Nissim, K.: On cutting a few vertices from a graph. Discrete Applied Mathematics 127, 643–649 (2003)
Feige, U., Seltser, M.: On the densest k-subgraph problem. Technical report, The Weizmann Institute, Rehovot (1997)
Flake, G., Lawrence, S., Giles, C.L., Coetzee, F.: Self-organization of the web and identification of communities. IEEE Computer, 35 (2002)
Flake, G., Tarjan, R., Tsioutsiouliklis, K.: Graph clustering techniques based on minimum cut trees. Technical Report 2002-06, NEC, Princeton (2002)
Ford, L., Fulkerson, D.: Maximal flow through a network. Can. J. Math. (1956)
Gallo, G., Grigoriadis, M.D., Tarjan, R.E.: A fast parametric maximum flow algorithm and applications. SIAM J. on Computing 18, 30–55 (1989)
Garg, N., Vazirani, V.V., Yannakakis, M.: Approximate max-flow min-(multi)cut theorems and their applications. SIAM J. on Computing (1996)
Kleinberg, J., Tardos, E.: Algorithm Design. Addison-Wesley, Reading (2005)
Kumar, R., Raghavan, P., Rajagopalan, S., Tomkins, A.: Trawling the web for emerging cyber-communities. In: WWW (1999)
Lawler, E.: Combinatorial Optimization: Networks and Matroids. Holt, Rinehard and Winston (1976)
Leighton, F.T., Rao, S.: Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms. Journal of the ACM, 46 (1999)
Radicchi, F., Castellano, C., Cecconi, F., Loreto, V., Parisi, D.: Defining and identifying communities in networks. Proc. Natl. Acad. Sci. USA (2004)
Shmoys, D.: Cut problems and their application to divide-and-conquer. In: Hochbaum, D. (ed.) Approximation Algorithms for NP-hard problems, pp. 192–235. PWD Publishing (1995)
Svitkina, Z., Tardos, E.: Min-max multiway cut. In: Jansen, K., Khanna, S., Rolim, J.D.P., Ron, D. (eds.) RANDOM 2004 and APPROX 2004. LNCS, vol. 3122, pp. 207–218. Springer, Heidelberg (2004)
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Hayrapetyan, A., Kempe, D., Pál, M., Svitkina, Z. (2005). Unbalanced Graph Cuts. In: Brodal, G.S., Leonardi, S. (eds) Algorithms – ESA 2005. ESA 2005. Lecture Notes in Computer Science, vol 3669. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11561071_19
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DOI: https://doi.org/10.1007/11561071_19
Publisher Name: Springer, Berlin, Heidelberg
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