All-pairs min-cut in sparse networks
Journal of Algorithms
Characterizations of k-terminal flow networks and computing network flows in partial k-trees
Proceedings of the sixth annual ACM-SIAM symposium on Discrete algorithms
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
Extensions and limits to vertex sparsification
Proceedings of the forty-second ACM symposium on Theory of computing
Metric Extension Operators, Vertex Sparsifiers and Lipschitz Extendability
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
On vertex sparsifiers with Steiner nodes
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
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Given a capacitated undirected graph G=(V,E) with a set of terminals K@?V, a mimicking network is a smaller graph H=(V"H,E"H) which contains the set of terminals K and for every bipartition [U,K-U] of the terminals, the cost of the minimum cut separating U from K-U in G is exactly equal to the cost of the minimum cut separating U from K-U in H. In this work, we improve both the previous known upper bound of 2^2^^^k[1] and lower bound of (k+1)[2] for mimicking networks, reducing the doubly-exponential gap between them to a single-exponential gap as follows:*Given a graph G, we exhibit a construction of mimicking network with at most k'th Hosten-Morris number (~2^(^(^k^-^1^)^@?^(^k^-^1^)^/^2^@?^)) of vertices (independent of the size of V). *There exist graphs with k terminals that have no mimicking network with less than 2^k^-^1^2 number of vertices.