An O(log k) Approximate Min-Cut Max-Flow Theorem and Approximation Algorithm
SIAM Journal on Computing
Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms
Journal of the ACM (JACM)
Approximate Max-Flow Min-(Multi)Cut Theorems and Their Applications
SIAM Journal on Computing
Minimizing Congestion in General Networks
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
The all-or-nothing multicommodity flow problem
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Multicommodity flow, well-linked terminals, and routing problems
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
The geometry of graphs and some of its algorithmic applications
SFCS '94 Proceedings of the 35th Annual Symposium on Foundations of Computer Science
Expander flows, geometric embeddings and graph partitioning
Journal of the ACM (JACM)
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
Vertex sparsifiers: new results from old techniques
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Edge Disjoint Paths in Moderately Connected Graphs
SIAM Journal on Computing
Metric Extension Operators, Vertex Sparsifiers and Lipschitz Extendability
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Vertex Sparsifiers and Abstract Rounding Algorithms
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Approximation Algorithms for the Edge-Disjoint Paths Problem via Raecke Decompositions
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
On mimicking networks representing minimum terminal cuts
Information Processing Letters
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Given an undirected graph G=(V,E) with edge capacities ce≥ 1 for e∈ E and a subset T of k vertices called terminals, we say that a graph H is a quality-q cut sparsifier for G iff T⊆ V(H), and for any partition (A,B) of T, the values of the minimum cuts separating A and B in graphs G and H are within a factor q from each other. We say that H is a quality-q flow sparsifier for G iff T⊆ V(H), and for any set D of demands over the terminals, the values of the minimum edge congestion incurred by fractionally routing the demands in D in graphs G and H are within a factor q from each other. So far vertex sparsifiers have been studied in a restricted setting where the sparsifier H is not allowed to contain any non-terminal vertices, that is V(H)=T. For this setting, efficient algorithms are known for constructing quality-O(log k/log log k) cut and flow vertex sparsifiers, as well as a lower bound of Ω(√log k) on the quality of any flow or cut sparsifier. We study flow and cut sparsifiers in the more general setting where Steiner vertices are allowed, that is, we no longer require that V(H)=T. We show algorithms to construct constant-quality cut sparsifiers of size O(C3) in time poly(n)• 2C, and constant-quality flow sparsifiers of size CO(log log C) in time nO(log C)• 2C, where C is the total capacity of the edges incident on the terminals.