Vertex Sparsification and Oblivious Reductions
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Given an undirected, capacitated graph $G = (V, E)$ and a set $K \subset V$ of terminals of size $k$, we construct an undirected, capacitated graph $G' = (K, E')$ for which the cut function approximates the value of every minimum cut separating any subset $U$ of terminals from the remaining terminals $K - U$. We refer to this graph $G'$ as a cut-sparsifier, and we prove that there are cut-sparsifiers that can approximate all these minimum cuts in $G$ to within an approximation factor that depends only polylogarithmically on $k$, the number of terminals. We prove such cut-sparsifiers exist through a zero-sum game, and we construct such sparsifiers through oblivious routing guarantees. These results allow us to derive a more general theory of Steiner cut and flow problems, and allow us to obtain approximation algorithms with guarantees independent of the size of the graph for a number of graph partitioning, graph layout, and multicommodity flow problems for which such guarantees were previously unknown.
DepartmentMassachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory; Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
SIAM Journal on Computing
Society for Industrial and Applied Mathematics
Moitra, Ankur. “Vertex Sparsification and Oblivious Reductions.” SIAM Journal on Computing 42, no. 6 (January 2013): 2400–2423. © 2013, Society for Industrial and Applied Mathematics.
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