About cookies on this site Our websites require some cookies to function properly (required). In addition, other cookies may be used with your consent to analyze site usage, improve the user experience and for advertising. For more information, please review your options. By visiting our website, you agree to our processing of information as described in IBM’sprivacy statement. To provide a smooth navigation, your cookie preferences will be shared across the IBM web domains listed here.
Publication
FOCS 2010
Conference paper
Metric extension operators, vertex sparsifiers and Lipschitz extendability
Abstract
We study vertex cut and flow sparsifiers that were recently introduced by Moitra [23], and Leighton and Moitra [18]. We improve and generalize their results. We give a new polynomial-time algorithm for constructing O(log k= log log k) cut and flow sparsifiers, matching the best known existential upper bound on the quality of a sparsi-fier, and improving the previous algorithmic upper bound of O(log2 k= log log k). We show that flow sparsifiers can be obtained from linear operators approximating minimum metric extensions. We introduce the notion of (linear) metric extension operators, prove that they exist, and give an exact polynomialtime algorithm for finding optimal operators. We then establish a direct connection between flow and cut sparsifiers and Lipschitz extendability of maps in Banach spaces, a notion studied in functional analysis since 1950s. Using this connection, we obtain a lower bound of Ω(√log k/log log k) for flow sparsifiers and a lower bound of Ω(√log k/log log k) for cut sparsifiers. We show that if a certain open question posed by Ball in 1992 has a positive answer, then there exist Õ(√log k) cut sparsifiers. On the other hand, any lower bound on cut sparsifiers better than Ω(√log k) would imply a negative answer to this question. © 2010 IEEE.