## Abstract

We study vertex cut and flow sparsifiers that were recently introduced by Moitra (2009), and Leighton and Moitra (2010). 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 sparsifier, and improving the previous algorithmic upper bound of *O*(log^{2}
*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 1930s. Using this connection, we obtain a lower bound of \(\Omega \left( {\sqrt {\log k/\log \log k} } \right)\) for flow sparsifiers and a lower bound of \(\Omega \left( {\sqrt {\log k} /\log \log k} \right)\) for cut sparsifiers. We show that if a certain open question posed by Ball in 1992 has a positive answer, then there exist \(\tilde O\left( {\sqrt {\log k} } \right)\) cut sparsifiers. On the other hand, any lower bound on cut sparsifiers better than \(\tilde \Omega \left( {\sqrt {\log k} } \right)\) would imply a negative answer to this question.

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## References

- [1]
S. Arora, J. Lee and A. Naor,

*Fréchet embeddings of negative type metrics*. Discrete & Computational Geometry**38**(2007), 726–739. - [2]
S. Arora, S. Rao and U. Vazirani,

*Expander flows, geometric embeddings, and graph partitionings*, in*Proceedings of the 36th Annual ACM Symposium on Theory of Computing*, ACM, New York, 2004, pp. 222–231. - [3]
K. Ball,

*Markov chains*, Riesz transforms and Lipschitz maps, Geometric and Functional Analysis**2**(1992), 137–172. - [4]
J. Bourgain,

*A counterexample to a complementation problem*, Compositio Mathematica**43**(1981), 133–144. - [5]
G. Calinescu, H. Karloff and Y. Rabani,

*Approximation algorithms for the 0-extension problem*, in*Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms*(Washington, DC, 2001), SIAM, Philadelphia, PA, 2001, pp. 8–16. - [6]
M. Charikar, T. Leighton, S. Li and A. Moitra,

*Vertex sparsifiers and abstract rounding algorithms*, in*Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science*(Las Vegas, NV, 2010), IEEE Computer Society, Los Alamitos, CA, 2010, pp. 265–274. - [7]
M. Englert, A. Gupta, R. Krauthgamer, H. Räcke, I. Talgam-Cohen and K. Talwar,

*Vertex sparsifiers: New results from old techniques*, in*Approximation, Randomization, and Combinatorial Optimization*, Lecture Notes in Computer Science, Vol. 6302, Springer, Berlin, 2010, pp. 152–165. - [8]
J. Fakcharoenphol, C. Harrelson, S. Rao and K Talwar,

*An improved approximation algorithm for the 0-extension problem*, in*Proceedings of the Fourteenth Annual ACMSIAM Symposium on Discrete Algorithms*(Baltimore, MD, 2003), ACM, New York, 2003, pp. 257–265. - [9]
J. Fakcharoenphol and K. Talwar,

*An improved decomposition theorem for graphs excluding a fixed minor*, in*Approximation, Randomization, and Combinatorial Optimization*, Lecture Notes in Computer Science, Vol. 2764, Springer, Berlin, 2003, pp. 36–46. - [10]
T. Figiel, W. Johnson and G. Schechtman,

*Factorizations of natural embeddings of ln p into L*_{ r },*I*, Studia Mathematica**89**(1988), 79–103. - [11]
B. Grünbaum,

*Projection constants*, Transactions of the American Mathematical Society,**95**(1960), 451–465. - [12]
U. Haagerup,

*The best constants in the Khintchine inequality*, Studia Mathematica**70**(1981), 231–283. - [13]
W. Johnson and J. Lindenstrauss,

*Extensions of Lipschitz mappings into a Hilbert space*, in*Conference in Modern Analysis and Probability*(New Haven, Conn., 1982), Contemporary Mathematics, Vol. 26, American Mathematical Society, Providence, RI, 1984, pp. 189–206. - [14]
A. Karzanov,

*Minimum 0-extension of graph metrics*, European Journal of Combinatorics**19**(1998), 71–101. - [15]
B. Kashin,

*The widths of certain finite-dimensional sets and classes of smooth functions*, Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya**41**(1977), 334–351. - [16]
M. D. Kirszbraun,

*Über die zusammenziehenden und Lipschitzchen Transformationen*, Fundamenata Mathematicae**22**(1934), 77–108. - [17]
J. Lee and A. Naor,

*Extending Lipschitz functions via random metric partitions*, Inventiones Mathematicae**160**(2005), 59–95. - [18]
J. Lee and A. Sidiropoulos,

*Genus and the geometry of the cut graph*, in*Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms*(Austin, TX, 2010) SIAM, Philadelphia, PA, 2010, pp. 193–201. - [19]
T. Leighton and A. Moitra,

*Extensions and limits to vertex sparsification*, in*Proceedings of the 2010 ACM International Symposium on Theory of Computing*, ACM, New York, 2010, pp. 47–56. - [20]
M. B. Marcus and G. Pisier,

*Characterizations of almost surely continuous p-stable random Fourier series and strongly stationary processes*, Acta Mathematica**152**(1984), 245–301. - [21]
E. J. McShane,

*Extension of range of functions*, Bulletin of the American Mathematical Society**40**(1934), 837–842. - [22]
M. Mendel and A. Naor,

*Some applications of Ball’s extension theorem*, Proceedings of the American Mathematical Society**134**(2006), 2577–2584. - [23]
A. Moitra,

*Approximation algorithms for multicommodity-type problems with guarantees independent of the graph size*, in*Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science*(Atlanta, GA, 2009) IEEE Computer Society, Los Alamitos, CA, 2009, pp. 3–12. - [24]
A. Naor, Y. Peres, O. Schrammand S. Sheffield,

*Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces*, Duke Mathematical Journal**134**(2006), 165–197. - [25]
J. von Neumann,

*Zur Theorie der Gesellshaftsphiele*, Mathematische Annalen**100**(1928), 295–320. - [26]
H. Räcke,

*Optimal hierarchical decompositions for congestion minimization in networks*, in*Proceedings of the 40th Annual ACM Symposium on Theory of Computing*(Victoria, BC, 2008), ACM, New York, 2008, pp. 255–264. - [27]
B. Randrianantoanina,

*Extensions of Lipschitz maps*, International Conference on Banach Spaces and Operator Spaces, 2007. - [28]
A. Schrijver,

*Combinatorial Optimization: Polyhedra and Efficiency*, Algorithms and Combinatorics, Vol. 24, Springer, Berlin, 2003. - [29]
M. Sion,

*On general minimax theorems*, Pacific Journal of Mathematics**8**(1958), 171–176.

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The conference version of this paper appeared at FOCS 2010.

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Makarychev, K., Makarychev, Y. Metric extension operators, vertex sparsifiers and Lipschitz extendability.
*Isr. J. Math.* **212, **913–959 (2016). https://doi.org/10.1007/s11856-016-1315-8

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