Coarse Differentiation and Multi-flows in Planar Graphs
APPROX '07/RANDOM '07 Proceedings of the 10th International Workshop on Approximation and the 11th International Workshop on Randomization, and Combinatorial Optimization. Algorithms and Techniques
Bilipschitz snowflakes and metrics of negative type
Proceedings of the forty-second ACM symposium on Theory of computing
Approximate Lasserre integrality gap for unique games
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We prove that the function d : \mathbb{R}^3\times \mathbb{R}^3\to [0,\infty ) given by d\left( {(x,y,z),(t,u,v)} \right)= \left( {[((t - x)^2+ (u - y)^2 )^2+ (v - z + 2xu - 2yt)^2 ]^{\frac{1} {2}}+ (t - x)^2+ (u - y)^2 } \right)^{\frac{1} {2}} .is a metric on \mathbb{R}^3 such that (\mathbb{R}^3, \sqrt d ) is isometric to a subset of Hilbert space, yet (\mathbb{R}^3, d) does not admit a bi-Lipschitz embedding into L_1. This yields a new simple counter example to the Goemans-Linial conjecture on the integrality gap of the semidefinite relaxation of the Sparsest Cut problem. The metric above is doubling, and hence has a padded stochastic decomposition at every scale. We also study the L_p version of this problem, and obtain a counter example to a natural generalization of a classical theorem of Bretagnolle, Dacunha-Castelle and Krivine (of which the Goemans-Linial conjecture is a particular case). Our methods involve Fourier analytic techniques, and a recent breakthrough of Cheeger and Kleiner, together with classical results of Pansu on the differentiability of Lipschitz functions on the Heisenberg group.