Approximating sparsest cut in graphs of bounded treewidth
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
A PRG for lipschitz functions of polynomials with applications to sparsest cut
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Sparsest cut on bounded treewidth graphs: algorithms and hardness results
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Improved Approximation of Linear Threshold Functions
Computational Complexity
| Hi-index | 0.00 |
We improve upon recent lower bounds on the minimum distortion of embedding certain finite metric spaces into $L_1$. In particular, we show that for every $n\ge1$, there is an $n$-point metric space of negative type that requires a distortion of $\Omega(\log\log n)$ for such an embedding, implying the same lower bound on the integrality gap of a well-known semidefinite programming relaxation for sparsest cut. This result builds upon and improves the recent lower bound of $(\log\log n)^{1/6-o(1)}$ due to Khot and Vishnoi [The unique games conjecture, integrality gap for cut problems and the embeddability of negative type metrics into $l_1$, in Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, IEEE, Piscataway, NJ, 2005, pp. 53-62]. We also show that embedding the edit distance metric on $\{0,1\}^n$ into $L_1$ requires a distortion of $\Omega(\log n)$. This result improves a very recent $(\log n)^{1/2-o(1)}$ lower bound by Khot and Naor [Nonembeddability theorems via Fourier analysis, in Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, IEEE, Piscataway, NJ, 2005, pp. 101-112].