1. Distance scales, embeddings, and efficient relaxations of the cut cone James R. Lee University of California, Berkeley

2. Distortion: Smallest number C such that… euclidean embeddings and geometry In general, we will consider euclidean embeddings and the geometry of high dimensional euclidean spaces. Why study this kind of geometry (in CS)? - Applicability of low-distortion Euclidean embeddings - Understanding semi-definite programs - Optimization, harmonic analysis, hardness of approximation, cuts and flows, Markov chains, expansion, randomness… Euclidean embedding: A map f : X ! R k for some metric space (X,d).

3. this paper Offers some improvements on two nice ideas… I. Measured descent Gluing Lemma II. Arora-Rao-Vazirani “Big Core” Theorem [Krauthgamer-L-Mendel-Naor 04]

4. goals and results Our initial motivation was the conjecture: Every n-point metric of negative type embeds into a Euclidean space (L 2 norm) with distortion, and this is tight. (known to imply a similar approximation for the general SparsestCut problem) We give improved embeddings of…

5. goals and results Our initial motivation was the conjecture: Every n-point metric of negative type embeds into a Euclidean space (L 2 norm) with distortion, and this is tight. Applications… Bandwidth, Euclidean volume-resp. embeddings (GL) Min-UnCut, 2CNF-deletion (BCT) [Agarwal, Charikar, Makarychev, Makarychev 04] vertex cover vanishing term (BCT) [Karakostas 04] general SparsestCut O(log n) 3/4 (BCT) [Chawla, Gupta, Racke 04] general SparsestCut O(log n log log n) 1 /2 (BCT, GL) [Arora, L, Naor 04]

6. the gluing lemma Every metric space (X,d) is composed of many scales (resolutions)… Often easy to construct an embedding for one scale:

7. the gluing lemma Often easy to construct an embedding for one scale: Idea of measured descent: Combine the component embeddings in a novel way to arrive at a final embedding which has low distortion. Problem: The component maps have to be of a very special form (Frechet embeddings) because the proof is non-black-box; it breaks the embeddings apart and puts them back together in complicated ways.

8. the gluing lemma G LUING L EMMA : We can construct an embedding of the same quality using the component maps as a black box. Idea of measured descent: Combine the component embeddings in a novel way to arrive at a final embedding which has low distortion. Problem: The component maps have to be of a very special form (Frechet embeddings) because the proof is non-black-box; it breaks the embeddings apart and puts them back together in complicated ways. Quantitatively, the distortion of the final map is. P ROOF :

9. metrics of negative type Metrics of negative type arise as follows: Let X µ R k be endowed with the distance function If (X,d) is a metric space, then X is a space of negative type. ·   x y z The family of negative type metrics is much bigger than the family of Euclidean metrics!

10. Improved version: Possible to choose the sets A and B “at random.” the “big core” theorem T HE ARV T HEOREM Let X µ S n-1 be an n-point subset of the unit sphere endowed with the squared Euclidean metric d(x,y) = ||x-y|| 2, and suppose that A B Then there exist two subsets A, B µ X such that | A |, | B | ¸  (n) and Good enough for lots of applications… but not others.

11. the “big core” theorem Randomized version: A B Choosing A and B “at random.” 1. Choose a random hyperplane. 2. Prune the “exceptions.” Pruning ) d(A,B) is large. The hard part is showing that |A|,|B| =  (n) whp. ARV yields. We obtain the optimal bound:.

12. the “big core” theorem The proof is a modification of the ARV geometric chaining argument… Idea: If the matchings are large in most directions, then a large subset of points must be matching against each other in most directions. T HE C ORE Theorem: (with param ) C ONTRADICTION ! Proof uses region growing to construct longer chains.

13. open questions Does a better gluing lemma exist for L 1 ? Can we pass from “deterministic” to “random” separators in a black-box manner? A B Q UESTIONS ?