1. geometric embeddings and graph expansion James R. Lee Institute for Advanced Study (Princeton) University of Washington (Seattle)

2. outline 1.Philosophy of geometric embeddings 2.Example: Finding balanced cuts in graphs 3.Four important open problems in the talk: not in the talk: No proofs (one slide). Mathematics borrows from high-dimensional convex geometry, functional analysis, harmonic analysis, differential geometry... (see other talks on my web page) so you should ask questions if something is confusing!

3. geometric embeddings in CS combinatorial problem geometric representation embedding nicer geometric space combinatorial solution

4. connections in CS geometric search clustering dimension reduction machine learning computational biology approximation algorithms divide and conquer network design graph layout tree decompositions geometric optimization semi-definite programming PCPs, unique games fourier analysis of boolean functions

5. graph expansion and the sparsest cut Input: A graph G =( V,E). S E(S, S) For a cut (S,S) let E(S,S) denote the edges crossing the cut. The sparsity of S is the value The SPARSEST CUT problem is to find the cut which minimizes  (S). This problem is NP-hard, so we try to find approximately optimal cuts. (approximation algorithms)

6. graph expansion and the sparsest cut Given a graph G =( V,E), we want to Clustering Divide & conquer algorithms

7. graph expansion and the sparsest cut Given a graph G =( V,E), we want to This is actually the EDGE EXPANSION problem. The full SPARSEST CUT problem is a weighted version

8. where is the geometry? Leighton-Rao (1988) approach via LP duality d is a metric on V if d(x,y) = d(y,x) and d(x,y) · d(x,z) + d(z,y) 8 x,y,z 2 V “cut metric” d(x,y) = 1 if x,y are on different sides of S d(x,y) = 0 otherwise S S

9. where is the geometry? Leighton-Rao (1988) approach via LP duality d is a metric on V if d(x,y) = d(y,x) and d(x,y) · d(x,z) + d(z,y) 8 x,y,z 2 V can minimize with a linear program dual of the multi-commodity flow LP - every edge has capacity 1 - send 1 unit of flow from x ! y for every x,y 2 V

10. finding cuts using embeddings Now we find a cut using LP relaxation + embeddings [Linial London Rabinovich 1992] S S cut metric d RnRn S S LP relaxation ? 1. Want to find a good cut in G. 2. Solve a linear program to get a metric d. 3. Embed the metric into a Euclidean space. 4. Use a geometric algorithm to find S. (random hyperplane cut)

11. The distortion of f is the smallest number D such that for all x,y 2 X: embeddings and distortion Given a metric space (X,d), a Euclidean embedding of X a mapping f : X ! R n. distortion measures how well f preserves the structure of X

12. The distortion of f is the smallest number D such that for all x,y 2 X: embeddings and distortion Given a metric space (X,d), a Euclidean embedding of X a mapping f : X ! R n. Depending on the application, sometimes we consider the L 1 norm or the L 2 norm. - Embeddings into L 2 are stronger than L 1 embeddings - L 1 embeddings are good enough for finding sparse cuts - We have many fewer techniques for analyzing L 1 embeddings

13. first results [Bourgain 1985] Every n-point metric space has a Euclidean embedding (L 2 norm) with distortion O(log n). [Linial-London-Rabinovich, Aumann-Rabani STOC’92] - Can use this to get an O(log n)-approximation for the SPARSEST CUT problem. - Bourgain’s result is tight (using expander graphs)

14. new results semi-definite programming special family of metric spaces “negative type” A metric space (X,d) is said to be negative type if we can write where x u 2 R n for every u 2 X.

15. embedding overview metric spaces have various scales

16. embedding overview Measured descent: New multi-scale embedding technique [Krauthgamer-L-Mendel-Naor FOCS’04] exploit non-trivial interaction between scales

17. embedding overview Measured descent: New multi-scale embedding technique [Krauthgamer-L-Mendel-Naor FOCS’04] -approximation algorithm for EDGE EXPANSION [Arora-Rao-Vazirani STOC’04] new techniques in high-dimensional convex geometry single-scale analysis via geometric chaining argument

18. embedding overview Measured descent: New multi-scale embedding technique [Krauthgamer-L-Mendel-Naor FOCS’04] -approximation algorithm for EDGE EXPANSION [Arora-Rao-Vazirani STOC’04] new techniques in high-dimensional convex geometry Gluing embeddings with “partitions of unity” [L SODA’05]

19. embedding overview Measured descent: New multi-scale embedding technique [Krauthgamer-L-Mendel-Naor FOCS’04] -approximation algorithm for EDGE EXPANSION [Arora-Rao-Vazirani STOC’04] new techniques in high-dimensional convex geometry Gluing embeddings with “partitions of unity” [L SODA’05] Improvements to the ARV geometric structure theorems [Chawla-Gupta-Racke SODA’05, L 05] upper bound [CGR 05]

20. embedding overview Measured descent: New multi-scale embedding technique [Krauthgamer-L-Mendel-Naor FOCS’04] -approximation algorithm for EDGE EXPANSION [Arora-Rao-Vazirani STOC’04] new techniques in high-dimensional convex geometry Gluing embeddings with “partitions of unity” [L SODA’05] Improvements to the ARV geometric structure theorems [Chawla-Gupta-Racke SODA’05, L 05] -approximation for SPARSEST CUT [Arora-L-Naor STOC’05, L 06] based on new Euclidean embedding theorems for “negative type” spaces

21. important problems: negative-type metrics analyze this semi-definite program - Analysis is equivalent to finding the best distortion of n-point “negative type” metrics into Euclidean space with the L 1 norm Upper bound: [Arora-L-Naor STOC’05, L 06] Lower bound: [Khot-Vishnoi FOCS’05] - Related to Fourier analysis of boolean functions, probabilistically checkable proofs (PCPs), unique games conjecture, geometric analysis...

22. important problems: edit distance AAG C T AA C T A C T A For two strings s,t 2 {A,C,G,T} d d EDIT (s,t) {minimum number of insert/delete character operations to change from s ! t} = - What is the distortion needed to embed d EDIT into a Euclidean space (with the L 1 norm)? (Applications to nearest-neighbor search, sketching, fast distance computations...) Upper bound: [Ostrovsy-Rabani STOC’05] Lower bound: [Krauthgamer-Rabani SODA’06]

23. important problems: vertex separators vertex cuts Earlier, we talked about edge cuts. We can also consider - Most important application: Finding low-treewidth decompositions (useful as a basic step in many algorithms) - Best approximation algorithms are from [Feige-Hajiaghayi-L STOC’05] Requires a stronger kind of embedding. We can only extend some of the known techniques.

24. important problems: planar multi-flows Max-flow / Min-cut theorem: In any graph G, for any two nodes s and t, the value of the value of the minimum s-t cut = value of the maximum s-t flow. What about multi-commodity flows? G s1s1 s2s2 s3s3 t1t1 t3t3 t2t2 - In general graphs, there is no max-flow/min-cut theorem for multi-flows. The gap can be log(k), k = # of flows - What about planar graphs? Conjecture: The max-flow/min-cut gap is only O( 1 ) for multi-flows on planar graphs.

25. important problems: planar multi-flows Max-flow / Min-cut theorem: In any graph G, for any two nodes s and t, the value of the value of the minimum s-t cut = value of the maximum s-t flow. Conjecture: The max-flow/min-cut gap is only O( 1 ) for multi-flows on planar graphs. This conjecture is equivalent to the question: If d(u,v) is the shortest-path metric on a planar graph G, does the metric space (G,d) embed into a Euclidean space (with the L 1 norm) with O( 1 ) distortion?

26. http://www.cs.berkeley.edu/~jrl conclusion - Embeddings are a fundamental tool in Computer Science - Very rich, exciting mathematics - Lots of important open problems at various levels of difficulty - Many applications to other parts of science AAGC T A A CT G s1s1 s2s2 s3s3 t1t1 t3t3 t2t2