geometric embeddings and graph expansionJames R. Lee

Institute for Advanced Study (Princeton) University of Washington (Seattle)

outline

1.Philosophy of geometric embeddings2.Example: Finding balanced cuts in graphs3.Four important open problems

in the talk:

not in the talk:

No proofs (one slide). Mathematics borrows from high-dimensional convexgeometry, functional analysis, harmonic analysis, differential geometry...(see other talks on my web page)

so you should ask questions if something is confusing!

geometric embeddings in CS

combinatorial problem

geometric representation

embedding

nicer geometric space

combinatorial solution

connections in CS

geometric searchclustering

dimension reductionmachine learning

computational biology

approximation algorithmsdivide and conquer

network designgraph layout

tree decompositions

geometric optimizationsemi-definite programming

PCPs, unique gamesfourier analysis of boolean functions

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 edgescrossing 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)

graph expansion and the sparsest cut

Given a graph G=(V,E), we want to

Clustering Divide & conquer algorithms

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

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) 8x,y,z 2 V

“cut metric”d(x,y) = 1 if x,y are on different sides of Sd(x,y) = 0 otherwiseS S

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) 8x,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

finding cuts using embeddings

Now we find a cut using LP relaxation + embeddings [Linial London Rabinovich 1992]

S S

cut metric d

Rn

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)

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 ! Rn.

distortion measures how well f preserves the structure of X

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 ! Rn.

Depending on the application, sometimes we consider the L1 norm or the L2 norm.

- Embeddings into L2 are stronger than L1 embeddings- L1 embeddings are good enough for finding sparse cuts- We have many fewer techniques for analyzing L1 embeddings

first results

[Bourgain 1985] Every n-point metric space has a Euclidean embedding (L2 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)

new results

semi-definite programming

special family ofmetric spaces

“negative type”

A metric space (X,d) is said to be negative type if we can write

where xu 2 Rn for every u 2 X.

embedding overview

metric spaces have various scales

embedding overview

Measured descent: New multi-scale embedding technique [Krauthgamer-L-Mendel-Naor FOCS’04]

exploit non-trivial interactionbetween scales

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 analysisvia geometric chaining argument

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]

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]

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

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 L1 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...

important problems: edit distance

A A G CT

A A CT

A CTA

For two strings s,t 2 {A,C,G,T}d

dEDIT(s,t)

{minimum number ofinsert/delete character operations

to change from s ! t}=

- What is the distortion needed to embed dEDIT into a Euclidean space (with the L1 norm)? (Applications to nearest-neighbor search, sketching, fast distance computations...)

Upper bound: [Ostrovsy-Rabani STOC’05]Lower bound: [Krauthgamer-Rabani SODA’06]

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.

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

s1

s2

s3

t1

t3

t2- 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.

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 L1 norm) with O(1) distortion?

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 scienceA A G CT

A A CT

Gs1

s2

s3

t1

t3

t2