Coarse Differentiation and Planar Multiflows

Prasad RaghavendraJames Lee

University of Washington

Embeddings

A function F : (X,dX) (Y,dY) is said to have distortion D if for any two points a, b in X

Low distortion embeddings have applications in Approximation Algorithms.

),())(),((),( badbFaFdDbad

XYX

Concurrent Multiflow

Input:Graph G = (V,E) with edge capacities.

Source-Destination pairs (s1,t1),(s2,t2),..(sk,tk)

Demands : D1 , D2 ,.. Dk

D1

D2

10

15

3

7

1

1

Maximize C, such that at least C fraction of all the demands can be simultaneously routed

Multiflow and L1 Embeddings

D1

D2

10

15

3

7

11

Sparsest Cut:

Minimize :

Capacity of Edges CutTotal Demand Separated

For single-source destination:Max Flow = Minimum Cut

For multiple sources and destinations :Worst ratio of Minimum Sparsest cut = L1 Distortion

Max Concurrent Flow [Linial-London-Rabinovich]

Planar Embedding Conjecture“There exists a constant C such that every

planar graph metric embeds in to L1 with distortion at most C.’’

Minor-Closed Embedding Conjecture[Gupta, Newman Rabinovich, Sinclair]

“For every non-trivial minor closed family of graphs F, there is a constant CF such that every graph metric in F embeds in to L1 with distortion at most CF “

For Planar Graphs, the Flow and Cut are within constants of

each other.

Our Result

A planar graph metric that requires distortion at least 2 to embed in to L1

-The previous best lower bound known was 1.5.[Okamura-Seymour, Andoni-Deza-Gupta-Indyk-Raskhodnikova]

The lower bound is tight for Series Parallel Graphs. -Matching upper bound in [Chakrabarti-Lee-Vincent]

Main Contribution is the use of Coarse Differentiation [Eskin-Fischer-Whyte] to obtain L1 distortion lower bounds.

(X,d)R2

Coarse Differentiation

0 1

[0,1] F

Find subsets of the domain [0,1] which are mapped to `near straight lines’By Classical Differentiation, find small enough sections that look like a straight lines.

ε-Efficient Paths [Eskin-Fischer-Whyte]

A path (u0,u1,…un) is said to be ε-Efficient if

By Triangle Inequality

Not ε-Efficient ε-Efficient

11 1 1 1 1 1 1

2.5 3.9

FDistortion D

Toy Version

0 1[0,1]

1/21/4 3/41/8 3/8 5/8 7/8

Aim : Find 3 points that are 0.5-efficient

F0

F1

Length of any such path ≤ 1

A Contradiction!

Cuts and L1 Embeddings

Fact:Every L1 metric can be expressed as a positive linear combination of Cut Metrics.

Cut Metric

1 if u, v are on different

d(u, v) = sides of the cut. d(u, v) = |1S(u) -1S (v)| 0

1

Monotone Cut

Cuts and ε-Efficient Paths

u0

u8

u1

u2

u3

u4

u5

u6

u7

Non-Monotone Cut

Path is ε-efficient

S

T

For an ε-efficient path P in an L1 embedding F,

The path P is monotone with respect to at least 1-2ε fraction of the cuts in F

Graph Constructions t

Embeds with distortion 4/3

K2,2

Argument

S T

Apply Coarse Differentiation on S-T Paths Find a K2,n copy with all S-T paths ε-efficient

K2,n Metric

s t

Observations:

• s and t are distance 2 from each other

•n vertices in between s and t(u1 , u2 ,… un)

• All the pairwise

distances are 2

D(s,t) = 2D(ui ,uj) = 2

2)1( nn

u1

un

u2

D(s,t) = average distance between D(ui ,uj )

Monotone Embeddings of K2,n

s t

Each cut separates at exactly one edge

along every path from s to t

|S|(n-|S|) ≤ n2/4 (ui, uj ) pairs are separated

S

s and t are separated.

u1

u2

un

D(s,t) ~ 2 · average distance between D(ui ,uj )

Among the n(n-1)/2 pairs of middle vertices, at most half are separated.

Thank You