1. Coarse Differentiation and Planar Multiflows
Prasad Raghavendra
James Lee
University of Washington
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2. 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.

3. Concurrent Multiflow
Input: Graph G = (V,E) with edge capacities. Source-Destination pairs (s1,t1),(s2,t2),..(sk,tk) Demands : D1 , D2 ,.. Dk 10 15 7 D1 1 1 3 D2
Maximize C, such that at least C fraction of all the demands can be simultaneously routed

4. Multiflow and L1 Embeddings10 15 3 7 1
Sparsest Cut:
Minimize :
Capacity of Edges Cut
Total 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]

5. Minor-Closed Embedding Conjecture[Gupta, Newman Rabinovich, Sinclair]
For Planar Graphs, the Flow and Cut are within constants of each other.
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 “

6. 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.

7. Coarse Differentiation
(X,d) R2 1
[0,1] F
By Classical Differentiation, find small enough sections that look like a straight lines.
Find subsets of the domain [0,1] which are mapped to `near straight lines’

8. ε-Efficient Paths [Eskin-Fischer-Whyte]
A path (u0,u1,…un) is said to be ε-Efficient if
By Triangle Inequality 1 1 1 1 1 1 1 1
3.9
2.5
Not ε-Efficient
ε-Efficient

9. Length of any such path ≤ 1
Aim : Find 3 points that are
0.5-efficient
Toy Version
1/8
1/4
3/8
1/2
5/8
3/4
7/8 1
[0,1] F
Distortion D
Length of any such path ≤ 1
A Contradiction!

10. Cuts and L1 Embeddings 1
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)|

11. Cuts and ε-Efficient Paths
Non-Monotone Cut u3 u0 u2 u4
Monotone Cut u6
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 u5 u8 u7
Path is ε-efficient

12. Embeds with distortion 4/3
Graph Construction s t
Embeds with distortion 4/3
K2,2

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

14. K2,n Metric D(s,t) = average distance between D(ui ,uj ) 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 s t u2 un
D(s,t) = 2
D(ui ,uj) = 2
D(s,t) = average distance between D(ui ,uj )

15. Monotone Embeddings of K2,n
Each cut separates at exactly one edge along every path from s to t
s and t are separated. u1 u2 S
|S|(n-|S|) ≤ n2/4
(ui, uj ) pairs are separated s t
Among the n(n-1)/2 pairs of middle vertices,
at most half are separated. un
D(s,t) ~ 2 · average distance between D(ui ,uj )

16. Thank You