Coarse Differentiation and Planar
Multiflows
Prasad Raghavendra
James 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
d X ( a, b)
 dY ( F (a ), F (b))  d X (a, b)
D
Low distortion embeddings have applications
in Approximation Algorithms.
Concurrent Multiflow
10
15
D1
7
1
1
3
D2
Input:
Graph G = (V,E) with edge
capacities.
Source-Destination pairs
(s1,t1),(s2,t2),..(sk,tk)
Demands : D1 , D2 ,.. Dk
Maximize C, such that at least C fraction of all the
demands can be simultaneously routed
Multiflow and L1 Embeddings
Sparsest Cut:
10
15
D1
3
7
1
1
D2
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]
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 “
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.
Coarse Differentiation
R2
(X,d)
0
1
[0,1]
F
By Classical
Find
subsetsDifferentiation,
of the domain [0,1]
find small
whichenough
are mapped
to `near straight
sections
that looklines’
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
1
1
1
2.5
Not ε-Efficient
1
1
1
1
3.9
ε-Efficient
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
F
1
0
[0,1]
1
F
Length of any such
pathD
Distortion
≤1
A Contradiction!
F
0
Cuts and L1 Embeddings
Fact:
Every L1 metric can be
expressed as a positive
linear combination of
Cut Metrics.
1
d(u, v) =

Cut Metric
1 if u, v are on different
sides of the cut.
0
d(u, v) = |1S(u) -1S (v)|
Cuts and ε-Efficient Paths
T
S
u1
u3
Non-Monotone Cut
u0
u2
u4
Monotone Cut
u6
u5
For an ε-efficient
path P in an L1 embedding F,
u8
u7
The path P is monotone with respect to at
Path is ε-efficient
least 1-2ε fraction of the cuts in F
s
t
Graph Construction
Embeds with distortion
4/3
K2,2
Apply Coarse Differentiation
on S-T Paths
Find a K2,n copy with all S-T
paths ε-efficient
S
Argument
T
K2,n Metric
u1
Observations:
u2
• s and t are distance 2 from
t each other
s
un
D(s,t) = 2
D(ui ,uj) = 2
•n vertices in between s and t
(u1 , u2 ,… un)
• All the n( n  1) pairwise
2
distances are 2
D(s,t) = average distance between D(ui ,uj )
Monotone Embeddings of K2,n
Each cut separates at
exactly one edge
along every path from
s to ts and t
u1
are
separated.
u2
s
|S|(n-|S|) ≤ n2/4
S
t
(ui, uj ) pairs are separated
Among the n(n-1)/2 pairs of middle vertices,
at most half are
un separated.
D(s,t) ~ 2 · average distance between D(ui ,uj )
Thank You