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