Coarse Differentiation and Planar Multiflows Prasad Raghavendra James Lee University of Washington TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA

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.

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

Multiflow and L1 Embeddings 10 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]

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 “

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 (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’

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

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!

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)|

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

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

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

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 )

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 )

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