Coarse Differentiation and Planar Multiflows

1. 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

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 Embeddings Sparsest Cut: Minimize : Capacity of Edges Cut Total Demand Separated 10 15 7 D1 1 1 3 For single-source destination: Max Flow = Minimum Cut For multiple sources and destinations : Worst ratio of Minimum Sparsest cut = L1Distortion Max Concurrent Flow [Linial-London-Rabinovich] D2

5. 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 CFsuch 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 F 0 1 [0,1] 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.

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

10. Cuts and L1 Embeddings Fact: Every L1 metric can be expressed as a positive linear combination of Cut Metrics. 1 Cut Metric 1 if u, v are on different d(u, v) = sides of the cut. d(u, v) = |1S(u) -1S (v)| 0

11. Cuts and ε-Efficient Paths T S u1 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. Graph Construction Embeds with distortion 4/3 K2,2 s t

13. 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 u1 • Observations: • s and tare distance 2 from each other • n vertices in between s and t • (u1 , u2 ,… un) • All the pairwise • distances are 2 u2 s t 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 stot sand 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