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Sparsest Cut

Sparsest Cut

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Sparsest Cut

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  1. S (G) = min |E(S, S)| |S|  (G) = min |E(S, S)| |S| S µ V S µ V c |S| ¸ c ¢ |V| Sparsest Cut G = (V, E) S c- balanced separator Both NP-hard

  2. Why these problems are important • Arise in analysis of random walks, PRAM simulation, packet routing, clustering, VLSI layout etc. • Underlie many divide-and-conquer graph algorithms (surveyed by Shmoys’95) • Related to curvature of Riemannian manifolds and 2nd eigenvalue of Laplacian (Cheeger’70) • Graph-theoretic parameters of inherent interest (cf. Lipton-Tarjan planar separator theorem)

  3. 3) Embeddings of finite metric spaces into l1 (Linial, London, Rabinovich’94) Previous approximation algorithms • Eigenvalue approaches (Cheeger’70, Alon’85, Alon-Milman’85)2c(G) ¸L(G) ¸ c(G)2/2 c(G) = minS µ V E(S, Sc)/ E(S) 2) O(log n) -approximation via multicommodity flows (Leighton-Rao 1988) • Approximate max-flow mincut theorems • Region-growing argument • Geometric approach; more general result

  4. log n log n Our results • O( ) -approximation to sparsest cut and conductance • O( )-pseudoapproximation to c-balanced separator (algorithm outputs a c’-balanced separator, c’ < c) • Existence of expander flows in every graph (approximate certificates of expansion)

  5. 1 0 0 1 1 Semidefinite LP Relaxations for c-balanced separator Min (i, j) 2 E Xij 0 · Xij· 1 Motivation: Every cut (S, Sc) defines a (semi) metric Xij2 {0,1} Xij + Xj k¸ Xik  i< j Xij¸ c(1-c)n2 There exist unit vectors v1, v2, …, vn2<n such that Xij = |vi - vj|2 /4

  6. Semidefinite relaxation (contd) Min (i, j) 2 E |vi –vj|2/4 |vi|2 = 1 |vi –vj|2 + |vj –vk|2¸ |vi –vk|28 i, j, k i < j |vi –vj|2¸ 4c(1-c)n2 Unit l22 space

  7. Vi Vj Vk s s s s l22 space Unit vectors v1, v2,… vn2<d |vi –vj|2 + |vj –vk|2¸ |vi –vk|28 i, j, k Angles are non obtuse Taking r steps of length s only takes you squared distance rs2 (i.e. distance r s)

  8. Example of l22 space: hypercube {-1, 1}k |u – v|2 = i |ui – vi|2 = 2 i |ui – vi| = 2 |u – v|1 In fact, every l1 space is also l22 Conjecture (Goemans, Linial): Every l22 space is l1 up to distortion O(1)

  9. log n Our Main Theorem Two subsets S and T are -separated if for every vi2 S, vj2 T |vi –vj|2¸ <d ¸  Thm: If i< j |vi –vj|2 = (n2) then there exist two sets S, T of size (n) that are  -separated for  = ( 1 )

  10. log n log n ) |E(R, Rc)| · SDPopt / · O( SDPopt) Main thm ) O( )-approximation v1, v2,…, vn2<d is optimum SDP soln; SDPopt = (I, j) 2 E |vi –vj|2 S, T :  –separated sets of size (n) Do BFS from S until you hit T. Take the level of the BFS tree with the fewest edges and output the cut (R, Rc) defined by this level  (i, j) 2 E |vi –vj|2¸ |E(R, Rc)| £

  11. Next 10-12 min: Proof-sketch of Main Thm

  12. -t2 /2 e log n 1 1 d d Pru[ projection exceeds 2 ] < 1/n2 d Projection onto a random line v <d u <u, v> ??

  13. If any vi2 Su and vj2 Tu satisfy |vi –vj|2·, delete them and repeat until no such vi, vj can be found 0.01 “Stretched pair”: vi, vj such that |vi –vj|2· and | h vi –vj, u i | ¸ 0.01 d d Algorithm to produce two  –separated sets <d Check if Su and Tu have size (n) u Tu Su If Su, Tu still have size (n), output them Main difficulty: Show that whp only o(n) points get deleted Obs: Deleted pairs are stretched and they form a matching.

  14. “Stretched pair”: vi, vj such that |vi –vj|2· and | h vi –vj, u i | ¸ 0.01 d log n O( 1 ) £ standard deviation  ) PrU [ vi, vj get stretched] = exp( - 1 )  = exp( - ) E[# of stretched pairs] = O( n2 ) £ exp(- ) logn “Matching is of size o(n) whp” : trivial argument fails

  15. Vi 0.01 d Suppose with probability (1) there is a matching of (n) stretched pairs Vj u Ball (vi , )

  16. |vfinal - vi| < r  | <vfinal – vi, u>| ¸ 0.01r 0.01 0.01 0.01 r d d d d  = O( r ) x standard dev. The walk on stretched pairs Vj vfinal    Vi r steps u Contradiction!!

  17. Reason: Isoperimetric inequality for spheres  Measure concentration (P. Levy, Gromov etc.) <d A : measurable set with (A) ¸ 1/4 A A : points with distance · to A (A) ¸ 1 – exp(-2 d) A

  18. log n S Our Thm: If G has expansion , then a d-regular expander flow can be routed in it where d=  Expander flows: Motivation “Expander” G = (V, E) Idea: Embed a d-regular (weighted) graph such that 8 S w(S, Sc) = (d |S|) (*) (certifies expansion = (d) ) S Graph w satisfies (*) iff L(w) = (1) [Cheeger] Cf. Jerrum-Sinclair, Leighton-Rao(embed a complete graph)

  19. Example of expander flow n-cycle Take any 3-regular expander on n nodes Put a weight of 1/3n on each edge Embed this into the n-cycle Routing of edges does not exceed any capacity ) expansion =(1/n)

  20. log n Formal statement : 90 >0 such that following LP is feasible for d = (G) Pij = paths whose endpoints are i, j 8i jp 2 Pij fp = d (degree) 8e 2 E p 3 e fp· 1 (capacity) 8S µ V i 2 S j 2 Scp 2 Pij fp¸0 d |S| (demand graph is an expander) fp¸ 0 8 paths p in G

  21. ) (d) ·(G) · O(d ) log n log n New result (A., Hazan, Kale; 2004) O(n2) time algorithm that given any graph G finds for some d >0 • a d-regular expander flow • a cut of expansion O( d ) Ingredients: Approximate eigenvalue computations; Approximate flow computations (Garg-Konemann; Fleischer) Random sampling (Benczur-Karger + some more) Idea: Define a zero-sum game whose optimum solution is an expander flow; solve approximately using Freund-Schapire approximate solver.

  22. Open problems • Improve approximation ratio to O(1); better rounding??(our conjectures may be useful…) • Extend result to other expansion-like problems (multicut, general sparsest cut; MIN-2CNF deletion) • Resolve conjecture about embeddability of l22 into l1 • Any applications of expander flows?

  23. A concrete conjecture (prove or refute) G = (V, E);  = (G) For every distribution on n/3 –balanced cuts {zS} (i.e., SzS =1) there exist (n) disjoint pairs (i1, j1), (i2, j2), ….. such that for each k, • distance between ik, jk in G is O(1/ ) • ik, jk are across (1) fraction of cuts in {zS} (i.e., S: i 2 S, j 2 Sc zS = (1) ) Conjecture ) existence of d-regular expander flows for d = 

  24. log n