1 / 46

Geometry and Expansion: A survey of recent results

Geometry and Expansion: A survey of recent results. Sanjeev Arora Princeton. ( touches upon: S. A., Satish Rao, Umesh Vazirani, STOC’04; S. A., Elad Hazan, and Satyen Kale, FOCS’04; S. A., James Lee, and Assaf Naor, STOC’05 + papers that are not mine). | E(S, S c )|.

barney
Download Presentation

Geometry and Expansion: A survey of recent results

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Geometry and Expansion: A survey of recent results Sanjeev Arora Princeton ( touches upon: S. A., Satish Rao, Umesh Vazirani, STOC’04; S. A., Elad Hazan, and Satyen Kale, FOCS’04; S. A., James Lee, and Assaf Naor, STOC’05 + papers that are not mine)

  2. | E(S, Sc)| (G) = min S S S µ V |S| |S| < |V|/2 | E(S, Sc)| c(G) = min S µ V |S| c |V| < |S| < |V|/2 Sparsest Cut / Edge Expansion G = (V, E) c- balanced separator Both NP-hard

  3. The three main characters Expansion Isoperimetry (continuous analog of expansion) Geometry (and geometric embeddings of finite metric spaces)

  4. Outline: • Graph partitioning problems: intro and history • New approximation algorithm via semidefinite programming (+ analysis using “Structure Theorem”) [A., Rao, Vazirani] • Outline of proof of “S. T.” • Uses of “S. T.” in geometric embeddings • Introduction to expander flows and O(n2) time algorithms • Open problems

  5. Why these problems are important • Analysis of random walks, PRAM simulation, packet routing, clustering, VLSI layout etc. • Underlie many divide-and-conquer graph algorithms (surveyed by Shmoys’95) • Discrete analog of isoperimetry; useful in Riemannian geometry (via 2nd eigenvalue of Laplacian (Cheeger’70) • Graph-theoretic parameters of inherent interest (cf. Lipton-Tarjan planar separator theorem)

  6. 3) Embeddings of finite metric spaces into l1 • Geometric approach; more general result (but still O(log n) approximation) Previous approximation algorithms • Eigenvalue approaches (Cheeger’70, Alon’85, Alon-Milman’85)Only yield factor n approximation. 2c(G) ¸ (G) ¸ c(G)2 /2 2) O(log n) -approximationvia LP (multicommodity flows) (Leighton-Rao’88) • Approximate max-flow mincut theorems • Region-growing argument (Linial, London, Rabinovich’94, AR’94)

  7. log n log n New results of [ARV’04] • 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) Disparate approaches from previous slide get “unified”

  8. Outline: • Graph partitioning problems: intro and history • New approximation algorithm via semidefinite programming (+ analysis using “Structure Theorem”) [A., Rao, Vazirani] • Outline of proof of “S. T.” • Uses of “S. T.” in geometric embeddings • Introduction to expander flows and O(n2) time algorithms • Open problems Next: Semidefinite relaxations for c-balanced separator (and how to round the solution)

  9. S | E(S, Sc)| c(G) = min S µ V |S| c |V| < |S| < |V|/2 Semidefinite relaxation for c-balanced separator |vi –vj|2/4 =1 |vi –vj|2 =0 +1 S -1 “cut semimetric” Find unit vectors in <n Assign {+1, -1} to v1, v2, …, vn to minimize (i, j) 2 E |vi –vj|2/4 Subject to i < j |vi –vj|2/4 ¸ c(1-c)n2 Triangle inequality |vi –vj|2 + |vj –vk|2¸ |vi –vk|28 i, j, k

  10. Vi Vj Vk Unit l22 space Unit vectors v1, v2,… vn2<d |vi –vj|2 + |vj –vk|2¸ |vi –vk|28 i, j, k non obtuse ! Example: Hypercube {-1, 1}k |u – v|2 = i |ui – vi|2 = 2 i |ui – vi| = 2 |u – v|1 In fact, l2 and l1 are subcases of l22

  11. log n Structure Theorem for l22 spaces [ARV’04] 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 9S, T of size (n) that are  -separated for  = ( 1 )

  12. log n ) |E(R, Rc)| · SDPopt / · O( SDPopt) Main thm ) O( )-approximation log n 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)| £

  13. log n Other new -approximation algorithms • MIN-2-CNF deletion and several graph deletion problems. [Agarwal, Charikar, Makarychev, Makarychev’04] • MIN-LINEAR ARRANGEMENT [Charikar, Karloff, Rao’04] • General SPARSEST CUT [A., Lee, Naor ’04] • Min-ratioVERTEX SEPARATORS and Balanced VERTEX SEPARATORS[ Feige, Hajiaghayi, Lee, ’04] All use the Structure Theorem (+ other ideas)

  14. T S • Outline: • Graph partitioning problems: intro and history • New approximation algorithm via semidefinite programming (+ analysis using “Structure Theorem”) [A., Rao, Vazirani] • Outline of proof of “S. T.” • Uses of “S. T.” in geometric embeddings • Introduction to expander flows and O(n2) time algorithms • Open problems (Algorithm to produce  -separatedsets S, T, of size (n) )

  15. 0.01 d “Stretched pair”: vi, vj such that |vi –vj|2· and | h vi –vj, u i | ¸ 0.01 d Algorithm to produce two  –separated sets <d Easy: Su and Tu likely to have size (n) u Tu Delete any vi2 Su, vj2 Tu s.t. |vi –vj|2 < . (till no such pair remains) 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.

  16. -t2 /2 e 1 1 d d = O( 1 ) Stretched pair: |vi –vj|2 < ; |<vi –vj, u>| > 0.01 d  Naïve analysis of random projection fails v <d u <u, v> ?? standard deviations E[# of stretched pairs] = n2 exp(-) À n

  17. Vi 0.01 d Proof by contradiction: Suppose matching of (n) size exists with probability (1)… ….stretched pairs are almost everywhere you look! Vj u Ball (vi , ) Idea: Put stretched pairs together; derive very improbable event

  18. Vi Vj Vk s s s s Walks in unit 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)

  19. r steps of length s ) squared distance rs2 (distance r s) s s 0.01 <vfinal –v0, u> ¸ r 0.01 0.01 d d d s s r Projection = £ standard deviation  Proof by contradiction (contd.) Claim: 9walk on stretched edges VERY UNLIKELY IF r large enough) Walk impossible (CONTRADICTION) Stretched pair: |vi –vj|2 < ; |<vi –vj, u> ¸ 0.01 d    …. u Why walk is possible: delicate argument; measure concentration |vfinal –v0|· r 

  20. Outline: • Graph partitioning problems: intro and history • New approximation algorithm via semidefinite programming (+ analysis using “Structure Theorem”) [A., Rao, Vazirani] • Outline of proof of “S. T.” • Geometric embeddings of metric spaces • Introduction to expander flows and O(n2) time algorithms • Open problems

  21. <k(with l2 norm) Finite metric space (X, d) f(x) y f d(x,y) x f(y) distortion of f is minimum C>1 such that d( x, y) · |f(x ) – f( y)|2·C d( x, y) 8 x, y Thm (Bourgain’85): For every n-point metric space, a map exists with distortion O(log n) [LLR’94]: Efficient algorithm to find the map; Proof that O(log n) cannot be improved in general Qs: Improve O(log n) for X = l22 (say) or l1 ?

  22. Embeddings and Cuts (LLR’94, AR’94) Recall: Cut semi-metric Fact: Metric (X, d) embeds isometricallyin l1 iff it can be written as a positive combination of cut semimetrics 1 0 Embedding l22 into l1 gives a way to produce cuts from SDP solution

  23. Status report of this area Best upperbound Best lowerbound Disproves Goemans-Linial conjecture log n [Bourgain’85] Uses fourier techniques developed for PCPs! log0.75 n [Chawla,Gupta,Racke ’04] Exactly the integrality gap of SDP for general SPARSEST CUT [LLR’94, AR’94] log0.5 n log log n [A., Lee, Naor’04]

  24. x Ai Upperbounds:Frechet’s recipe to embed metric space (X, d) into Rk Pick k suitable subsets A1, A2, …, Ak of X Map x 2 X to (d(x, A1), d(x, A2), … , d(x, Ak)) Note: d(x, A1) – d(y, A1) · d(x, y) In recent embeddings, Ai’s are chosen using S.T.and “Measured descent” idea of [Krauthgamer, Lee, Naor, and Mendel’04]

  25. Embedding lowerbounds (Khot-Vishnoi’05) Explicit unit- l22 space (X, d) that requires distortion log log log n into l1 Main observation: Need good handle on cut structure of X Use hypercube as building block ! Cut ´ Boolean Function Number of cut edges = average sensitivity (Fourier analysis a la KKL, Friedgut, Hastad, Bourgain etc. ) isoperimetric theorems)

  26. Outline: • Graph partitioning problems: intro and history • New approximation algorithm via semidefinite programming (+ analysis using “Structure Theorem”) [A., Rao, Vazirani] • Outline of proof of “S. T.” • Uses of “S. T.” in geometric embeddings • Introduction to expander flows and O(n2) time algorithms • Open problems

  27. log n S Our Thm: If G has expansion , then a D-regular expander flow exists 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|) (*) S (certifies expansion = (D) ) Weighted Graph w satisfies (*) iff L(w) = (1) [Cheeger] Cf. Jerrum-Sinclair, Leighton-Rao(embed a complete graph) Can be found in O(n2) time (A., Hazan, Kale ’04)

  28. 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)

  29. OPEN PROBLEMS • Better approximation factor than O( )? (For general SPARSEST CUT, log log n “lowerbound” ) • Better distortion bound for embedding l22 into l1?( upperbound v/s loglog n lowerbound.) • Combinatorial approximation algorithms for other problems ? (similar to one for SPARSEST CUT from [A., Hazan, Kale] ) • O(m) time algorithm for SPARSEST CUT instead of O(n2)? (not known even for Leighton-Rao’88 O(log n) approximation) • Other applications of expander flows? (Useful in some geometric results [Naor, Rabani, Sinclair’04])

  30. Looking forward to more progress… Thanks !

  31. ) (D) ·(G) ·O(D ) log n log n New Result (A., Hazan, Kale;FOCS’04) 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.

  32. Expander flows: LP view · 1 LP feasible )¸(D) · D Thm [ARV]:90 s.t. the LP is feasible with D = /√log n G

  33. Open problems (circa April’04) O(n2) time; [A., Hazan, Kale] • Better running time/combinatorial algorithm? • 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; of l1 into l2 • Any applications of expander flows? Integrality gap is (log n) [Charikar] log3/4 n distortion; [Chawla,Gupta, Racke] Yes [Naor,Sinclair,Rabani] Better embeddings of lp into lq [Lee]

  34. Various new results O(n2) time combinatorial algorithm for sparsest cut (does not use semidefinite programs) [A., Hazan, Kale’04] New results about embeddings: (i) lp into lq[J. Lee’04] (ii) l22 and l1 into l2[CGR’04] (approx for general sparsest cut) Clearer explanation of expander flows and their connection to embeddings [NRS’04]

  35. log n Formal statement : 90 >0 s.t. foll. 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

  36. 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 = 

  37. log n log n

  38. 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)

  39. 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

  40. 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 Many other NP-hard problems have similar relaxations.

  41. If any vi2 Su and vj2 Tu satisfy |vi –vj|2·, delete them and repeat until no such vi, vj remain 0.01 d “Stretched pair”: vi, vj such that |vi –vj|2· and | h vi –vj, u i | ¸ 0.01 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.

  42. T S Next 10-12 min: Proof-sketch of Structure Thm ( algorithm to produce  -separated S, T of size (n);  = 1/ )

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

  44. |vfinal - vi| < r  = O( r ) x standard dev. | <vfinal – vi, u>| ¸ 0.01r 0.01 0.01 0.01 r d d d  d Generating a contradiction: the walk on stretched pairs Contradiction if r is large enough! Vj vfinal    Vi r steps u

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

  46. Expander flows (approximate certificates of expansion)

More Related