Download presentation

Presentation is loading. Please wait.

Published byLillian Baker Modified over 4 years ago

1
Geometry and Expansion: A survey of some results Sanjeev Arora Princeton ( touches upon: S. A., Satish Rao, Umesh Vazirani, STOC04; S. A., Elad Hazan, and Satyen Kale, FOCS04; S. A., James Lee, and Assaf Naor, STOC05 & JAMS08 S.A., S. Kale STOC 2007. + papers that are not mine)

2
Outline: Graph partitioning problems: intro and history New approximation via expander flows. 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 Open problems

3
Sparsest Cut / Edge Expansion S S G = (V, E) c- balanced separator Both NP-hard G) = min S µ V | E(S, S c )| |S| |S| < |V|/2 c (G) = min S µ V | E(S, S c )| |S| c |V| < |S| < |V|/2

4
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 Shmoys95) Discrete analog of isoperimetry; useful in Riemannian geometry (via 2 nd eigenvalue of Laplacian ( Cheeger70 ) Graph-theoretic parameters of inherent interest (cf. Lipton-Tarjan planar separator theorem)

5
Previous approximation algorithms 1)Eigenvalue approaches ( Cheeger70, Alon85, Alon-Milman85 ) Only yield factor n approximation. 2c(G) ¸ (G) ¸ c(G) 2 /2 2) O(log n) - approximation via LP (multicommodity flows ) ( Leighton-Rao88 ) Approximate max-flow mincut theorems Region-growing argument ( Linial, London, Rabinovich94, AR94 ) 3) Embeddings of finite metric spaces into l 1 Geometric approach; more general result (but still O(log n) approximation)

6
New results of [ARV04] 1.O( ) -approximation to sparsest cut and conductance 2.O( )-pseudoapproximation to c-balanced separator (algorithm outputs a c-balanced separator, c < c) 3.Existence of expander flows in every graph (approximate certificates of expansion) log n Disparate approaches from previous slide get unified Subsequent work: [AHK05],[AK07],[S09]: O(m + n 1.5 ) time!

7
The three main characters ExpansionIsoperimetry (continuous analog of expansion) Geometry (and geometric embeddings of finite metric spaces)

8
Identifying sparse cuts via traffic flows

9
Approach 1: traffic congestion identifies sparse cuts [SM87]: Stress a network by passing traffic flow through it. Look at congested edges to identify sparse cuts [LR88] O(log n) approximation to sparsest cut. Route 1 unit of traffic between every pair of nodes [ARV04] Traffic flow is like embedding a weighted graph. w ij = amount of traffic from i to j Solve a math program to find the right flow pattern ([AHK05] Do it in O(n 2 ) time)

10
Expander traffic flows [ARV04] G = (V, E) S S A D-regular flow graph s.t. 8 S w(S, S c ) = ( D |S|) Weighted Graph w satisfies (*) iff L (w) = (1) [Cheeger] (*) Our Thm: If G has expansion, then a D-regular expander flow exists in it where D= (certifies expansion = (D) )

11
Formal statement : 9 0 >0 s.t. foll. LP is feasible for D = (G) log n f p ¸ 0 8 paths p in G 8i j p 2 P ij f p = D (degree) P ij = paths whose endpoints are i, j 8S µ V i 2 S j 2 S c p 2 P ij f p ¸ 0 D |S| (demand graph is an expander) 8e 2 E p 3 e f p · 1 (capacity) WHY IS THIS FEASIBLE???

12
Feasibility Criterion for LP on prev. slide (via Farkass Lemma) Existence of such i, j proved in [ARV04]. When fail to find such i, j, we find a cut of small expansion

13
Overall approximation algorithm via flows Try to solve above LP to find D-regular expander flow If succeed, have verified that expansion is D/10. If fail, then use [ARV04] ideas to find a cut of capacity Note: Before finding this cut already had D/2-regular flow

14
Next: The SDP-based approach to Graph partitioning (ARV04)

15
c-balanced separator c (G) = min S µ V | E(S, S c )| |S| c |V| < |S| < |V|/2 S S Assign {+1, -1} to v 1, v 2, …, v n to minimize (i, j) 2 E |v i –v j | 2 /4 Subject to i < j |v i –v j | 2 /4 ¸ c(1-c)n 2 +1 |v i –v j | 2 /4 =1 Semidefinite relaxation for Find unit vectors in < n |v i –v j | 2 + |v j –v k | 2 ¸ |v i –v k | 2 8 i, j, k Triangle inequality cut semimetric |v i –v j | 2 =0

16
Unit l 2 2 space Unit vectors v 1, v 2,… v n 2 < d |v i –v j | 2 + |v j –v k | 2 ¸ |v i –v k | 2 8 i, j, k ViVi VkVk VjVj non obtuse ! Example: Hypercube {-1, 1} k |u – v| 2 = i |u i – v i | 2 = 2 i |u i – v i | = 2 |u – v| 1 In fact, l 2 and l 1 are subcases of l 2 2

17
Structure Theorem for l 2 2 spaces [ARV04] Subsets S and T are -separated if for every v i 2 S, v j 2 T |v i –v j | 2 ¸ ¸ Thm: If i< j |v i –v j | 2 = (n 2 ) then 9 S, T of size (n) that are -separated for = ( 1 ) <d<d log n

18
Main thm ) O( )-approximation log n v 1, v 2,…, v n 2 < d is optimum SDP soln; SDP opt = (i, j) 2 E |v i –v j | 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, R c ) defined by this level (i, j) 2 E |v i –v j | 2 ¸ |E(R, R c )| £ ) |E(R, R c )| · SDP opt / · O( SDP opt ) log n S d(S, i) i j d(S, j)

19
Other new -approximation algorithms MIN-2-CNF deletion and several graph deletion problems. [Agarwal, Charikar, Makarychev, Makarychev04]. Weighted version of S.T. MIN-LINEAR ARRANGEMENT [Charikar, Karloff, Rao04] General SPARSEST CUT [A., Lee, Naor 04] Min-ratio VERTEX SEPARATORS and Balanced VERTEX SEPARATORS [ Feige, Hajiaghayi, Lee, 04] log n All use the Structure Theorem (+ other ideas)

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. Uses of S. T. in geometric embeddings Introduction to expander flows and O(n 2 ) time algorithms Open problems S T (Algorithm to produce -separated sets S, T, of size (n) )

21
Algorithm to produce two –separated sets <d<d u SuSu TuTu 0.01 d Easy: S u and T u likely to have size (n) If S u, T u still have size (n), output them Main difficulty: Show that whp only o(n) points get deleted d Stretched pair: v i, v j such that |v i –v j | 2 · and | h v i –v j, u i | ¸ 0.01 Obs: Deleted pairs are stretched and they form a matching. Delete any v i 2 S u, v j 2 T u s.t. |v i –v j | 2 <. (till no such pair remains)

22
Naïve analysis of random projection fails <d<d v u ?? 1 d 1 d e -t 2 /2 Stretched pair: |v i –v j | 2 | > 0.01 d standard deviations E[# of stretched pairs] = n 2 exp(- ) À n = O( 1 )

23
ViVi Ball (v i, ) u VjVj 0.01 d Proof by contradiction: Suppose matching of (n) size exists with probability (1)… ….stretched pairs are almost everywhere you look! Idea: Put stretched pairs together; derive very improbable event

24
Walks in unit l 2 2 space Unit vectors v 1, v 2,… v n 2 < d |v i –v j | 2 + |v j –v k | 2 ¸ |v i –v k | 2 8 i, j, k ViVi VkVk VjVj Angles are non obtuse Taking r steps of length s only takes you squared distance rs 2 (i.e. distance r s) ss ss

25
Proof by contradiction (contd.) s s s s r steps of length s ) squared distance rs 2 (distance r s) Stretched pair: |v i –v j | 2 ¸ 0.01 Claim: 9walk on stretched edges u d 0.01 d d …. ¸ r 0.01 d |v final –v 0 | · r Projection = r £ standard deviation VERY UNLIKELY IF r large enough ) Walk impossible (CONTRADICTION) Why walk is possible: delicate argument; measure concentration

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. Geometric embeddings of metric spaces Open problems

27
Finite metric space (X, d) x y d(x,y) < k (with l 2 norm) f 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 (Bourgain85): For every n-point metric space, a map exists with distortion O(log n) [LLR94]: Efficient algorithm to find the map; Proof that O(log n) cannot be improved in general Qs: Improve O(log n) for X = l 2 2 (say) or l 1 ? f(x) f(y)

28
Embeddings and Cuts (LLR94, AR94) Recall: Cut semi-metric 1 0 Fact: Metric (X, d) embeds isometrically in l 1 iff it can be written as a positive combination of cut semimetrics Embedding l 2 2 into l 1 gives a way to produce cuts from SDP solution

29
Status report of this area l 1 into l 2 log 0.5 n [Enflo69] l 2 2 into l 1 1.16 [Zatloukal04] Superconstant [Khot, Vishnoi04] (logn) 0.01 [Cheeger,Kleine r, 08] l 2 2 into l 2 log 0.5 n [Enflo69] Best lowerbound Best upperbound Exactly the integrality gap of SDP for general SPARSEST CUT [LLR94, AR94] log n [Bourgain85] log 0.75 n [Chawla,Gupta,Racke 04] log 0.5 n log log n [A., Lee, Naor04] Uses fourier techniques developed for PCPs! Disproves Goemans-Linial conjecture Uses new metric differentiation techniques

30
Upperbounds: Frechets recipe to embed metric space (X, d) into R k Pick k suitable subsets A 1, A 2, …, A k of X Map x 2 X to (d(x, A 1 ), d(x, A 2 ), …, d(x, A k )) AiAi x In recent embeddings, A i s are chosen using S.T.and Measured descent idea of [Krauthgamer, Lee, Naor, and Mendel04] Note: d(x, A 1 ) – d(y, A 1 ) · d(x, y)

31
Embedding lowerbounds (Khot-Vishnoi05) Explicit unit- l 2 2 space (X, d) that requires distortion log log log n into l 1 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)

32
OPEN PROBLEMS Better approximation factor than O( )? (log log n lowerbound assuming UGC ) Better distortion bound for embedding l 2 2 into l 1 ?.) ( upperbound v/s lowerbound Combinatorial approximation algorithms for other problems ? (similar to one for SPARSEST CUT from [A., Hazan, Kale] ) Other applications of expander flows? (Useful in some geometric results [Naor, Rabani, Sinclair04]) Ways to use spectral ideas a la [ABS10] for SPARSEST CUT?

33
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)

34
Other extensions of flow-based techniques Generalization to problems other than sparsest cut [A., Kale07] Primal-dual approach to SDP. Very fast algorithms for O(log n) approximation: O(n 1.5 + m) time (Faster than [LR88] type algorithms!) Very simple algorithms; use only maxflow and eigenvalue computations [KRV06]

35
Looking forward to more progress… Thanks !

36
New Result (A., Hazan, Kale;FOCS04) O(n 2 ) time algorithm that given any graph G finds for some D >0 a D-regular expander flow a cut of expansion O( D ) log n 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. ) D) · (G) ·O(D ) log n

37
Expander flows: LP view LP feasible ) ¸ (D) G G · D · 1 Thm [ARV]: 9 0 s.t. the LP is feasible with D = /log n Thm [ARV]: 9 0 s.t. the LP is feasible with D = /log n

38
Open problems (circa April04) 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 l 2 2 into l 1 ; of l 1 into l 2 Any applications of expander flows? O(n 2 ) time; [A., Hazan, Kale] log 3/4 n distortion; [Chawla,Gupta, Racke] Integrality gap is (log n) [Charikar] Yes [Naor,Sinclair,Rabani] Better embeddings of l p into l q [Lee]

39
Various new results O(n 2 ) time combinatorial algorithm for sparsest cut (does not use semidefinite programs) [A., Hazan, Kale04] New results about embeddings: (i) l p into l q [J. Lee04] (ii) l 2 2 and l 1 into l 2 [CGR04] (approx for general sparsest cut) Clearer explanation of expander flows and their connection to embeddings [NRS04]

40
A concrete conjecture (prove or refute) G = (V, E); = (G) For every distribution on n/3 –balanced cuts {z S } (i.e., S z S =1) there exist (n) disjoint pairs ( i 1, j 1 ), ( i 2, j 2 ), ….. such that for each k, distance between i k, j k in G is O(1/ ) i k, j k are across (1) fraction of cuts in {z S } ( i.e., S: i 2 S, j 2 S c z S = (1) ) Conjecture ) existence of d-regular expander flows for d =

41
log n

42
Example of l 2 2 space: hypercube {-1, 1} k |u – v| 2 = i |u i – v i | 2 = 2 i |u i – v i | = 2 |u – v| 1 In fact, every l 1 space is also l 2 2 Conjecture (Goemans, Linial): Every l 2 2 space is l 1 up to distortion O(1)

43
LP Relaxations for c-balanced separator Motivation: Every cut (S, S c ) defines a (semi) metric 1 1 1 0 0 X ij 2 {0,1} i< j X ij ¸ c(1-c)n 2 X ij + X j k ¸ X ik 0 · X ij · 1 Semidefinite There exist unit vectors v 1, v 2, …, v n 2 < n such that X ij = |v i - v j | 2 /4 Min (i, j) 2 E X ij

44
Semidefinite relaxation (contd) Min (i, j) 2 E |v i –v j | 2 /4 |v i | 2 = 1 |v i –v j | 2 + |v j –v k | 2 ¸ |v i –v k | 2 8 i, j, k i < j |v i –v j | 2 ¸ 4c(1-c)n 2 Unit l 2 2 space Many other NP-hard problems have similar relaxations.

45
Algorithm to produce two –separated sets <d<d u SuSu TuTu 0.01 d Check if S u and T u have size (n) If any v i 2 S u and v j 2 T u satisfy |v i –v j | 2 · repeat until no such v i, v j remain delete them and If S u, T u still have size (n), output them Main difficulty: Show that whp only o(n) points get deleted d Stretched pair: v i, v j such that |v i –v j | 2 · and | h v i –v j, u i | ¸ 0.01 Obs: Deleted pairs are stretched and they form a matching.

46
Next 10-12 min: Proof-sketch of Structure Thm ( algorithm to produce -separated S, T of size (n); = 1/ ) S T

47
Matching is of size o(n) whp : naive argument fails d Stretched pair: v i, v j such that |v i –v j | 2 · and | h v i –v j, u i | ¸ 0.01 O( 1 ) £ standard deviation ) Pr U [ v i, v j get stretched] = exp( - 1 ) = exp( - ) log n E[# of stretched pairs] = O( n 2 ) £ exp(- )log n

48
Generating a contradiction: the walk on stretched pairs u ViVi VjVj 0.01 d d r steps 0.01 d r |v final - v i | < r | | ¸ 0.01r d = O( r ) x standard dev. v fina l Contradiction if r is large enough!

49
Measure concentration (P. Levy, Gromov etc.) <d<d A A : measurable set with (A) ¸ 1/4 A : points with distance · to A A A ) ¸ 1 – exp(- 2 d) Reason: Isoperimetric inequality for spheres

50
Expander flows (approximate certificates of expansion)

Similar presentations

OK

What have we learnt about graph expansion in the new millenium? Sanjeev Arora Princeton University & Center for Computational Intractability.

What have we learnt about graph expansion in the new millenium? Sanjeev Arora Princeton University & Center for Computational Intractability.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google