Arora Rao Vazirani Approximation for Expansion

Transcription

1 Proof, belefs, ad algorthms through the les of sum-of-squares 1 Arora Rao Vazra Approxmato for Expaso I ths lecture, we cosder the problem of fdg a set wth smallest possble expaso 1 a gve graph. Earler we saw that degree-2 pseudo-dstrbutos allow us to acheve o-trval approxmato guaratee, whch turs out to be closely related to Cheeger s equalty from spectral graph theory (see lecture 2). We also showed that ths guaratee s tght for degree-2 pseudo-dstrbutos (see lecture 3). I ths lecture, we vestgate the questo f hgher-degree pseudo-dstrbutos allow us to acheve better approxmato guaratee. A dcato that hgher degree mght deed help s that certa lear programmg approaches, that are captured by hgherdegree pseudo-dstrbutos, acheve approxmato guaratees for expaso that are better tha those from Cheeger s equalty some rage of parameters (Leghto ad Rao [1988]). (I partcular, those lear programmg approaches certfy tght lower bouds o the expaso of cycle graphs, for whch o good degree-2 sos certfcates exst.) 1 I the cotext of ths algorthm, the computatoal problem beg solved s typcally called Sparsest Cut. We wll refer to ths problem as M Expaso order to be cosstet wth earler lectures. We wll show that degree-4 pseudo-dstrbutos acheve approxmato guaratees for expaso (Arora et al. [2004]) that for a large rage of parameters are sgfcatly stroger tha both the guaratees of degree-2 pseudo-dstrbutos ad the guaratees of prevous lear programmg approaches. Our exposto (especally the proof of the structure theorem ) dffers sgfcatly from others the lterature. However, t borrows heavly from lectures otes of Rothvoss [2016]. Pseudo-dstrbutos for expaso Let G be a d-regular graph wth vertex set []. 2 Let f G : {0, 1} R be the fucto that outputs for every vector x {0, 1} the umber of edges f G (x) cut by x (vewed as a bpartto of the vertex set), f G (x) = (x x ) 2. (1) {,} E(G) 2 The treatmet of expaso here also works for o-regular graph. We assume regularty order to be cosstet wth earler deftos the cotext of Cheeger s equalty ad degree-2 sum-of-squares. We detfy the vector x {0, 1} wth the subset S = { [] x = 1} dcated by x. For every, [], the fucto (x x ) 2 dcates whether ad are o dfferet sdes of the bpartto correspodg to x. Recall the expaso of G, ϕ(g) = m x {0,1} f G (x). (2) d x ( x )

2 Boaz Barak ad Davd Steurer 2 Here, x = =1 x s the sze of the set dcated by x. Note that actual probablty dstrbutos µ : {0, 1} R 0 satsfy E µ f G ϕ(g) E µ(x) d x ( x ) ad equalty s acheved by dstrbutos supported o sets wth mmum expaso. If we cosder degree-4 pseudo-dstrbutos, we ca fd polyomal tme oe such that the rato of Ẽ µ f G ad Ẽ µ(x) d x ( x ) s as small as possble certaly o more tha ϕ(g). The followg theorem shows that ths rato ca ever be much smaller tha ϕ(g) ad that we ca extract from the pseudo-dstrbuto a set wth small expaso at most ϕ(g) O( log ). 1. Theorem (Arora Rao Vazra approxmato for M Expaso). Let G be a d-regular graph wth vertex set [] ad let µ : {0, 1} R be a degree-4 pseudo-dstrbuto. The, ( ) Ẽ µ f G ϕ(g) O log d Ẽ µ(x) x ( x ). (3) Furthermore, there exsts a polyomal-tme algorthm that gve G ad µ fds a vector x {0, 1} wtessg ths boud o ϕ(g). A brds-eye vew of the proof The proof of Theorem 1 cossts of several part. We descrbe each of them at a hgh level ad expla how they ft together. The frst compoet s a algorthm called rego growg. Gve two subsets A, B [] ths algorthm tres to fd the best cut of the graph betwee A ad B guded by the pseudo-dstrbuto µ. As we wll see, ths algorthm acheves a O(1/ )-approxmato (compared to the pseudo-dstrbuto) f the sets A, B are -separated the sese that A, B Ω() ad E µ(x) (x x ) 2 for every A ad B. The secod compoet s a structure theorem for pseudodstrbutos that shows that for some Ω(1)/ log such -separated sets always exst. 3 To prove ths structure theorem we apply the quadratc samplg lemma (from lecture 1) order to obta Gaussa varables X 1,..., X wth the same frst two momets as the pseudodstrbuto. A atural frst attempt to costruct separated sets from those varables s to choose A to be all vertces [] such that X s sgfcatly below ts mea, say X E X 1 ad to choose B to be all vertces [] such that X s sgfcat above ts mea, 3 The actual proof requres a small case dstcto here whch we are gog to gore for ths hgh-level dscusso.

3 Proof, belefs, ad algorthms through the les of sum-of-squares 3 say X E X + 1. Ideed, ths costructo esures that A ad B are -separated for some Ω(1)/ log (whch tur gves a O(log ) approxmato). To mprove over ths frst attempt, we post-process the sets A ad B by teratvely removg pars A ad B that volate the separato codto,.e., E(X X ) 2. The heart of the proof s to upper boud the umber of dsot pars of ths form. We wll show that ts expectato s at most O() ( log ) Ω(1). It follows that f we choose = c log for a suffcetly small costat c > 0, the oly a small fracto of vertces are removed the post-processg phase expectato. To upper boud ther expected umber, we relate the above dsot pars to maxma of Gaussa processes defed terms of the varables X 1,..., X. A crucal gredet s a upper boud o the varace (ad cocetrato) of the maxmum of a Gaussa process ad the fact that the Gaussa varables X 1,..., X satsfy squared tragle equaltes E(X X ) 2 E(X X k ) 2 + E(X k X ) 2 (because the pseudo-dstrbuto µ satsfes ths equalty ad X 1,..., X have the same frst two momets as µ). Rego growg Let G be a d-regular graph wth vertex set []. Let µ : {0, 1} R be a degree-4 pseudo-dstrbuto. Let d(, ) = Ẽ µ(x) (x x ) 2 be the probablty uder µ that ad are o dfferet sdes of the bpartto. For a subset S [], we let d(, S) = m s S d(, s). Sce the equalty (x x ) 2 + (x x k ) 2 (x x k ) 2 holds over the hypercube for all,, k [] ad ths equalty as a deg-4 sos proof, we have the followg tragle equalty for all,, k [] d(, ) + d(, k) d(, k). (4) The followg lemma s the ma tool for extractg a set wth small expaso out of a degree-4 pseudo-dstrbuto. 2. Lemma (Rego growg). Suppose for > 0 there exsts a subset A [] such that A =1 d(, A),=1 d(, ). The,. ϕ(g) 1/ Ẽ µ f G Ẽ µ(x) d x ( x ). (5) Furthermore, oe of the sets A t = { [] d(, A) t} for t 0 wtesses ths boud o the expaso.

4 Boaz Barak ad Davd Steurer 4 Proof. Cosder the followg dstrbuto µ over the hypercube: choose t uformly at radom the terval [0, 1] output the vector x {0, 1} wth x = 1 f ad oly f d(, A) t. The, every par, [] wth d(, A) d(, A) satsfes E µ (x ) (x x )2 = P {d(, A) t d(, A)} t [0,1] = d(, A) d(, A) (6) By the tragle equalty Eq. (4), d(, A) d(, A) d(, ). Therefore, E µ f G Ẽ µ f G. At the same tme, sce x A ad P µ (x){ x = 0 } = P t [0,1] {t < d(, A)} = d(, A), E µ (x ) x ( x ) A =1 d(, A) d(, ), [] = Ẽ x ( x ). µ(x) (7) It follows that the expaso of G satsfes ϕ(g) E µ f G E µ (x ) x ( x ) 1 Ẽ µ f G Ẽ µ(x) x ( x ). (8) Furthermore, there exsts a vector x {0, 1} the support of µ that wtesses ths boud o the expaso. It remas to show that we ca always fd a subset A as above for Ω(1/ log ). For smplcty, we frst focus o the followg specal case, 4, [] d(, ) (9) I ths specal case, we show that for some Ω(1/ log ) there exst sets A, B [] wth A, B Ω() such that d(, ) for all A ad B. Ideed for ths choce of A, the codto of the rego growg lemma s satsfed, 4 If the graph has a sparsest cut such that the smaller sde has sze at least 0.2, we ca requre ths equalty for the pseudo-dstrbuto wthout loss of geeralty. Furthermore, t turs out that eve the geeral case ca be reduced to ths specal case. We preset ths reducto at the ed of these otes. A =1 d(, A) A B Ω( 2 ) Ω( ) d(, ). (10), [] By Lemma 2, we obta a O(1/ ) O( log )-approxmato for the expaso of G.

5 Proof, belefs, ad algorthms through the les of sum-of-squares 5 Structure theorem The quadratc samplg lemma allows us to traslate the task of fdg sets A ad B as above to a questo about otly-dstrbuted Gaussa varables. Wthout loss of geeralty we may assume that E µ(x) x = Let X = (X 1,..., X ) be a Gaussa vector that matches the frst two momets of x uder the pseudo-dstrbuto µ. The, X satsfes E X = 0, E X 2 1, ad E(X X ) 2 = d(, ) for all, []. 5 The defto of expaso Eq. (2) s symmetrc the sese that both x ad ts complemet 1 x have the same value. Therefore, we ca symmetrze the pseudo-dstrbuto the same way as for Max Cut. The followg theorem about such Gaussa vectors shows that sets A ad B as above exst ad that we ca fd them effcetly. Together wth Lemma 2 ths theorem mples Theorem 1 the specal case that the pseudo-dstrbuto satsfes Eq. (9) 3. Theorem (Structure). Let X = (X 1,..., X ) be a cetered Gaussa vector wth E X 2 1 for all []. Suppose d(, ) = E(X X ) 2 satsfes the equaltes Eq. (4) ad Eq. (9). The, there exsts sets A, B [] wth A, B Ω() such that ( m d(, ) Ω 1/ ) log. (11) A, B Furthermore, we ca fd such sets A ad B polyomal tme. It wll useful to rephrase the above theorem terms of vertex separators certa graphs assocated wth the Gaussa vector X. For > 0, let H be the drected graph wth vertex set [] such that E(H) = {(, ) [] 2 d(, ) }. (12) We say that subsets A, B [] form a vertex separator f o edge of H goes drectly betwee A ad B,.e., E(H) A B =. We say such a vertex separator s good f A B Ω( 2 ). Wth ths termology place, we see that order to prove Theorem 3 t s eough to show that there exsts Ω ( 1/ log ) such that H has a good vertex separator. Proof of structure theorem Let X = (X 1,..., X ) be a cetered Gaussa vector wth E X 2 1 for all []. Let d(, ) = E(X X ) 2 for all, []. Suppose that d satsfes Eq. (4) ad Eq. (9). For > 0, let H be the graph wth vertex set [] ad edge set Eq. (12). We cosder the followg radomzed algorthm to fd vertex separators for H:

6 Boaz Barak ad Davd Steurer 6 Fgure 1: I ths graph, the set of vertces the mddle s a vertex separator. Removg ths set breaks the graph to two parts A ad B that are ot coected by a edge. If we added, the red edge to the graph, the set of vertces the mddle would o loger be a vertex separator. 1. Choose subsets A 0, B 0 [] such that A 0 = { X 1} ad B 0 = { X 1}. (13) 2. Fd a maxmal matchg M the edges E(H) A 0 B Output the sets A = A 0 \ V(M) ad B = B 0 \ V(M). Note that the obects costructed by the algorthm, partcular the sets A ad B ad the matchg M, are otly dstrbuted wth the Gaussa vector X. We emphasze that the edges of the matchg M are drected. I partcular, the evet (, ) E mples that X X 2. We observe that the output (A, B) of the algorthm forms a vertex separator H. Ideed, by the maxmalty of M, every edge (, ) E(H) A 0 B 0 has at least oe edpot V(M), whch meas that (, ) s ot cotaed A B. We also observe that f we fd the maxmal matchg step 2 by greedly passg over the edges a caocal order depedet of the radomess of X, the for every x R, the matchg produced the evet X = x s the reverse of the matchg produced the evet X = x. It follows that for every vertex [], P{ has outgog edge M} = P{ has comg edge M}. (14) The followg lemma characterzes the success of the above algorthm terms of the expected sze of M. 4. Lemma (Vertex separator or large matchg). Ether the above algorthm outputs a good vertex separator for the graph H wth probablty Ω(1), or the radom matchg M produced by the algorthm satsfes E M Ω().

7 Proof, belefs, ad algorthms through the les of sum-of-squares 7 Proof. A smlar aalyss as the Goemas Wllamso algorthm for Max Cut shows that there exsts a absolute costat c > 0 such that P{X 1 ad X 1} c d(, ) for every, []. Thus, by equalty Eq. (9), E A 0 B 0 0.1c 2. (15) Therefore, E A B E A 0 B 0 E M 0.01c 2 E M. (16) It follows that E M c E M. (Otherwse, E A B Ω(2 ), whch would meas that the algorthm outputs a good separator for H wth probablty Ω(1), cotradctg the premse of the lemma.) We wll aalyze the expected sze of the radom matchg M by relatg t to the maxmum of a Gaussa process Lemma (Maxmum of Gaussa process). The expected maxmum of the Gaussa process {X X, []} satsfes 6 We remark that the lemma oly uses the tragle equaltes Eq. (4) but ot equalty Eq. (9). Ω(1) (E M /)3 E max, [] X X 2 log. (17) Together Lemma 4 ad Lemma 5 allow us to prove Theorem 3. Proof of Theorem 3. We prove the cotrapostve: f the above algorthm fals to fd a good vertex separator H, the the dstace parameter of the graph satsfes Ω(1/ log ). 7 Let M be the radom matchg defed by the algorthm. Sce the algorthm fals to fd a good vertex separator Lemma 4 gves a lower boud o the expected sze of the matchg, E M / Ω(1). O the other had, Lemma 5 allows us to upper boud the same quatty E M / O( log ) 1/3. We coclude log Ω(1), whch gves the desred lower boud Ω(1)/ log. 7 That meas that f we choose a suffcetly small costat factor tmes 1/ log, the algorthm wll succeed fdg a vertex separator for the correspodg graph H. It remas to prove Lemma 5. The proof cosders the followg famly of Gaussa processes ad relates ther expected maxma terms of the expected sze of the matchg M. For [] ad k N, let Y (k) be the maxmum of the Gaussa process {X X H k ()}, Y (k) = max X X. (18) H k () Here, H k () deotes the set of vertces that ca be reached from by at most k steps the graph H. Defe the followg potetal fucto, Φ(k) = E Y (k). (19) =1

8 Boaz Barak ad Davd Steurer 8 Note that Φ(k) allows us to lower boud E max, [] X X because Φ(k)/ E max, [] X X. The followg lemma s the key gredet for the proof of Lemma Lemma (chag). For every k N, Φ(k + 1) Φ(k) + E M O() max [], H k+1 () ( E(X X ) 2) 1/2. (20) 8 We remark that the proof of ths lemma ether uses the tragle equaltes Eq. (4) or the equalty Eq. (9). Ths chag lemma drectly mples Lemma 5. Proof of Lemma 5. By the tragle equalty Eq. (4), E(X X ) 2 k for all [] ad H k (). By Lemma 6, t follows that there exsts a absolute costat C 1 such that for every k N, Φ(k + 1) Φ(k) + E M C k. For all k k 0 = (1/4C 2 ) (E M /) 2 /, ths equalty smplfes to Φ(k + 1) Φ(k) E M. (21) If we uroll ths recurrece relato, we obta the desred lower boud E max, [] X X Φ(k 0 + 1)/ 1 2 k 0(E M /) = (1/8C 2 ) (E M /) 3 /. (22) The proof of Lemma 6 uses the followg remarkable fact about the varace of maxma of Gaussa processes: The varace of the maxmum of a collecto of Gaussa varables s bouded by the maxmum varace of a varable the collecto Theorem (Varace of maxma of Gaussa processes). Let Z = (Z 1,..., Z t ) be cetered Gaussa varables. The, the varace of the maxmum of Z 1,..., Z t s bouded by 9 I cotrast, the expectato of the maxmum typcally depeds also o the umber of varables the collecto. V[max{Z 1,..., Z t }] O(1) max{v[z 1 ],..., V[Z t ]}. (23) The proof of ths equalty also shows Gaussa-lke cocetrato aroud the mea. I that form t s sometmes called Borell s equalty ad follows from Gaussa cocetrato equaltes for Lpschtz fuctos (Sudakov ad Crel so [1974]; Borell [1975]). 10 Proof of Lemma 6. The startg pot of the proof s the followg equalty amog radom varables for all (, ) E(H), 10 See ths book chapter by Pollard for dfferet proofs of ths cocetrato equalty. Y (k+1) Y (k) + X X. (24)

9 Proof, belefs, ad algorthms through the les of sum-of-squares 9 The equalty holds because H k+1 () H k (). Hece, f Y (k) = X h X for some h H k (), the Y (k+1) X h X = Y (k) + X X. Let N [] [] be a arbtrary matchg of the vertces ot matchg M. Sce matchg edges (, ) M satsfy X X 2, we ca derve the followg set of /2 equaltes, (, ) M. Y (k+1) Y (k) + 2 ad (, ) N. 2 1Y(k+1) Y(k+1) If we sum up these /2 equaltes, we obta the equalty Y (k+1) =1 L 1 2 Y(k) (25) Y (k) R + 2 M. (26) =1 Here, L = 1 f has a outgog edge M, L = 0 f has a comg edge M, ad L = 1/2 f s ot matched M. Smlarly, R = 1 f has a comg edge M, R = 0 f has a outgog edge M, ad R = 1/2 f s ot matched M. Sce every vertex s equally lkely to have a comg edge ad a outgog edge M (see dscusso after the descrpto of the algorthm), t follows that E L = E R = 1/2 for all, []. Usg a geeral boud o the varace of maxma of Gaussa processes, 11 E Y (k+1) L E Y (k+1) [ E L V Smlarly, E Y (k) R E Y (k) [ E R V Y (k) Y (k+1) ] ( V[L ] O(1) max H k+1 () (27) Y(k). 11 Here, we use the geeral equalty E XY E X E Y VX VY, whch we leave as a exercse to the reader. E(X X ) 2) 1/2. ] ( V[R ] O(1) max E(X X ) 2) 1/2. H k () (28) By takg the expectato of equalty Eq. (26) ad applyg the above devato bouds, t follows as desred that E Y (k+1) E Y (k) ( + 4 M O() max E(X X ) 2) 1/2. =1 =1 [], H k+1 () (29) Reducto to the well-spread case We descrbe ow how to prove Theorem 1 case equalty Eq. (9) s ot satsfed. Recall the dstace d(, ) = E µ(x) (x x ) 2. Let δ > 0 be the average dstace δ = 1 2, [] d(, ). (30)

10 Boaz Barak ad Davd Steurer 10 The followg lemma shows that f there exsts a cluster of radus sgfcatly smaller tha δ that cotas a costat fracto of vertces, the the codto of Lemma 2 s satsfed for Ω(1) (whch meas that we get a O(1)-approxmato). 8. Lemma (Heavy cluster). Suppose there exsts [] such that A = { [] d(, ) δ/4} satsfes A Ω(). The, A =1 d(, A) Ω(1), [] d(, ). Proof. By the tragle equalty, δ 1 d(, A) + d(, A) + δ/2 δ/ d(, A). (31), [] =1 Thus, A =1 d(, A) δ/4 A Ω(1) δ2, whch s the desred boud. The followg lemma shows that f the codto of the prevous lemma s ot satsfed, the after restrctg to a costat fracto of the vertces there are Gaussa radom varables that we ca apply the structure theorem to order to obta sets A, B [] wth A, B Ω() such that d(, ) Ω(1/ log ) δ for all A ad B. 9. Lemma (Heavy cluster or well-spread). Ether there exsts a vertex [] that satsfes the codto of Lemma 8 or there exsts a subset U [] wth U Ω() ad a cetered Gaussa vector Y = (Y 1,..., Y ) such that E Y 2 1 for all U,, U E(Y Y ) , there exsts α Ω(δ) such every, U satsfes d(, ) = α E(Y Y ) 2. Proof. Suppose that o vertex [] satsfes the codto of Lemma 8. So every [] satsfes B(, δ/4) 0.1, where B(, δ/4) s the set of vertces [] such that d(, ) δ/4. Let k [] be such that =1 d(, k) δ. Let U [] be the set of vertces [] such that d(, k) 2δ. By Markov s equalty, U /2. The, d(, ) δ/4 U \ B(, δ/4) δ/4 0.4 = 0.1δ. (32), U [] By symmetry, we may assume Ẽ µ(x) x = Therefore, f we let X 1,..., X be the Gaussa varables wth the same frst two momets as the pseudo-dstrbuto µ. The, the varables Y 1,..., Y wth Y = (X X k )/ 2δ satsfy E Y 2 1 for all [],, U E(Y Y ) ad E(Y Y ) 2 = d(, )/2δ.

11 Proof, belefs, ad algorthms through the les of sum-of-squares 11 Refereces Saeev Arora, Satsh Rao, ad Umesh V. Vazra. Expader flows, geometrc embeddgs ad graph parttog. I STOC, pages ACM, Chrster Borell. The Bru-Mkowsk equalty Gauss space. Ivet. Math., 30(2): , ISSN Frak Thomso Leghto ad Satsh Rao. A approxmate max-flow m-cut theorem for uform multcommodty flow problems wth applcatos to approxmato algorthms. I FOCS, pages IEEE Computer Socety, Thomas Rothvoss. Lecture otes o the ARV algorthm for sparsest cut. CoRR, abs/ , V. N. Sudakov ad B. S. Crel so. Extremal propertes of half-spaces for sphercally varat measures. Zap. Nauč. Sem. Legrad. Otdel. Mat. Ist. Steklov. (LOMI), 41:14 24, 165, Problems the theory of probablty dstrbutos, II.