Lecture 6 spectral method for planted clique

Overview Apply matrix perturbation Pesut and Matrix concentration

result to the planted clique problem AKS98

spectrol algorithm

Let G G n E k Let Kdenote the planted k clique and

A denote the adjacency matrix

Ai fl f i'J Ek

Berck aw

Let W be

Wii f2 5 1 if jO i j

Algorithm

1 Find the top eigenvector U of Wz Let Tc be the setof K indices i with the largest Heil

3 clean up Define 92 the setof vertices in Ghang241kneighbors in Te ie Te fi 8 dream 23

ofedgesfrom tonodesin K

Tin AKS98 If K2CDTfor a large constant Cthen Rf Eek 1

Pt step show Tc is approximately correct

TKAKI z a Elk Whp forsome E ECC

Fix some small so that we will specify later

Let W 33T where 3 1 3 I sick3 ien

Now W't is rank 1 and is its top eigenvectorwith Conespondy eigenvalue di wit K

Now View W W't t W W By Davis Kahan's SinoTheorem if x ht AzCW o then

Min Il ut sull HW WHop_SESH dint MW µ aq gtfoWLOG assure the sign s

iNotethat 11W Witkop 11W EEN Hop THEENT Witkop

11W ECW Hop t 1

E co Tn I whplmacmfennetyutaq.ge

By Weyl's inequality

IWW HzCh WWII EHW WHopECoJn11Thus dilutt WW Z K co In 170 WhenC co

Co IntlHunk EC G Jn T EE whp

for C big enough spectroegap

We then claimthat Hu whee implies

Te h Kl Z Ct E k forsome E

To see this note that IKI 4Tcl K so KIKI 4ktKIFurthermore

E z Hu VIK Eek Ui Fk tEkUiIf Mil E for all i f k Then

E z EepyUi El z 1KIKI

KIKI e AKE

If 7 IITs s t IUj l ITT Then Bydef of K

1417 for all I C Ts Thus

E Z Eq Ui Z 4 ITalk I

IKIN E 4kt

In either case we conclude that

1 KAKI z a d k with E 4E

Step We show 42 k Whp In thefolky

proof we suppose the high prob event

11h Hk EC holds

If i Ek then

dkcilzdknkc.DZ TAkl Iza eYky

Thus i C 92 when E's 4If idk then

dkciledkciltdklkciltdkciltlklklf.dk

Ci sikAlso dKci Bin Kik By Hoeffding'sinqua

for all e E'T

P9 did it 2 Et e k3EP 9 did it ZIK 3E e

rock

Applyy unionbound over all idk gies that

IP9 I idk dkCil Z I d K

E n eNIK

In conclusion under the event that flu VIKEE and

decile d K fall idk we have gZ KFinally

IPC'Etkle Ipf Hu ult E TIP 7 idk dkcilzH.dkc IPS 11WEEudllopicoJn n e

NM

E 2 e MM n eNUT o

B

Renard Improving the constant

Next we show the constant C in KICJn canbe made

arbitrarily smolt atte price of increase the time Coplexity

This partis generic and applies to any algorithmwhichsucceedswhen

Fix a subset of vertices S C En Isles Define

14 15 as the setof common neighbors of S i e

14 151 9VE VIS trues u u 3Let's say 5 2 Look at the induced subgraph G G HISD

oops If S E K then G G 114101,4212 2

Conditional on NHS

and I Isil nBinnKiki 112 2NHS o o O O O O

G I Hoch when12091

We see that after this subgraph operationn Ik K z

1 2

The upgraded algorithm

Forany S subset S EV run the existyalgorithm on

a I G IN IS andoutput Q Repeat until SOA

is a k clique and output SUQ

when search over S we are sure to find SEK

Thus K S z C guaratees that Q E K

Thus QU S K

Pick S 21092 then the success condition reducesto

K z f Jn The extra search time is at most

5 ans that is still poly in n

Pemaek Eigen spectral gapLet's plot the empirical eigenuake distributionof W

Xi wdi W W't

t Xi l w Eun

paintsemi circle law