Approximation Algorithms for Unique Games via Orthogonal Separators

Transcription

1 Approximaion Algorihms for Unique Games via Orhogonal Separaors Lecure noes by Konsanin Makarychev. Lecure noes are based on he papers [CMM06a, CMM06b, LM4]. Unique Games In hese lecure noes, we define he Unique Games problem and describe an approximaion algorihm for i. Definiion. (Unique Games). Given a consrain graph G = (V, E) and a se of permuaions π uv on [k] = {,..., k} (for all edges (u, v)), assign a value (label) x u from he se of labels [k] = {,..., k} o each verex u so as o saisfy he maximum number of consrains of he form π uv (x u ) = x v. Observe ha if he insance is compleely saisfiable, hen i is very easy o find a soluion ha saisfies all consrains: We simply need o guess he label for one verex, and hen propagae he values o all oher verices in he graph. However, even if a very small fracion of all consrains is violaed, hen i is very hard o find a good soluion. Kho [Kho0] conjecured ha if he opimal soluion saisfies 99% of all consrains, hen i is N P-hard o find a soluion saisfying even a % of all consrains. The conjecure is known as Kho s Unique Games conjecure. We sae i formally below. Definiion. (Unique Games Conjecure [Kho0]). For every posiive ε and δ, here exiss a k such ha given an insance of Unique Games wih k labels, i is N P-hard o disinguish beween he following wo cases: There exiss a soluion saisfying ( ε) fracion of all consrains. Every assignmen saisfies a mos δ fracion of all consrains. I is unknown wheher he conjecure is rue or false. The bes approximaion algorihms for Unique Games find soluions saisfying O( ε log k) and O(ε log n log k) fracion of all consrains, if he opimal soluion saisfies ε fracion of all consrains. Theorem.3 (Charikar, Makarychev, Makarychev [CMM06a]). There exiss an approximaion algorihm ha given a Unique Games insance saisfying ( ε) fracion of all consrains, finds a soluion saisfying O( ε log k) fracion of all consrains. Theorem.4 (Chlamáč, Makarychev, Makarychev [CMM06b]). There exiss an approximaion algorihm ha given a Unique Games insance saisfying ( ε) fracion of all consrains, finds a soluion saisfying O(ε log n log k) fracion of all consrains. Noe ha he approximaion of Theorem.3 canno be improved assuming he Unique Games conjecure is rue [KKMO07]. We prove Theorem.3 using he echnique of orhogonal separaors from [CMM06b]. Below, we denoe he number of violaed consrains by OP T. We assume ha OP T ε E, and show how o find a soluion violaing a mos O( ε log k) E consrains. In he nex secion, we presen a sandard SDP relaxaion for Unique Games (wihou l riangle inequaliies). Then, in Secion 3, we inroduce a echnique of orhogonal separaors. However, we pospone he proof of exisence of orhogonal separaors o Secion 5. In Secion 4, we presen he approximaion algorihm and prove Theorem.3. Finally, in Secion 6, we give some useful bounds on he Gaussian disribuion.

2 SDP Relaxaion In he SDP relaxaion, we have a vecor ū i for every verex u V and label i [k]. In he inended inegral soluion, he vecor ū i is he indicaor of he even he verex u has he label i. Tha is, if x u is he opimal labeling, hen he corresponding inegral soluion is as follows: { ū, if x u = i; i = 0, oherwise. Observe ha if a consrain (u, v) is saisfied hen ū i = v π uv(i) for all i. If he consrain is violaed, hen ū i = v π = 0 for all bu exacly wo i s: uv(i) ū x u =, bu v π = 0; and uv(x u) v x v =, bu ū = 0. πuv (x v) Thus, { ū i v 0, if assignmen x π uv(i) u saisfies consrain (u, v); =, if assignmen x u violaes consrain (u, v). Therefore, he number of violaed consrains equals (u,v) E ū i v π uv(i). () Our goal is o minimize his expression. Noe ha for a fixed verex u, one and only one ū i equals. Hence, ū i, ū j = 0, if i j; and i ū i =. We now wrie he SDP relaxaion. min (u,v) E ū i v πuv(i) ū i, ū j = 0 for all u V and i j ū i = for all u V This is a relaxaion, since for ū i = ū i, he SDP value equals he number of violaed consrains (see ()); and ū i is a feasible soluion for he SDP. Usually, such relaxaions conain an exra consrains he l riangle inequaliy. Bu we will noe use i here. 3 Orhogonal Separaors Overview Le X be a se of vecors in l of lengh a mos. We say ha a disribuion over subses of X is an m- orhogonal separaor of X wih l disorion D, probabiliy scale α > 0 and separaion hreshold β <, if he following condiions hold for S X chosen according o his disribuion:. For all ū X, Pr(ū S) = α ū.

3 . For all ū, v X wih ū, v β max( ū, v ), 3. For all ū, v X, Pr(ū S and v S) α min( ū, v ). m Pr(I S (ū) I S ( v)) αd ū v min( ū, v ) + α ū v, where I S is he indicaor of he se S i.e. I S (ū) =, if ū S; I S (ū) = 0, if ū / S. In mos cases, i is convenien o use a slighly weaker (bu simpler) bound on Pr(I S (ū) I S ( v)). 3. For all ū, v X, Pr(I S (ū) I S ( v)) αd ū v max( ū, v ), The propery (3 ) follows from (3), since ū v = ū v ( ū + v ) ū v max( ū, v ). The las inequaliy follows from he (regular) riangle inequaliy for vecors ū, v and (ū v). Our algorihm for Unique Games relies on he following heorem. Theorem 3. (Chlamac, Makarychev, Makarychev [CMM06b]). There exiss a polynomial-ime randomized algorihm ha given a se of vecors X in he uni ball and parameer m, generaes a m-orhogonal separaor wih l disorion D = O ( log m ), probabiliy scale α poly(/m) and separaion hreshold β = 0. 4 Approximaion Algorihm We are now ready o presen he approximaion algorihm. Inpu: An insance of Unique Games. Oupu: Assignmen of labels o verices.. Solve he SDP. Le X = {ū i : u V, i [k]}.. Mark all verices as unprocessed. 3. while (here are unprocessed verices) (a) Produce an m-orhogonal separaor S X wih disorion D and probabiliy scale α as in Theorem 3., where m = 4k and D = O( log n log m). (b) For all unprocessed verices u : Le S u = {i : u i S}. If S u conains exacly one elemen i, hen assign he label i o u, and mark he verex u as processed. 4. If he algorihm performs more han n/α ieraions, assign arbirary values o any remaining verices (noe ha α /poly(k)). 3

4 Lemma 4.. The algorihm saisfies he consrain beween verices u and v wih probabiliy O(D ε uv ), where ε uv is he SDP conribuion of he erm corresponding o he edge (u, v): ε uv = k ū i v πuv(i). i= Proof. If D ε uv /8, hen he saemen holds rivially, so we assume ha D ε uv < /8. For he sake of analysis we also assume ha π uv is he ideniy permuaion (we can jus rename he labels of he verex v, his clearly does no affec he execuion of he algorihm). A he end of an ieraion in which one of he verices u or v assigned a value we mark he consrain as saisfied or no: he consrain is saisfied, if he he same label i is assigned o he verices u and v; oherwise, he consrain is no saisfied (here we conservaively coun he number of saisfied consrains: a consrain marked as no saisfied in he analysis may poenially be saisfied in he fuure). Consider one ieraion of he algorihm. There are hree possible cases:. Boh ses S u and S v are equal and conain only one elemen, hen he consrain is saisfied.. The ses S u and S v are equal, bu conain more han one or none elemens, hen no values are assigned a his ieraion o u and v. 3. The ses S u and S v are no equal, hen he consrain is no saisfied (a conservaive assumpion). Le us esimae he probabiliies of each of hese evens. Using he fac ha for all i j he vecors ū i and ū j are orhogonal, and he firs and second properies of orhogonal separaors we ge (below α is he probabiliy scale): for a fixed i, Pr ( S u = ; i S u ) = Pr (i S u ) Pr (i S u and j S u for some j i) Here we used ha j [k] ū j =. Then, Pr ( S u = ) = Pr (i S u ) j [k] j i α ū i j [k] j i α ū i α 4k α ū i α 4k. Pr (i, j S u ) α min( ū i, ū j ) 4k ū j j [k] j i Pr ( S u = ; i S u ) [ α ū i α ] 3/4 α. 4k The probabiliy ha he consrain is no saisfied is a mos Pr (S u S v ) Pr (I S (u i ) I S (v i )). By he hird propery of orhogonal separaors (see propery (3 )): Pr (S u S v ) αd ū i v i max( ū i, v i ). 4

5 By Cauchy-Schwarz, Pr (S u S v ) αd αd = αd ε uv. ū i v i ε uv Finally, he probabiliy of saisfying he consrain is a leas max( ū i, v i ) ū i + v i }{{} = Pr ( S u = and S u = S v ) 3 4 α αd ε uv α. Since he algorihm performs n/α ieraions, he probabiliy ha i does no assign any value o u or v before sep 4 is exponenially small. A each ieraion he probabiliy of failure is a mos O(D ε uv ) imes he probabiliy of success, hus he probabiliy ha he consrain is no saisfied is O(D ε uv ). We now show ha he approximaion algorihm saisfies O( ε log k) fracion of all consrains. Proof of Theorem.3. By Lemma 4., he expeced number of unsaisfied consrains is equal o O( ε uv log k). (u,v) E By Jensen s inequaliy for he funcion, εuv log k E E (u,v) E (u,v) E ε uv log k = SDP E log k. Here, SDP = (u,v) E ε uv denoes he SDP value. If OP T ε E, hen SDP OP T ε E. Hence, he expeced cos of soluion is upper bounded by O( ε log k) E. 5 Orhogonal Separaors Proofs Proof of Theorem 3.. In he proof, we denoe he probabiliy ha a Gaussian N (0, ) random variable X is greaer han a hreshold by Φ(). We use he following algorihm for generaing m-orhogonal separaors wih l disorion: Assume w.l.o.g. ha all vecors ū lie in R n. Fix m = m and = Φ (/m ) (i.e., fix such ha Φ() = /m ). Sample independenly a random Gaussian n dimensional vecor g N (0, I) in R n and a random number r in [0, ]. Reurn he se S = {ū : ū, g ū and ū r}. We claim ha S is an m-orhogonal separaor wih l disorion O( log m), probabiliy scale α = /m and β = 0. Le us verify ha S saisfies he required condiions.. For every nonzero vecor ū X, we have Pr(ū S) = Pr( ū, g ū and r ū ) = Pr( ū/ ū, g ) Pr(r ū ) = ū /m α ū. 5

6 Here we used ha ū/ ū, g is disribued as N (0, ), since ū/ ū is a uni vecor. If ū = 0, hen Pr(r ū ) = 0, hence Pr(ū S) = 0.. For every ū, v X wih ū, v = 0, we have Pr(ū, v S) = Pr( ū, g ; v, g ; r ū and r v ) = Pr( ū, g ū and v, g v ) Pr(r min( ū, v )). The random variables ū, g and v, g are independen, since ū and v are orhogonal vecors. Hence, Pr(ū, v S) = Pr( ū, g ū ) Pr( v, g v ) Pr(r min( ū, v )). Noe ha ū/ ū is a uni vecor, and ū/ ū, g N (0, ). Thus, Pr( ū, g ū ) = Pr( ū/ ū, g ) = /m. Similarly, Pr( v, g v ) = /m. Then, Pr(r min( ū, v )) = min( ū, v ), since r is uniformly disribued in [0, ]. We ge Pr(ū, v S) = min( ū, v ) m = α min( ū, v ). m 3. If I S (ū) I S ( v) hen eiher ū S and v / S, or ū / S and v S. Thus, Pr(I S (ū) I S ( v)) = Pr(ū S; v / S) + Pr(ū / S; v S). We upper bound he boh erms on he righ hand side using he following lemma (swiching ū and v for he second erm) and obain he desired inequaliy. Lemma 5.. If ū v, hen oherwise, Proof of Lemma 5.. We have Pr(ū S; v / S) αd ū v min( ū, v ) + α ū v ; Pr(ū S; v / S) αd ū v min( ū, v ). Pr(ū S; v / S) = Pr( ū, g ū ; r ū ; v / S). The even { v / S} is he union of wo evens { v, g v and r v } and {r v }. Hence, Pr(ū S; v / S) Pr( ū, g ū ; v, g < v ; r min( ū, ū )) () + Pr( ū, g ū ; v r ū ). Le g u = ū/ ū, g and g v = v/ v, g. Boh g u and g v are sandard N (0, ) Gaussian random variables. Thus, Pr(g u ) = Pr(g v ) = /m = α. We rewrie () as follows: Pr(ū S; v / S) = Pr(g u ; g v < ) Pr(r min( ū, v )) + Pr(g u ) Pr( v r ū ) = Pr(g u ; g v < ) min( ū, v ) + α Pr( v r ū ). (3) 6

7 To finish he proof we need o esimae Pr(g u ; g v < ) and Pr( v r ū ). Since r is uniformly disribued in [0, ], Pr( v r ū ) = ū v, if ū v > 0; and Pr( v r ū ) = 0, oherwise. We use Lemma 6. o upper bound Pr(g u ; g v < ): Pr(g u ; g v < ) O ( cov(gu, g v ) log m /m ). (4) The covariance of g u and g v equals cov(g u, g v ) = ū/ ū, v/ v and ū v = ū + v ū, v. Hence, cov(g u, g v ) = ū + v ū v ū v We plug his bound in (4) and ge Pr(g u ; g v < ) α = ū v ( ū + v ū v ) ū v = ū v ( ū v ) ū v ū v ū v O( log m ). Now, Lemma 5. follows form (3). This concludes he proof of Lemma 5. and Theorem 3.. ū v ū v. 6 Gaussian Disribuion In his secion, we prove several useful esimaes on he Gaussian disribuion. Le X N (0, ) be one dimensional Gaussian random variable. Denoe he probabiliy ha X by Φ(): Φ() = Pr(X ). The firs lemma gives a very accurae esimae on Φ() for large. Lemma 6.. For every > 0, π ( + ) e < Φ() < π e. Proof. Wrie Thus, Φ() = = π e x dx = π e π π e x x e x x dx. Φ() < π e. e x x dx On he oher hand, π e x x dx < π e x dx = Φ(). 7

8 Hence, and, consequenly, Φ() > π e Φ() > Φ(), π( + ) e. Lemma 6.. Le X and Y be Gaussian N (0, ) random variables wih covariance cov(x, Y ) = ε. Pick he hreshold > such ha Φ() = /m for m > 3. Then Pr(X and Y ) = O(ε log m/m). Proof. If ε or ε /, hen we are done, since ε log m = Ω(ε) = Ω() and So we assume ha ε and ε < /. Le Pr(X and Y ) Pr(X ) = m. ξ = X + Y ε ; η = X Y. ε Noe ha ξ and η are N (0, ) Gaussian random variables wih covariance 0. Hence, ξ and η are independen. We have Pr ( X and Y ) = Pr ( ε ξ + εη and ε ξ εη ). Denoe by E he following even: Then, E = { ε ξ + εη and ε ξ εη }. Pr ( X and Y ) = Pr(E and εη ) + Pr(E and εη ). Observe ha he second probabiliy on he righ hand side is very small. I is upper bounded by Pr(εη ), which, in urn, is bounded as follows: Pr(εη ) = π e x /ε We now esimae he firs probabiliy: ( ε e ε dx = Φ(/ε) O Pr(E and εη ) = E η [Pr(E and η /ε η)] = /ε Pr(E η = x) e x / dx π 0 ) ( ε e O ) = O(ε/m). = /ε Pr( ε ξ [ εx, + εx]) e x / dx. π 0 The densiy of he random variable ε ξ in he inerval ( εx, + εx) for x [0, /ε] is a mos π( ε ) e ( εx) ( ε ) e ( εx) e e εx e e x, 8

9 here we used ha ε / and ε. Hence, Therefore, Pr( εx ε ξ + εx) εx e e x. Pr(E and εη ) ε e π /ε 0 xe x e x ε e dx π xe x e x dx. 0 } {{ } O() The inegral in he righ hand side does no depend on any parameers, so i can be upper bounded by some consan (e.g. one can show ha i is upper bounded by π). We ge This finishes he proof. Pr(E and εη ) O(ε e ) = O(ε Φ()) = O(ε log m/m). References [CMM06a] Moses Charikar, Konsanin Makarychev, and Yury Makarychev. Near-opimal algorihms for unique games. In ACM Symposium on Theory of Compuing, STOC, 006. [CMM06b] Eden Chlamac, Konsanin Makarychev, and Yury Makarychev. How o play unique games using embeddings. In IEEE Symposium on Foundaions of Compuer Science, FOCS, 006. [Kho0] Subhash Kho. On he power of unique -prover -round games. In ACM Symposium on Theory of Compuing, STOC, 00. [KKMO07] Subhash Kho, Guy Kindler, Elchanan Mossel, and Ryan O Donnell. Opimal inapproximabiliy resuls for MAX-CUT and oher -variable CSPs? SIAM J. Compu., 37():39 357, 007. [LM4] Anand Louis and Konsanin Makarychev. Approximaion algorihm for sparses k-pariioning. In ACM-SIAM Symposium on Discree Algorihms, SODA, 04. 9