Quantum and Randomized Lower Bounds for Local Search on Vertex-Transitive Graphs

Quan tum and Randomized Lo w er Bounds for Lo cal Searc h on V ertex-T ransitiv e Graphs Hang Dinh and Alexand e r Russell Department of Computer Science & Engineering Universit y of Connecticut Storrs, CT 06269, USA { hangdt, acr } @engr .uconn.edu Abstract. W e study the problem of l o c al se ar ch on a graph. Given a real-v alued blac k -box function f on the graph’s vertices, this is the p ro blem of determining a lo cal minimum of f —a vertex v for whic h f ( v ) is n o more than f ev aluated at any of v ’s neigh b ors. In 1983, A ldous gav e the first strong lo wer bounds fo r the problem, sho wing that any randomized algorithm requires Ω (2 n/ 2 − o (1) ) queries to determine a lo cal minima on th e n -dimensional hyp ercube. The next ma jor step forw ard w as not until 2004 when Aaronson, in tro ducing a new metho d for query complexity b ounds, b oth strengthened this lo wer b ound to Ω (2 n/ 2 /n 2 ) and gav e an analogous lo w er bound on the quan tu m q uery complexit y . While these bound s are very strong, they are kn o wn only for narrow fa milies of gra ph s (hyp ercu bes and grids). W e sho w how to generalize Aaronson’s t ec h niques in order to giv e randomized (and q u an tum) lo wer b ounds on th e query complexity of lo cal search for the family of vertex-transitive graph s . In particular, w e show that for an y vertex-transitive graph G of N vertices and d i ameter d , the rand omized and quantum query complex ities for local searc h on G are Ω “ √ N d log N ” and Ω “ 4 √ N √ d l o g N ” , resp ectiv ely . 1 In tro duction The lo c al se ar ch pro ble m is that of determining a local minimum o f a function defined on the v ertices of a gr aph. Spec ific a lly , g iven a rea l-v alued black-box function f on the vertices of a graph, this is the problem of determining a vertex v at whic h f ( v ) is no more than f ev aluated at any of v ’s neighbors . The problem pro vides an abstract f ra mew ork for studying local search heuristics that hav e b een wide ly applied in combinatorial optimization, heuristics that t ypica lly combine random selection with steep est de s cen t. The per formance of these heuristic algorithms, as recog nized in [1], “wa s gener al ly c onsider e d to b e satisfactory, p artly b ase d on ex p erienc e, p artly b ase d on a b elief in some physic al or biolo gic al analo gy, . . . ” Ideally , of course, we would ev aluate practical results in the co n text of crisp theoretical bounds on the complexity of these problems! Mor eo ver, as p oin ted out in [2], the co mplexit y of the lo cal search problem is als o central for understanding a series of complexit y classes whic h are sub classes of the total function class TFNP , including PPP (Polynomial Pigeoho le Principle), PO DN (P olynomial Odd- Deg ree No de), a nd PL S (Polynomial Lo cal Search). Lo cal sea r c h has been the sub ject of a sizable b ody o f theor etical work, in which co mp lexity is typically measured by query c omplexity : the to ta l n umber o f queries made to the black-box function f in or der to find a lo cal minimum. The fir s t strong lower bounds were establis he d in 1983 by Aldous [3 ], who showed that 2 n/ 2 − o ( n ) queries are necessar y , in g eneral, in or der for a ra ndo mized algor it hm to find a lo cal minimum of a function on the hyper c ub e { 0 , 1 } n . His pro of constr ucts a rich collection of unimo dal functions (that is, functions with a unique minim um) using hitting times of r andom walks. Llewellyn et a l. [6] improved the bound for deterministic query complexity to Ω (2 n / √ n ) using an adv ersa ry a rgumen t c hara cterized b y vertex cuts. With the adven t o f quantum computing, these black-box problems r eceiv ed renewed interest [2, 1 1, 8, 9, 10]. Most notably , Aaronso n [2] intro duced a query low er bo und metho d tuned for such problems, the r elational adversary metho d . Though his principal mo tiv a tio n was, no doubt, to provide quant u m low er bo unds for lo cal sear c h, his tec hniques felicitously demonstrated impr oved b ounds on randomized quer y complexity . In particular, he established a Ω (2 n/ 2 n − 2 ) low er bound for randomized lo cal sear c h on the Bo olean hypercub e { 0 , 1 } n and the first non trivia l lower b ound of Ω  n d/ 2 − 1 / ( d lo g n )  for randomized lo cal search o n a d -dimensional grid [ n ] d with d > 3 . These t wo low er bounds of Aaronson’s ha ve b een recen tly im proved by Zhang [11]: refining Aaronson’s framework, he establis he d randomized query complexity low er b ounds of Θ (2 n/ 2 n 1 / 2 ) o n the hypercub e and Θ ( n d/ 2 ) on the grid [ n ] d , d > 4. Additionally , Zhang’s method can be applied to certain class e s of pr oduct graphs, though it provides a rather complicated relationship b et ween the low er bo und and the product decomp osition. 1 A remaining h urdle in this dir ect line of resea rc h w as to es ta blish strong bo unds for grids of small dimension. Sun and Y ao [9] hav e addressed this pro blem, proving that the quantum quer y c o mplexit y is Ω  n 1 / 2 − c  for [ n ] 2 and Ω ( n 1 − c ) for [ n ] 3 , for any fixed constant c > 0 . F o cusing on g eneral gr a phs, Santha and Szegedy [8] esta blished quant um low er b ounds of Ω (log N ) and Ω  8 p s δ / log N  for lo cal sear c h o n connected N -vertex g raphs with maxima l deg ree δ a nd sep ar ation n umb er s . 2 W e remar k that s/δ ≤ N and these bo unds for g eneral graphs ar e, naturally , muc h weak er than those obtained for the hig hly structured families of gra phs ab o ve. In this a rticle, we show o ff the flex ibility of Aaronso n’s framework by extending it to ar bitrary vertex- transitive gra phs. Recall that a gra ph G = ( V , E ) is vertex-tr ansitive if the a uto morphism gro up of G ac ts transitively on the vertices: for any pair o f vertices x, y ∈ V , there is a gra ph automo rphism φ : V → V for which φ ( x ) = y . In par ticula r, all Cayley graphs ar e vertex-tra nsitiv e, so this class of gr aphs contains the hypercub es of previous in teres t and the lo oped grids (tor i) . Our low er bounds dep end only on the size and dia meter o f the gr aph: Theorem 1. L et G b e a c onne cte d, vertex-tr ansitive gr aph with N vertic es and diameter d . Then RLS ( G ) = Ω √ N d log N ! , and QLS ( G ) = Ω 4 √ N √ d log N ! wher e R LS ( G ) and QLS ( G ) ar e the r andomize d and quant um query c omplexities of lo c al se ar ch on G , r esp e c- tively. Thu s the v ertex transitive graphs, compromising b et ween the sp ecific families of graphs addr essed by [2, 11] and the g eneral res ult s of Santha and Szegedy , still provide enoug h str uctu re to supp ort s trong lower bo unds. 2 Definitions and Notation As in [2, 11], w e fo cus on the lo cal search problem stated pr e cisely as follows: g iv en a graph G = ( V , E ) and a bla c k-b o x function f : V → R , find a lo cal minimum of f on G , i.e. find a vertex v ∈ V such that f ( v ) ≤ f ( w ) for all neighbo rs w o f v . While the gra ph is known to the algo r ithm, the v a lues of f ma y only be accessed through an ora c le. F or an a lg orithm A that solves the lo cal search pr oblem on G , let T ( A , G ) be the max im um n umber of queries made to the black-box function by A b efore it returns a lo cal minimum, this maximum taken ov er a ll functions f on G . Given a graph G , the randomized quer y complexity for L o cal Search on G is defined as min A T ( A , G ), where the minimum r anges ov er all randomized algorithms A that output a lo cal minim um with proba bilit y a t least 2 / 3 . The quantum query co mplexit y is defined similarly , 1 Zhang [11]’s general low er b ounds for a pro duct graph G w × G c inv olve the length L of the longest self-av oiding path in the “clock” graph G c , and parameters p ( u, v , t )’s of a regular random w alk W on G w , where p ( u, v , t ) is the probability t h at t he random w alk W starting at u ends u p at v after exactly t steps. In particular, h e sho wed that RLS ( G w × G c ) = Ω „ L P L/ 2 t =1 max u,v p ( u,v ,t ) « and QLS ( G w × G c ) = Ω „ L P L/ 2 t =1 √ max u,v p ( u,v ,t ) « . 2 Santha and S zegedy define the sep ar ation numb er s ( G ) of a graph G = ( V , E ) to b e: s ( G ) = max H ⊂ V min S ⊂ H, | H | / 4 ≤| S |≤ 3 | H | / 4 | ∂ H S | , where ∂ H S = { v ∈ H \ S : ∃ u ∈ S, ( v , u ) ∈ E } is the b oundary of S in the sub- graph of G restricted to H . 2 except that in the qua n tum case, T ( A , G ) is the max im um num b er of unitar y quer y transformations of the error -bounded quantum a lgorithm A . The randomized (resp. q uan tum) query co mplexit y for lo cal search on G will b e denoted b y RLS ( G ) (resp. QLS ( G )). As mentioned in the intro duct ion, we fo cus on the vertex-tra nsitiv e graphs, those whose a ut omo rphism groups act tra nsitiv ely on their vertex sets. Perhaps the mo s t imp ortan t sub class of the vertex-trans itiv e graphs a re the Ca yley g raphs. Let G be a group (finite, in this article, and written multiplicativ ely) and Γ a set o f g enerators for G . The Cayley gra ph C ( G, Γ ) is the g raph with vertex set G and edges E = { ( g , g γ ) | g ∈ G , γ ∈ Γ ∪ Γ − 1 } . Note that with this definition for the edges, ( a, b ) ∈ E ⇔ ( b, a ) ∈ E even when G is nonab elian, and we may consider the graph to b e undirected. If X is a sequence of vertices in a gra ph, we write X i → j to denote the subsequenc e o f X fro m p osition i to p osition j ( i ≤ j ). If X = ( x 1 , . . . , x t ) is a seq uence of v er t ices in a Cayley gr aph and g is a gro up element, then we use g X to deno t e the se q uence ( g x 1 , . . . , g x t ). More generally , for an y automor phis m σ of a vertex-transitive graph G and any sequence X = ( x 1 , . . . , x t ) of vertices in G , we let σ X denote the se q uence ( σ ( x 1 ) , . . . , σ ( x t )). The distance b et ween tw o vertices u, v of a graph G shall be denoted by ∆ G ( u, v ); when G is unders t o o d from co n tex t we abbreviate to ∆ ( u, v ). The statistical dista nce betw een tw o distributions D 1 and D 2 on the same set Ω is defined as the distance in total v ar iation: k D 1 − D 2 k t . v . = max E ⊂ Ω | D 1 ( E ) − D 2 ( E ) | = 1 2 X ω ∈ Ω | D 1 ( ω ) − D 2 ( ω ) | . W e s a y that the distribution D 1 is δ -close to dis t ributio n D 2 if k D 1 − D 2 k t . v . ≤ δ . 3 Generalizing Aaronson’s Snak es Aaronson’s [2] application of the quantum and relational adversary metho ds to lo cal s e arc h pro blems inv olved certain families o f walks on a gr a ph he called “snakes.” W e b egin by presenting Aaro ns on’s snake metho d, adjusted to suit our gene r alization. Thro ughout this s e c t ion, let G b e a g raph. A snake X of length L is a s equence ( x 0 , . . . , x L ) of vertices in G such that ea c h x i +1 is either equal to x i or a neighbor of x i . The subsequence X 0 → j shall b e referred to as the j - le ngth “head” of the snake X . Supp ose D x 0 ,L is a distribution ov er snakes of length L starting at x 0 , a nd X is a snake drawn from D x 0 ,L . In Aa r onson’s parlance, the snake X “flicks” its tail by c ho osing a p osition j uniformly at random from the set { 0 , . . . , L − 1 } , and then drawing a new snake Y from D x 0 ,L conditioned on the even t that Y 0 → j = X 0 → j , that is, that Y has the same j - length head as X . In order to simplify the pro of for vertex-transitive gr aphs b elo w, we consider a generaliza tion in which a snake flic ks its tail according to a dis t ributio n D L , which may be nonuniform , on the se t { 0 , . . . , L − 1 } . W e shall relax, also, Aaronso n’s original condition that, aside from adjace nt rep etition of a vertex v , snakes be no n-self-in tersecting. Let X = ( x 0 , . . . , x L ) b e a snake. Define the function f X on G as follows: for each vertex v o f G , f X ( v ) = ( L − max { i : x i = v } if v ∈ X , L + ∆ ( x 0 , v ) if v 6∈ X . In other words, f X ( x L ) = 0, and for a n y i < L , f X ( x i ) = L − i if x i 6∈ { x i +1 , . . . , x L } . Cle arly f X has a unique lo cal minim um at x L . Let X and Y b e snakes o f length L star ting at x 0 . A vertex v is called a disagr e ement b et ween X and Y if v ∈ X ∩ Y and f X ( v ) 6 = f Y ( v ). W e sa y X and Y are c onsistent if there is no disagreement b et ween X and Y . Observe that so lo ng as X a nd Y are co nsisten t, f X ( v ) 6 = f Y ( v ) ⇐ ⇒ set X ( v ) 6 = set Y ( v ) for all vertices v , where set X is the function o n G defined as set X ( v ) = 1 if v ∈ X a nd 0 otherwise. Fix a dis tr ibut ion D x 0 ,L for snakes of length L starting at x 0 and a distribution D L on the set { 0 , . . . , L − 1 } . With these in place, we let P r j,X [ · ] denote the proba bilit y o f an even t over the distribution determined by independently selecting j accor ding to D L and X from D x 0 ,L . 3 W e r ecord Aa ronson’s definition of go o d sna kes, replacing the uniform distribution on the set { 0 , . . . , L − 1 } with the distribution D L , and requiring a go o d snake’s endp oint to b e different from those of mos t other snakes . Definition 1. A snake X ∈ D x 0 ,L is ǫ -go o d w.r.t. distribution D L if it satisfies the fol lowing: 1. X is 0 . 9 -consisten t : Pr j,Y [ X and Y ar e c onsistent, and x L 6 = y L | Y 0 → j = X 0 → j ] ≥ 0 . 9 . 2. X is ǫ -hitting : F or al l v ∈ G , P r j,Y [ v ∈ Y j +1 → L | Y 0 → j = X 0 → j ] ≤ ǫ . Our low er bounds will dep end on the following adaptation of Aaronson’s theorem of [2]: Theorem 2. As s ume a snake X dr awn fr om D x 0 ,L is ǫ -go o d w.r.t. D L with pr ob ability at le ast 0 . 9 . Then RLS ( G ) = Ω (1 /ǫ ) and QLS ( G ) = Ω ( p 1 /ǫ ) . Pr o of. T o b egin, we reduce the lo cal s earc h pr o blem to a dec ision problem. F or each snake X ∈ D x 0 ,L and a bit b ∈ { 0 , 1 } , define the function g X,b on G as follows: g X,b ( v ) = ( f X ( v ) , − 1) for a ll vertices v 6 = x L , a nd g X,b ( x L ) = (0 , b ). Then, a n input of the decision proble m for lo cal search on G is an order ed pair ( X , g X,b ), where X ∈ D x 0 ,L and b ∈ { 0 , 1 } is an answer bit. Howev er, the “ snak e par t ” X in the input cannot b e queried—it appea rs in the input a s a bo okkeeping to ol. Given such an input ( X , g X,b ), the decision problem is to output the answer bit b . Obser v e that the ra ndo mized (resp. qua n tum) quer y complexity of the decision problem is a low er b ound for that o f the original lo cal search pro blem. This inco rpora tion of X into the input o f the decision pro blem induces a natur al one- to-one cor respondence b et ween an input s e t of the same answer bit and the set of snakes app earing in the input set. (Thanks to Sco tt Aa ronson for suggesting this conv ention to us!) In Aaro nson’s o riginal version, since the input par t X is o mit ted, the snakes must be non-self-intersecting in order to obtain such a one-to-one corr espondence. Santha and Szegedy [8] have presented a n alter nate a pproac h for eliminating self-int er s ecting snakes while following Aaronso n’s pro of scheme, thoug h their technique only applies to the qua n tum case. The remaining part of the proo f , whic h esta blishes lo wer bo unds for the decision problem using the rational and quantum adversary methods , is similar to Aaronso n’s pro of with the exception of so me tec hnical deta ils due to the adjustments in the definition o f go o d snakes. W e hav e relegated the full pro of to the app endix. 4 Lo w er B ou nds for V ertex-T ransitiv e Graphs F or s implicity , we first apply the snake fra mew ork for Cayley gra ph s, and then extend the a pproac h for vertex-transitive graphs. 4.1 Lo w er b ounds for C ayley graphs Consider a Cayley gr aph C ( G, Γ ) of group G determined by a generating set Γ . Our g oal is to design a go od snake distribution fo r C ( G, Γ ). O ur sna k es will c onsist o f a ser ie s of “ch unks” so that the endp oin t o f each ch unk lo oks almost random g iv en the preceding chu nks. The lo cations at which a sna k e flicks its tail will b e c hosen randomly from the lo cations of the ch unks’ endp oin ts. Each c hunk is a n “ex t ended” sho rtest path connecting its endp oin t with the endp oin t of the previo us one. The relev ant pr o perties of these snakes depe nds on the length of each ch unk a s well a s the n umber o f ch unks in e a c h sna k e. T o determine these parameters , we b egin with the following definitions. Let B ( s ) be the ball of radius s cen tered at the group identit y , i.e., B ( s ) is the set of vertices v for whic h ∆ (1 , v ) ≤ s . W e say that Cayley g raph C ( G, Γ ) is s -mixing if there is a distribution over the ball B ( s ) that is O  s/ | G | 3 / 2  -close to the unifor m distribution over G . Clearly , every Cayley gra ph of diameter d is d -mix ing . Now we a s sume C ( G, Γ ) is s - mixing , and let D s be a distribution ov er B ( s ) so that the ex t ensio n of D s to b e over G is δ -close to the unifor m distr ibut ion over G , wher e s ≤ p | G | and δ = 0 . 1 s/ | G | 3 / 2 . F or e a c h group element g ∈ B ( s ), we fix a shor tes t path (1 , g 1 , . . . , g r ) in C ( G, Γ ) from the gro up identit y to g (here r = ∆ (1 , g )). Then let S ( g ) denote the s equence ( g 1 , g 2 , . . . , g s ), where g i = g for i ≥ r . 4 Fix ℓ = p | G | / (200 s ) a nd let L = ( ℓ + 1) s . W e formally define our snake distributio n D x 0 ,L for snakes X = ( x 0 , . . . , x L ) as follows. F or any k ∈ { 0 , . . . , ℓ } , choos e g k independently ac cording to the distribution D s , and let the k th “ch unk” ( x sk +1 , . . . , x sk + s ) b e iden tical to the sequence x sk S ( g k ). Prop osition 1. A snake X dr awn fr om D x 0 ,L δ -mixes by s steps in the sense that for any k and any t ≥ s , x sk + t is δ -close to uniform over G given x sk . W e define distribution D L on { 0 , . . . , L − 1 } as the unifor m distribution o n the se t { s, 2 s, . . . , ℓs } . So , unlike Aaronson’s snakes whose tails may be flic ked at any lo cation, our sna k es can not “ break” in the middle of any ch unk a nd only flick their tails at the ch unk endpo in ts. T o s ho w that most of our sna k es are g oo d, we star t by showing that most snakes X a nd Y are co nsisten t and hav e different endpoints . Prop osition 2. L et j b e chosen ac c or ding to D L . L et X , Y b e dr awn fr om D x 0 ,L c onditione d on Y 0 → j = X 0 → j . Then Pr X,j,Y [ X and Y ar e c onsistent, and x L 6 = y L | Y 0 → j = X 0 → j ] ≥ 0 . 9 999 − 2 | G | . Pr o of. Fix j ∈ { s, 2 s, . . . , ℓs } . Supp ose v is a disa greemen t b et ween X a nd Y , letting t = max { i : x i = v } and t ′ = max { i : y i = v } , then t 6 = t ′ and t ′ , t ≥ j . W e ca n’t hav e b oth t < j + s and t ′ < j + s , b ecause otherwise we would ha ve v 6 = x j + s and v 6 = y j + s which implies tha t b oth t − j a nd t ′ − j equa l the distance from x j to v . If there is a disa greemen t, there m ust e x ist t a nd t ′ such that x t = y t ′ and either t ≥ j + s or t ′ ≥ j + s . In the case t ≥ j + s , we have x t is δ -close to uniform given y t ′ , which implies Pr X,Y j → L [ x t = y t ′ ] ≤ δ + 1 | G | ≤ 2 | G | . Similarly , in the case t ′ ≥ j + s , we also ha ve P r X,Y j → L [ x t = y t ′ ] ≤ 2 | G | . Summing up for all p ossible pairs of t and t ′ yields Pr X,Y j → L [there is a disag reemen t betw een X and Y ] ≤ 2( L − s ) 2 | G | ≤ 0 . 0001 . Averaging o ver j pro duces Pr X,j,Y [ X and Y ar e not consistent | Y 0 → j = X 0 → j ] ≤ 0 . 000 1 . T o co mp lete the pro of, observe that Pr X,j,Y [ x L = y L | Y 0 → j = X 0 → j ] ≤ δ + 1 | G | ≤ 2 | G | . since y L is δ -close to uniform given x L . By Marko v’s inequality , w e obtain: Corollary 1. L et X b e dr awn fr om D x 0 ,L . Then Pr X [ X is 0 . 9 - c onsistent ] ≥ 1 − 0 . 0001 + 2 / | G | 0 . 1 = 0 . 999 − 20 | G | . W e now turn our attent ion to bo unding the hitting pro babilit y when a snake flicks its ta il. F ollowing Aaronson, we in tro duce a notion of ǫ -sp arseness for snakes and show tha t (i) if a snake is ǫ -sparse then it is O ( ǫ )-hitting, and that (ii) most snakes a r e ǫ - sparse. F or mally , w e define: 5 Definition 2. F or e ach x ∈ G , let P ( x ) = P r g ∈ D s [ x ∈ S ( g )] . A snake X dr awn fr om D x 0 ,L is c al le d ǫ -sparse if for al l vertex v ∈ G , ℓ X k =1 P ( x − 1 sk v ) ≤ ǫℓ . Int uitively , the spar seness of a snake means that if the sna k e flicks a random ch unk, it is unlikely to hit any fixed vertex. Prop osition 3. F or ǫ ≥ 2( L − s ) | G | , if snake X is ǫ - s p arse then X is 2 ǫ - h itting. Pr o of. Fix a s nak e X , a nd fix j ∈ { s, 2 s . . . , ℓs } . Let Y b e drawn from D x 0 ,L conditioned on the e vent that Y 0 → j = X 0 → j . Since y t is δ -close to uniform for all t ≥ j + s , Pr Y [ v ∈ Y j + s → L | Y 0 → j = X 0 → j ] ≤ ( L − s )( δ + 1 | G | ) ≤ 2( L − s ) | G | . On the other hand, Pr Y [ v ∈ Y j +1 → j + s | Y 0 → j = X 0 → j ] = P r g ∈ D s [ v ∈ x j S ( g )] = P ( x − 1 j v ) . Hence, Pr j,Y [ v ∈ Y j +1 → L | Y 0 → j = X 0 → j ] ≤ 1 ℓ ℓ X k =1 P ( x − 1 sk v ) + 2( L − s ) | G | ≤ 2 ǫ . It remains to show that a snake dr a wn from D x 0 ,L is ǫ -spars e with high proba bilit y . Firstly , we consider for the “ideal” case in which the endpoints of the ch unks in a sna k e are independently uniform. Lemma 1. L et u 1 , . . . , u ℓ b e indep endently and uniformly r andom vertic es in G . If s | G | ≤ ǫ 2 / 6 then Pr u 1 ,...,u ℓ " ℓ X i =1 P ( u i ) > 2 ℓ ǫ # ≤ 2 − ℓǫ . Pr o of. W e will use a Cherno ff bound to show that ther e are very few u i ’s for which P ( u i ) is larg e. T o do this, we fir st nee d a n upp er b ound on the exp ectation of P ( u i ). Let u b e a uniformly random vertex in G . F o r any given g ∈ G , we hav e Pr u [ u ∈ S ( g )] = ∆ (1 ,g ) | G | ≤ s | G | . Averaging over g ∈ D s yields P r g,u [ u ∈ S ( g )] ≤ s | G | , where g is chosen fro m D s independently to u . Since E u [ P ( u )] = Pr u,g [ u ∈ S ( g )], we hav e E u [ P ( u )] ≤ s | G | . Let Z = | { i : P ( u i ) ≥ ǫ } | . By Mar k ov’s inequality , E [ Z ] = ℓ Pr u [ P ( u ) ≥ ǫ ] ≤ ℓ E u [ P ( u )] ǫ ≤ ℓs | G | ǫ = µ . By a Chernoff b ound, for any λ ≥ 2 e Pr u [ Z ≥ λµ ] ≤  e λ − 1 λ λ  µ =  e λ  λµ e − µ ≤ 2 − λµ − µ . Note that if Z < λµ then ℓ X i =1 P ( u i ) ≤ ( ℓ − Z ) ǫ + Z ≤ ℓǫ + λµ . Setting λµ = ℓ ǫ , which satisfies λ ≥ 2 e due to the ass um ption that s | G | ≤ ǫ 2 / 6, we have Pr u 1 ,...,u ℓ " ℓ X i =1 P ( u i ) > 2 ℓ ǫ # ≤ Pr u 1 ,...,u ℓ [ Z ≥ ℓ ǫ ] ≤ 2 − ℓǫ . 6 In or der to apply this to our scenar io without strict indep endence, we r ecord the following fact ab out distance in total v ar iation. Prop osition 4. L et X 1 , . . . , X n and Y 1 , . . . , Y n b e discr ete r andom va riables so that X i and Y i have the same value r ange. L et ( X i | A 1 , . . . , A i − 1 ) denote the distribution of X i given that X 1 ∈ A 1 , . . . , X i − 1 ∈ A i − 1 ; similarly let ( Y i | A 1 , . . . , A i − 1 ) denote the distribution of Y i given that Y 1 ∈ A 1 , . . . , Y i − 1 ∈ A i − 1 . Then k ( X 1 , . . . , X n ) − ( Y 1 , . . . , Y n ) k t.v. ≤ k X 1 − Y 1 k t.v. + n X i =2 ∆ i wher e ∆ i = max A 1 ,...,A i − 1 k ( X i | A 1 , . . . , A i − 1 ) − ( Y i | A 1 , . . . , A i − 1 ) k t.v. . A detailed pro of of Prop osition 4 can be found in the a ppendix. Lemma 2. Su p p ose s | G | ≤ ǫ 2 / 6 . Then a snake X dr awn fr om D x 0 ,L is 2 ǫ -sp arse with pr ob ability at le ast 1 − | G | 2 − ℓǫ − 1 / 2 000 . Pr o of. The pro of for the lemma follows immediately by observing tha t for any vertex v , the v ar ia bles x − 1 s v , . . . , x − 1 sℓ v sa t isfy that x − 1 s ( k +1) v is δ -close to uniform given x − 1 sk v . By Prop osition 4 ,      Pr X " ℓ X k =1 P ( x − 1 sk v ) > 2 ℓǫ # − Pr u 1 ,...,u ℓ " ℓ X i =1 P ( u i ) > 2 ℓ ǫ #      ≤ ℓδ ≤ 1 2000 | G | . F ro m Lemma 1, Pr X " ℓ X k =1 P ( x − 1 sk v ) > 2 ℓǫ # ≤ 2 − ℓǫ + 1 2000 | G | . Summing up ov er v ∈ G gives Pr X [ X is not 2 ǫ -spa rse ] ≤ | G | 2 − ℓǫ + 1 / 2 000. W e nee d to choose ǫ such that | G | 2 − ℓǫ ≤ 1 / 2000 , or ǫ ≥ log | G | + O (1) ℓ . Corollary 2. A snake X dr awn fr om D x 0 ,L is O  s log | G | √ | G |  -hitting with pr ob ability at le ast 0 . 99 9 . Putting all the pieces together and applying Theo rem 2 , we ha ve Theorem 3. F or s = O ( p | G | ) , if Cayley gr aph C ( G, Γ ) is s - mixing, then RLS ( C ( G, Γ )) = Ω p | G | s lo g | G | ! , QLS ( C ( G, Γ )) = Ω 4 p | G | p s lo g | G | ! . In particular, any Cayley gra ph C ( G, Γ ) o f diameter d ha s RLS ( C ( G, Γ )) = Ω p | G | d log | G | ! , QLS ( C ( G, Γ )) = Ω 4 p | G | p d log | G | ! . F or compar ison, applying Aldous’s ra ndomized upper bo und [3] and Aar onson’s quantum upp er b ound [2] for arbitra r y Cayley graph C ( G, Γ ), we have RLS ( C ( G, Γ )) = O  p | G || Γ |  and QLS ( C ( G, Γ )) = O  3 p | G | 6 p | Γ |  . F or example, for constant degree e xpanding Cayley graphs, this ra ndomized lower bo und is tight to within O (log 2 | G | ) of Aldous’s upp er b ound. 7 R andom Cayley gr aphs. In fact, it can b e show ed that mos t Cayley graphs ar e s -mixing fo r s = Ω (log | G | ). Let g 1 , . . . , g s be a sequence of gro up elements. F ollowing [5], we call an element of the form g a 1 1 · · · g a s s , wher e a i ∈ { 0 , 1 } , a subpr o duct o f the s equence g 1 , . . . , g s . A ra ndom subpr o duct of this sequence is a subpro duct obtained by indep enden tly choo sing a i as a fair co in flip. A s e quence g 1 , . . . , g s is called a se quenc e of δ - uniform Er d¨ os-R´ enyi (E-R) gener ators if its ra ndom subpro ductors are δ -uniformly distributed ov er G in the sense that (1 − δ ) 1 | G | ≤ Pr a 1 ,...,a s [ g a 1 1 · · · g a s s = g ] ≤ (1 + δ ) 1 | G | for all g ∈ G . Theorem 4. (Er d¨ os and R´ enyi, S e e also [5]) F or s ≥ 2 lo g | G | + 2 log (1 /δ ) + λ , a se quenc e of s r andom elements of G is a se quenc e of δ -uniform E-R gener ators with pr ob ability at le ast 1 − 2 − λ . Clearly , any Cayley gra ph determined by an s - length seq uence of δ -uniform E-R gener a tors is s -mixing. So applying our lower b ounds for arbitrar y Cayley g raphs and the E-R theor em, we ha ve Prop osition 5. L et s ≥ 5 lo g | G | − 2 lo g s + λ . With pr ob ability at le ast 1 − 2 − λ , a r andom Cayley gr aph C ( G, Γ ) determine d by a se quenc e of s r andom gr oup elements has O ( p | G | s ) ≥ RLS ( C ( G, Γ )) ≥ Ω p | G | s lo g | G | ! and O ( 3 p | G | 6 √ s ) ≥ QLS ( C ( G, Γ )) ≥ Ω 4 p | G | p s lo g | G | ! . 4.2 Extending to V ertex-T ransitive Graphs Our a pproac h ab ov e for Cayley graphs ca n b e easily extended to vertex-transitive g raphs. W e shall describe here how to define a snake distribution D x 0 ,L similar to that for a Cayley graph. Co nsider a vertex-transitive graph G = ( V , E ) with N = | V | , a nd let d b e the dia meter of G . W e fix an ar bitrary vertex v 0 ∈ V . F or each vertex v ∈ V , we a lso fix a n extended shortest path S ( v ) = ( v 1 , . . . , v d ) of length d fro m v 0 to v . ( v 0 is omitted in S ( v ) for technical rea s ons.) That is, ( v 0 , . . . , v r ) is the actual shor test path from v 0 to v , where r = ∆ ( v 0 , v ), a nd v i = v fo r all i ≥ r . Since the automo rphism gr oup of G acts transitively on V , we can fix an automorphism σ x , for e ac h x ∈ V , so tha t σ x ( v 0 ) = x . Hence, for any x, v ∈ V , the sequence σ x S ( v ) is the extended shortest path from x to σ x ( v ). So now we can determine the k th ch unk o f a s nak e as the sequence σ x dk S ( u k ), wher e x dk is the endp oin t of the ( k − 1)th ch unk of the snak e, and u k is an indep enden tly and unifor mly ra ndom vertex. Le t P ( x ) = P r u [ x ∈ S ( u )], where u is c hosen fr om V uniformly at random. The co ndit ion for a snake X = ( x 0 , . . . , x ( ℓ +1) d ) to b e ǫ -sp arse is now rede fined as ℓ X k =1 P  σ − 1 x dk ( v )  ≤ ℓǫ for all v ∈ V . Observe that, given x dk , the endp oin t x dk + k = σ x dk ( u k ) of the k th ch unk is a unifor mly r andom vertex, since σ x dk is a bijectiv e. Also clearly , σ x 6 = σ y for any x 6 = y b ecause σ x and σ y send v 0 to differen t places. This means there is a one-to- one corresp ondence x ↔ σ x betw een V a nd the s e t of automorphisms { σ x : x ∈ V } . Therefore, if x is uniformly distributed ov er V , then s o is the vertex at any given p osition in σ x S , for any sequence S of vertices. It follo ws that in o ur snake X = ( x 0 , . . . , x L ), for all t ≥ k , x dk + t is unifor mly distributed ov er V given x dk . With this snak e distribution, we can similarly follow the proo f for Cayley graphs to prove the low er b ounds for v ertex- transitiv e graphs as given in Theo rem 1 . Ac kno wledgem en ts W e gratefully a c knowledge Scott Aar onson for discussing his previous work with us and showing us the trick for removing the requirement of snake non-s e lf -intersection. W e would lik e to thank anonymous referees for many helpful co mmen ts . 8 References [1] K a ren Aardal, Stan van Hoesel, Jan Karel Lenstra, and Leen Stougie. A decade of combinatori al optimization. In CW I T r acts 122 , pages 5–14. 1997. [2] S co tt Aaronson. Low er b ounds for local searc h by quantum arguments. In STOC ’04: Pr o c e e dings of the 36th Ann ual ACM Symp osium on The ory of Computing , 2004. [3] D a v i d Aldous. Minimization algori thms and random walk on the d -cub e. Annals of Pr ob abili t y , 11(2):403–41 3, 1983. [4] A ndris Ambainis. Quantum low er b o un ds by qu an tum arguments. In STOC ’ 00: Pr o c e e dings of the thirty-se c ond annual ACM symp osium on The ory of c omputing , 2000. [5] L´ aszl´ o Babai. Lo cal expansion of vertex-transitive graphs and rand om generation in finite groups. In STOC ’91: Pr o c e e dings of the twenty-thir d annual ACM symp osium on The ory of c omputing , pages 164–174, New Y ork, NY, U SA, 1991. ACM. [6] D o nn a Crystal Llew ellyn, Craig T ov ey , and Mic hael T rick. Lo cal optimization on graphs. Discr ete Appl. Math. , 23(2):157– 178, 1989. [7] Bo jan Mohar and Cartsen Thomasse n. Gr aphs on Surfac es . The Johns Hopkins U niv ersity Press, 2001. [8] Miklos Santha and Mario Szegedy . Quantum and classical query complexities of lo cal searc h are p olynomiall y related. In STOC ’04: Pr o c e e di ng s of the thirty-sixth annual ACM symp osium on The ory of c omputing , pages 494–501 , N ew Y ork, NY , U SA, 2004. A CM. [9] X ia oming Su n and Andrew C. Y ao. On the quantum query complexity of lo cal searc h in tw o and three dimensions. In F OCS ’06: Pr o c e e dings of the 47th Annual IEEE Symp osium on F oundations of Computer Scienc e , pages 429– 438, W ashington, DC, U SA, 2006. IEEE Computer S o ciet y . [10] Y ves F. V erhoeven. Enh a nced algorithms for local searc h. Inf. Pr o c ess. L ett. , 97(5):171–176, 2006. [11] Sh engyu Zhang. New up per an d lo wer b ounds for randomized and quantum lo cal search. In STOC ’06: Pr o c e e dings of the thirty-eighth annual ACM symp osium on The ory of c omputing , pages 634–643, New Y ork, NY, U SA, 2006. ACM. 9 A App endix A.1 Quan tum and R elational Adv ersary Metho ds The quantum a dv ersar y metho d [4] is a p o werful to ol underlying many pro ofs o f quantum low er bo unds. The clas s ical counterpart applied ab o ve is the rela tional a dv ersary metho d [2]. The central int uition of these adversary metho ds is to make it ha rd to distinguish “r elated” input sets. T echnically , co nsider tw o input sets A and B for a function F : I n → [ m ] so that F ( A ) 6 = F ( B ) for all A ∈ A and B ∈ B . Her e , an input to function F is a black-b o x function A : [ n ] → I . The or acle for a n input A answers q ueries of the form A ( x ) = ?. If A and B a re the t wo inputs that have the same v alue a t every queryable lo cation, then w e m ust hav e F ( A ) = F ( B ). Define a “ relation” function R ( A, B ) ≥ 0 on A × B . Two inputs A and B ar e said to b e related if R ( A, B ) > 0. Then, for A ∈ A , B ∈ B , and a queryable lo cation x ∈ [ n ], le t M ( A ) = P B ′ ∈B R ( A, B ′ ) , M ( B ) = P A ′ ∈A R ( A ′ , B ) M ( A, x ) = P B ′ ∈B : A ( x ) 6 = B ′ ( x ) R ( A, B ′ ) , M ( B , x ) = P A ′ ∈A : A ′ ( x ) 6 = B ( x ) R ( A ′ , B ) . Int uitively , the fraction M ( A, x ) / M ( A ) (resp. M ( B , x ) / M ( B )) mea sures how hard it is to distinguish input A (resp. B ) with rela ted inputs in B (resp. A ) by queying at lo cation x . F or mally , if there are such input sets A , B and r e lation function R ( A, B ), then Theorem 5. (Amb ainis) The numb er of quantu m queries ne e de d t o evaluate F with pr ob ability at le ast 0 . 9 is Ω ( M geom ) , wher e M geom = min A ∈A ,B ∈B ,x R ( A,B ) > 0 ,A ( x ) 6 = B ( x ) s M ( A ) M ( A, x ) M ( B ) M ( B , x ) . Theorem 6. (Aa r onson) The numb er of r andomize d queries ne e de d t o evaluate F with pr ob ability at le ast 0 . 9 is Ω ( M max ) , wher e M max = min A ∈A ,B ∈B ,x R ( A,B ) > 0 ,A ( x ) 6 = B ( x ) max  M ( A ) M ( A, x ) , M ( B ) M ( B , x )  . A.2 Pro ofs Con tinued pro of fo r Aaronson’s theorem (Theorem 2) Pr o of. T o apply the quantum and relational adversary metho d for the decision problem, define the input sets A = { ( X , g X, 0 ) : X ∈ D ∗ } a nd B = { ( Y , g Y , 1 ) : Y ∈ D ∗ } , wher e D ∗ denotes the set of ǫ -go od snakes drawn fr om D x 0 ,L . F o r s imp licity , we write A X as ( X , g X, 0 ), and B Y as ( Y , g Y , 1 ). F o r A X ∈ A a nd B Y ∈ B , define relation function R ( A X , B Y ) = w ( X, Y ) if X and Y are c onsisten t and x L 6 = y L , and R ( A X , B Y ) = 0 otherwise, wher e w ( X , Y ) is determined as fo llows. Let p ( X ) be the probability of dr a wing snake X fro m D x 0 ,L , and let w ( X , Y ) = p ( X ) Pr j,Z [ Z = Y | Z 0 → j = X 0 → j ] . Claim. F or any snakes X , Y ∈ D x 0 ,L , w e hav e w ( X , Y ) = w ( Y , X ). Pr o of. (of the claim) Fix j ∈ { 0 , . . . , L − 1 } and let q j ( X, Y ) = Pr Z [ Z = Y | Z 0 → j = X 0 → j ]. W e wan t to show p ( X ) q j ( X, Y ) = p ( Y ) q j ( Y , X ) . Assume X 0 → j = Y 0 → j , otherwise q j ( X, Y ) = q j ( Y , X ) = 0. Then letting Z be drawn from D x 0 ,L and le t E be the even t Z 0 → j = X 0 → j = Y 0 → j , we have p ( X ) q j ( X, Y ) = Pr Z [ E ] · Pr Z [ Z j +1 → L = X j +1 → L | E ] · P r Z [ Z j +1 → L = Y j +1 → L | E ] = Pr Z [ E ] · Pr Z [ Z j +1 → L = Y j +1 → L | E ] · Pr Z [ Z j +1 → L = X j +1 → L | E ] = p ( Y ) q j ( Y , X ) . 10 completing the pro of for the claim. As in Aaronson’s orig inal pro of, we won’t be able to tak e the whole input sets A and B defined above b ecause of the fact that not all sna k es ar e g oo d. Instead, we will take only a subset of each of these input sets that would b e hard enough to distinguish. This is done by apply ing Lemma 8 in [2], which states a s fo llo ws. Lemma 3. L et p (1) , . . . , p ( m ) b e p ositive r e als such t h at P i p ( i ) ≤ 1 . L et R ( i , j ) , for i, j ∈ { 1 , . . . , m } , b e nonn e gative r e als satisfying R ( i, j ) = R ( j, i ) and P i,j R ( i, j ) ≥ r . Then t h er e exists a nonempty su bse t U ∈ { 1 , . . . , m } such t h at P j ∈ U R ( i, j ) ≥ r p ( i ) / 2 for al l i ∈ U . T o apply this le mm a, we need a low er b ound for the sum P X,Y ∈D ∗ R ( A X , B Y ). Let E ( X , Y ) deno te the even t that snakes X and Y ar e co ns isten t and x L 6 = y L . F o r a n y X ∈ D ∗ , we have X Y : E ( X,Y ) w ( X , Y ) = p ( X ) Pr j,Y [ E ( X , Y ) | Y 0 → j = X 0 → j ] ≥ 0 . 9 p ( X ) . Hence, since a snake drawn from D x 0 ,L is go od with pro babilit y at least 0 . 9 , X X,Y : E ( X,Y ) w ( X , Y ) ≥ 0 . 9 X X ∈D ∗ p ( X ) ≥ 0 . 9 × 0 . 9 ≥ 0 . 8 . By the union b ound, X X,Y ∈D ∗ R ( A X , B Y ) ≥ X X,Y : E ( X,Y ) w ( X , Y ) − X X 6∈D ∗ p ( X ) − X Y 6∈D ∗ p ( Y ) ≥ 0 . 8 − 0 . 1 − 0 . 1 = 0 . 6 . So, by Lemma 3, there exists a nonempty subset e D ⊂ D ∗ so that for all X , Y ∈ e D , X Y ′ ∈ e D R ( A X , B Y ′ ) ≥ 0 . 3 p ( X ) , X X ′ ∈ e D R ( A X ′ , B Y ) ≥ 0 . 3 p ( Y ) . So now we take the input sets e A = n A X : X ∈ e D o and e B = n B Y : Y ∈ e D o . W e have shown that M ( A X ) ≥ 0 . 3 p ( X ) a nd M ( B Y ) ≥ 0 . 3 p ( Y ) for any A X ∈ e A and B Y ∈ e B . Since the s nak e par t in the inputs can not be queried, w e only ca re ab out the measure for distinguishing A X , B Y with their rela ted inputs b y q uerying the function part (i.e. g X, 0 or g Y , 1 ) in the inputs. F ormally , w e fo cus on low er-b ounding M ( A X , v ) a nd M ( B Y , v ) for inputs A X ∈ e A , B Y ∈ e B for which R ( A X , B Y ) > 0 and g X, 0 ( v ) 6 = g Y , 1 ( v ). W e remark that since R ( A X , B Y ) > 0, the event E ( X , Y ) m ust ho ld, which implies that for a ll vertex v , g X, 0 ( v ) 6 = g Y , 1 ( v ) ⇐ ⇒ f X ( v ) 6 = f Y ( v ) ⇐ ⇒ set X ( v ) 6 = set Y ( v ) . Applying the quantum and r andomized a dv ersary method, we will hav e RLS ( G ) ≥ Ω ( M max ) and QLS ( G ) ≥ Ω ( M geom ), where M max = min A X ∈ e A ,B Y ∈ e B ,v R ( A X ,B Y ) > 0 ,set X ( v ) 6 = set Y ( v ) max  M ( A X ) M ( A X , v ) , M ( B Y ) M ( B Y , v )  M geom = min A X ∈ e A ,B Y ∈ e B ,v R ( A X ,B Y ) > 0 ,set X ( v ) 6 = set Y ( v ) s M ( A X ) M ( A X , v ) M ( B Y ) M ( B Y , v ) . 11 Let A X ∈ e A , B Y ∈ e B be inputs for which set X ( v ) 6 = s e t Y ( v ) . Then v 6∈ X o r v 6∈ Y . Assuming the ca se v 6∈ X , w e will show M ( A X , v ) is small. W e have M ( A X , v ) ≤ X Y ′ ∈ e D : set X ( v ) 6 = set Y ′ ( v ) w ( X , Y ′ ) ≤ X Y ′ : v ∈ Y ′ p ( X ) Pr j,Z [ Z = Y ′ | Z 0 → j = X 0 → j ] = p ( X ) Pr j,Z [ v ∈ Z | Z 0 → j = X 0 → j ] = p ( X ) Pr j,Z [ v ∈ Z j +1 → L | Z 0 → j = X 0 → j ] (sinc e v 6∈ X ) ≤ p ( X ) ǫ (since X is ǫ -hitting) . In the case v 6∈ Y , we ca n also obtain M ( B Y , v ) ≤ p ( Y ) ǫ due to symmetry . Hence, max  M ( A X ) M ( A X , v ) , M ( B Y ) M ( B Y , v )  ≥ 0 . 3 /ǫ s M ( A X ) M ( A X , v ) M ( B Y ) M ( B Y , v ) ≥ p 0 . 3 /ǫ . The latter inequa lit y is obtained due to the fact that M ( A X , v ) ≤ M ( A X ) and M ( B Y , v ) ≤ M ( B Y ). Consequently , M max = Ω (1 /ǫ ) and M geom = Ω ( p 1 /ǫ ), completing the pr oof for Theore m 2. Pro of of Prop osition 4 Pr o of. W e prove b y induction on n . T he ca se n = 2 can be ea sily obtained by applying the following simple fact: F act 1 Le t x 1 , x 2 , y 1 , y 2 b e any r e al nu mb ers in [0 , 1] . Then | x 1 x 2 − y 1 y 2 | = | ( x 1 − y 1 ) x 2 + ( x 2 − y 2 ) y 1 | ≤ | x 1 − y 1 | x 2 + | x 2 − y 2 | y 1 ≤ | x 1 − y 1 | + | x 2 − y 2 | . In particular , applying the a b ov e fact, w e have for any pair of even ts ( A, B ),    Pr[ X 1 ∈ A, X 2 ∈ B ] − Pr[ Y 1 ∈ A, Y 2 ∈ B ]    ≤    Pr[ X 1 ∈ A ] − Pr[ Y 1 ∈ A ]    +    Pr[ X 2 ∈ B | X 1 ∈ A ] − Pr[ Y 2 ∈ B | Y 1 ∈ A ]    . Recall that by definition of total v a riation, k ( X 1 , X 2 ) − ( Y 1 , Y 2 ) k t.v. = max A,B    Pr[ X 1 ∈ A, X 2 ∈ B ] − Pr[ Y 1 ∈ A, Y 2 ∈ B ]    and ∆ 2 = max A,B    Pr[ X 2 ∈ B | X 1 ∈ A ] − Pr[ Y 2 ∈ B | Y 1 ∈ A ]    . Hence, k ( X 1 , X 2 ) − ( Y 1 , Y 2 ) k t.v. ≤ k X 1 − Y 1 k t.v. + ∆ 2 . Now we can apply this re s ult and get k ( X 1 , . . . , X n ) − ( Y 1 , . . . , Y n ) k t.v. ≤ k ( X 1 , . . . , X n − 1 ) − ( Y 1 , . . . , Y n − 1 ) k t.v. + ∆ n which establishes the prop osition b y induction. 12 A.3 Upp er Bou nds fo r Lo ca l Searc h V ar io us upper b ounds for b oth q ua n tum and classic a l query complexities have been given for genera l g raphs. F or a n y g raph G of N vertices and ma ximal degree δ , it has b een show ed that RLS ( G ) = O ( √ N δ ) [3] a nd QLS ( G ) = O ( N 1 / 3 δ 1 / 6 ) [2]. The idea for designing lo cal search algorithms in [3, 2] is r andom sampling follow ed by stee p est descent. More sp ecifically , these algo rithms star t o ff by sampling a subset of vertices, find the b est vertex v (i.e., the one with the minimum f v alue) in the sa mpled set, a nd finally p erforming steep est descen t b eginning at the chosen v ertex v . Zhang [11] later intro duced new quant um and ra ndomized algorithms for lo cal search on general gra phs , providing upper b ounds that depend on the graph diameter and the expansion s peed. While Zhang’s upp er bo unds can o nly w or k well for graphs with slow expansio n sp eed, such as h yp ecubes, many vertex-transitive graphs, unfortunately , do not po ssess this prop ert y . Also, Zhang’s r andomized upper b ound is no b e t ter tha n O  N d log log d  , and his quantum upp er b ound is no b etter tha n O  q N d (log log d ) 1 . 5  , exc e pt for the line or cycle gr aphs, where d is the diameter of the gr a ph. This means Zhang’s upp er b ounds do not s eem to bea t Aaronson and Aldous’s b ounds, especia lly for graphs with small degrees and small diameters . Note tha t there are Cayley gr aphs o f non- a belian simple gr o ups which have cons ta n t degr ees and hav e diameters no larger tha n O (log N ). While Zhang’s upp er b ounds fail for gr aphs of small dia meters, Aldous and Aarons o n’s upper b ounds fail for gr aphs of lar g e degr ees. So, a question to ask is whether there is a b etter upper bo und for graphs with lar ge deg rees and small dia meter s? Recently , V erho even [10] has pro posed a nother deterministic a lgorithm and enha nced Zhang’s quantum algorithm, improving upp er b ounds on determinis tic and qua ntum query c o mplexities of Lo cal Search that depe nd on the graph’s degr ees and genus. Pr ecisely , he show ed that for any N -vertex gr aph G of genus g and ma ximal degree δ , the deterministic (thus, ra ndomized) and query complexities o f Lo cal Search on G are δ + O ( √ g ) √ N a nd O ( √ δ ) + O ( 4 √ g ) 4 √ N log log N , resp ectively . How ever, these b ounds fail for the class of g r aphs we are c aring ab out: vertex-transitive gr aphs, since every vertex-transitive graph is r e g ular and it has b een shown that the ge nus o f an N -vertex m - e gde connected gr aph is at least  m 6 − N 2 + 1  (see [7, p114]). 13