PRODUCT RULE WINS A COMPETITIVE GAME

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1 PROCEEDIGS OF THE AMERICA MATHEMATICAL SOCIETY Volume 00, umber 0, Pages S XX PRODUCT RULE WIS A COMPETITIVE GAME ADREW BEVERIDGE, TOM BOHMA, ALA FRIEZE, AD OLEG PIKHURKO Abstract We cosider a game that ca be viewed as a radom graph process The game has two players ad begis with the empty graph o vertex set [] Durig each tur a pair of radom edges is geerated ad oe of the players chooses oe of these edges to be a edge i the graph Thus the players guide the evolutio of the graph as the game is played Oe player cotrols the eve rouds with the goal of creatig a so-called giat compoet as quickly as possible The other player cotrols the odd rouds ad has the goal of keepig the giat from formig for as log as possible We show that the product rule is a asymptotically optimal strategy for both players 1 Itroductio Cosider the followig radom graph game There are two players, Creator ad Destroyer, ad the game is played o vertex set [] The game begis with the empty graph, which we deote G 0 The game is played i a sequece of turs that alterate betwee Creator ad Destroyer Durig tur i a pair of radom edges e i, f i is geerated ad the player who cotrols roud i chooses oe of these edges to be a edge i the graph, addig it to G i 1 to form the graph G i We defie a giat compoet to be ay coected compoet with at least / log vertices A giat compoet is usually defied as a compoet with Ω vertices We choose a threshold of / log here i order to defie the game without referece to asymptotics i Whe we state our formal results, we will take, ad it will tur out that our choice here is somewhat arbitrary Creator s goal is to achieve the formatio of a giat compoet i G i as quickly as possible while Destroyer s goal is to keep a giat from formig for as log as possible Oe way to measure the success of the players is to compare with the phase trasitio i the Erdős-Réyi radom graph; that is, we might say that if G c has a giat where c is a costat less tha 1/2 the Creator wis ad if G c does ot have a giat where is a c is a costat greater tha 1/2 the Destroyer wis If we fix the strategies for the two players the this game gives us a example of a Achlioptas process, a radom graph process i which we choose oe edge from each pair i a sequece e 1, f 1 ; e 2, f 2 ; of pairs of radom edges where the choices are determied by some fixed algorithm ad are made o-lie For discussio of Achlioptas processes ad results o the timig of the emergece of the giat compoet i these ad related models see [1], [2], [3], [4], [5] ad [7] Tom Bohma supported i part by SF grat DMS Ala Frieze supported i part by SF grat CCR Oleg Pikhurko supported i part by SF grat DMS c 1997 America Mathematical Society

2 2 BEVERIDGE, BOHMA, FRIEZE, AD PIKHURKO Let G be a fixed graph with coected compoets C 1,, C r We defie the susceptibility of G to be XG = 1 r C i 2 where C i is the umber of vertices i compoet C i ote that the susceptibility gives the expected umber vertices i the coected compoet cotaiig a vertex chose uiformly at radom This parameter plays a importat role i percolatio theory Motivated i part by this coectio, Specer itroduced the product rule, which is the Achlioptas process that chooses the edge that miimizes the chage i the susceptibility [6] ote that the chage i the susceptibility whe we add to G a edge e that coects distict compoets C i ad C j is 2 C i C j / Thus to miimize the icrease i the susceptibility we choose the edge from the pair e i, f i that miimizes the product of the sizes of the compoets joied This together with a arbitrary covetio for breakig ties ad dealig with edges that fall withi a sigle compoet defies the product rule It is believed that the product rule is a early optimal Achlioptas process with respect to delayig the appearace of the giat compoet for as log as possible for discussio of this issue ad some examples of Achlioptas processes that appear to do slightly better tha the product rule see [7] Very little is kow about the graph evolutio give by the product rule; for example, the timig ad ature of the appearace of a giat compoet i the product rule are itriguig ope issues I order to state our results o the game defied above, we must make a distictio betwee the product rule that maximizes the chage i the susceptibility ad the product rule as defied above, which miimizes the chage i susceptibility Suppose e i, f i is a pair of radom edges ad G i 1 is a fixed graph Suppose further that e i coects compoets i G i 1 cosistig of a ad b vertices, respectively, while f i coects compoets cosistig of c ad d vertices, respectively Destroyer s product rule is defied by { e i if ab < cd 11 Destroyer s product rule chooses if ab > cd ote that Destroyer s product rule is idetical to the product rule defied above Aalogously, we defie Creator s product rule by stipulatig that { f i if ab < cd 12 Creator s product rule chooses if ab > cd For both of these rules ties ie ab = cd ad edges that fall withi a sigle compoet are hadled arbitrarily We assume throughout that edges are chose uiformly ad idepedetly at radom from [] 2 Of course, this covetio will lead to the appearace of loops ad multiple edges i our process As the umber of these is bouded by, say, log whp, they ca be removed from the process without alterig our results Theorem 11 Let G i be the graph produced through i rouds of our competitive game a If Destroyer employs Destroyer s product rule ad c < 1/2 the whp the largest compoet of G c has Olog vertices e i f i

3 PRODUCT RULE WIS 3 b If Creator employs Creator s product rule ad c > 1/2 the whp the largest compoet of G c has Ω vertices We prove this theorem by applyig techiques developed by Specer ad Wormald i a proof that the evolutio of the susceptibility determies the timig of the phase trasitio for a certai atural class of Achlioptas processes This result was idepedetly proved by Bohma ad Kravitz [4], ad the methods developed i [4] ca also be used to prove Theorem 11 However, the boud o the compoet sizes i part a would ot be as strog, so we opt for the methods of Specer ad Wormald here Suppose we measure success i our competitive game relative to the Erdős-Réyi phase trasitio Our methods ca be used to show ot oly that the product rule does ot lose the game as we saw i Theorem 11 but also that the product rule wis whe the other player uses a strategy that varies sigificatly from the product rule Theorem 12 Let G i be the graph produced through i rouds of our competitive game Let the radom variables A i ad B i be the umber of times Creator ad Destroyer, respectively, do ot follow their respective product rules durig the first i turs of the game Let ǫ > 0 be a costat a If Destroyer employs Destroyer s product rule ad Creator uses a rule for which A /2 > ǫ whp the there exists a costat c > 1/2 such that whp the largest compoet of G c has Olog vertices b If Creator employs Creator s product rule ad Destroyer uses a rule for which B /2 > ǫ whp the there exists a costat c < 1/2 such that whp the largest compoet of G c has Ω vertices Theorem 12 implies, for example, that the phase trasitio for the product rule ie we apply Destroyer s product rule i every roud occurs after 1/2 + δ edges for some fixed δ > 0 Furthermore, Theorem 12 ca be used to aalyze the followig variatio o Achlioptas processes Suppose that there is oe player who simultaeously builds two graphs G ad H o the same vertex set As before, two radom edges arrive i each roud The player has to decide o-lie which edge is added to G, while the other edge goes to H The objective of the player is to cotrol the birth of the giat i both graphs; for example, to delay it i G ad to create it i H It follows from Theorem 12 that whp the player ca strictly beat the Erdős-Réyi threshold i both graphs I the example give, the player uses Destroyer s product rule with respect to G i odd rouds ad Creator s product rule with respect to H i eve rouds To see that mistakes are made by a imagiary oppoet i the formatio of both G ad H, cosider the set of vertices X that are isolated i both G ad i H ad the set of vertices Y that are isolated i either G or H Both X ad Y are a positive proportio of the vertices from, say, step /100 util step /2 Therefore, there will be a positive proportio of rouds whe e i lies i X ad f i lies i Y I such rouds the player is makig the correct choice i both graphs 2 Prelimiaries We begi with some backgroud take from [6] ad [7] The startig poit for our aalysis is a differetial equatio that describes the evolutio of the susceptibility i the Erdős-Réyi radom graph Suppose G is a fixed graph o vertex

4 4 BEVERIDGE, BOHMA, FRIEZE, AD PIKHURKO set [] with coected compoets C 1,, C r Let e be a radom edge, ad set G + = G + e If e coects compoets C i ad C j the the chage i the susceptibility is 2 C i C j / Thus 21 E[XG + XG] = C i C j 2 C i C j 2 i j r 2 = 2 C i 2 r C i = 2X2 G 2 r C i 4 3 Sice the sum of the fourth powers of the compoet sizes divided by 3 is egligible whe all compoets are small, we expect the susceptibility of the Erdős-Réyi radom graph G,c to be cocetrated aroud fc where f is the solutio to the differetial equatio f = 2f 2 f0 = 1 ote that fx = 1 1 2x This gives a ice explaatio for the fact that the giat compoet suddely appears i the Erdős-Réyi graph whe there are about /2 radom edges Ie at about /2 radom edges the susceptibility blows up ad this should reflect a dramatic chage i the compoet structure Of course, this is a formal proof of either the cocetratio of the susceptibility aroud this trajectory or the sudde emergece of the giat See Bohma ad Kravitz [4] or Specer ad Wormald [7] for proper developmets alog these lies A key piece of the Specer ad Wormald proof is the followig defiitio ad Theorem For a vertex v of graph G we let C v be the coected compoet of G that cotais v We say that G has a K, α compoet tail if 1 {v : C v s} Ke αs for all s ote that if G has a K, α compoet tail ad αα > 1 the max C v < α log v for sufficietly large The radom graph G,p is the radom graph o vertex set [] i which each of the 2 possible edges appears idepedetly with probability p Theorem 21 Specer, Wormald Let L, K, α > 0 be costats Let G be a graph o vertices with a K, α compoet tail Let H be the radom graph H = G,d/ where d is a fixed costat Set G + = G H a If dl < 1 ad XG < L the there exist K +, α + such that whp G + has a K +, α + compoet tail b If dl > 1 ad XG > L the whp G + has a compoet with Ω vertices 3 Proof of Theorem 11 Our key observatio is that if the product rule is employed by either player the the expected chage i the susceptibility i two turs of the game is bouded by the expected chage i the susceptibility of the Erdős-Réyi radom graph whe two radom edges are added

5 PRODUCT RULE WIS 5 Lemma 31 Let G be a fixed graph o vertex set [] with compoets C 1,,C r Let e 1, f 1, e 2, f 2 [] 2 be idepedet radom edges If G + = G + {a, b} where a {e 1, f 1 } is chose accordig to Destroyer s product rule ad b {e 2, f 2 } is chose arbitrarily the E[XG + ] XG + 4 X2 G X3 G Proof For each edge e [] 2 set e = 2 C 1 C 2 if e jois distict compoets C 1, C 2 of G ad set e = 0 if e lies withi a coected compoet For each positive iteger m let p m be the probability that e m where e [] 2 is a radom edge We have E[ a ] = Pr e1 m ad f1 m = p 2 m m 1 m 1 E[ b ] Pr e2 m or f2 m = 2pm p m 1 m 1 2 m ote that E[XG + XG] may be larger tha E[ a ] + E[ b ]/ as we must accout for the possibility that a ad b itersect a commo compoet of G Whe this happes the additioal cotributio to E[XG + XG] is 2 C a C b / where C a is the compoet of G that itersects a but ot b ad C b is the compoet of G that itersects b but ot a If we fix the compoets C a ad C b there are 4 vertices amog the edges e 1, f 1 that could fall i C a ad 4 vertices amog the edges e 2, f 2 that could fall i C b Therefore, apply 21 we have E[XG + XG] E[ a] + E[ b] + 16 [ C i i j k 1 = 2 4 m 1 m 1 1 C k 2 2 p 2 m + 2p m p 2 m p m + 32X3 G 2 r C i X3 G 2 k ] C j 2 C i C j C k 2 C i 2 C j 2 2 ote that we apply 21, droppig the sum of the fourth powers of the compoet sizes as we are establishig a upper boud i,j Lemma 32 Let G be a fixed graph o vertex set [] with compoets C 1,,C r Let e 1, f 1, e 2, f 2 [] 2 be idepedet radom edges If G + = G + {a, b} where a {e 1, f 1 } is chose accordig to Creator s product rule ad b {e 2, f 2 } is chose arbitrarily the E[XG + ] XG + 4 X2 G 4 r r 2 3 C i C i 3

6 6 BEVERIDGE, BOHMA, FRIEZE, AD PIKHURKO Proof We follow the proof of Lemma 31, otig that here we must accout for the possibility that edges a ad b joi the same pair of compoets from G The probability that oe of the edges e 1, f 1 jois compoets C i ad C j is at most 4 C i C j Therefore, we have 2 E[XG + XG] E[ a] + E[ b] 2 4 Ci C j 2 C i C j 2 i<j ow we apply 21 to achieve the stated boud, otig that here we caot drop the sum of fourth powers of the compoet sizes as we are establishig a lower boud We are ow ready to apply the so-called differetial equatios method 31 Destroyer Let c < 1/2 be a costat ad assume that Destroyer employs Destroyer s product rule We start by defiig some costats Let β be a costat such that 31 c + β < 1 2 Let η be a costat such that 32 4ηf c + β = 4η 1 2c + β < 1 Let 0 = c 0 < c 1 < c 2 < < c = c be costats that satisfy 33 c i+1 c i < η for i = 0,, 1 We view these c i s as ladmarks i the process, ad we show that whp at each of these ladmarks our graph is very well behaved To be precise, we will show that whp we have i G ci has a K i, α i compoet tail where K i ad α i are costats, ad ii XG ci f c i + iβ 1 = 1 2c i + iβ for i = 1,, To establish these coditios, we go by iductio o i Assume G ci satisfies i ad ii First, we establish i for G ci+1 Let H be the graph o vertex set [] with edge set {e k : c i < k c i+1 } {f k : c i < k c i+1 } Sice G ci satisfies ii, we have applyig 32 XG ci 4η 4ηf c i + iβ 4ηfc + β < 1 It the follows from part a of Theorem 21 that there exist costats K i+1, α i+1 such that whp G ci + G,4η/ has a K i+1, α i+1 compoet tail Sice, usig stadard techiques, the umber of edges i G,4η/ is cocetrated aroud 2η ad we have 33, it follows that G ci + H has a K i+1, α i+1 compoet tail whp Sice G ci+1 is cotaied i G ci + H, the graph G ci+1 also has a K i+1, α i+1 compoet tail whp The proof that G ci+1 satisfies ii is more delicate ad requires the assumptio that G ci+1 satisfies i whp as we have already established We apply the socalled differetial equatios method for radom graph process to the susceptibility

7 PRODUCT RULE WIS 7 for a excellet referece for this method see [8] Set K = c i+1 c i 2 For k = 0,,K set X k = XG ci +2k For k = 0,,K set We have Y 0 = X 0 f c i + Y k = X k f i + 1β c i + + 2k i + 1β f c i + iβ f c i + i + 1β < γ < 0 ote that this iequality defies γ Our aim is to show that {Y i } is a supermartigale with bouded differeces To achieve this, we apply the stadard techique of itroducig a stoppig time Let T be the smallest idex k such that G ci +2k has a compoet with more tha α log vertices where α α i+1 > 1, or Y k γ/2 Fially, let Z k = Y mi{k,t } Lemma 33 If is sufficietly large the the sequece {Z i } is a supermartigale Proof If k T the we determiistically have Z k+1 = Z k Suppose k < T It follows that Y k < γ/2 Let F k be the filtratio of our probability space give by the set of edges geerated through the first k turs of the game Applyig Lemma 31 ad the covexity of f we have E[Z k+1 Z k F k ] = E[Y k+1 Y k F k ] = E[X k+1 X k F k ] [ f i + 1β c i + 4 X2 k X3 k 2 f 4 [ i + 1β f c i + + 2k 2 f = 1 0 for sufficietly large + 2k + 2 f i + 1β c i + c i + [ γ X3 k 4γf Lemma 34 For k = 0,, K we have γ 2 i + 1β + 2k ] X3 k + 2k i + 1β c i + + 2k Z k+1 Z k 6α 2 log 2 c i + ] i + 1β + 2k Proof If k T the we trivially have Z k+1 = Z k If k < T the the largest compoets i G ci +2k has at most α log vertices The maximum chage i the susceptibility whe we add two edges to such a graph is 6α 2 log 2 / ]

8 8 BEVERIDGE, BOHMA, FRIEZE, AD PIKHURKO We have Pr XG ci+1 > f c i+1 + i + 1β Pr T K Pr Z K γ/2 + Pr G ci+1 does ot have K i+1, α i+1 compoet tail The secod probability here is o1 as we oted above We apply a Azuma- Hoeffdig type iequality see for example Lemma 42 of [8] to boud the first probability usig Lemmas 33 ad 34 To be precise, { γ 2 } Pr Z K Z 0 γ/2 exp 72α 4 log 4 Thus ii holds whp 32 Creator Here we follow the aalysis of the previous sectio to get to a poit where the susceptibility is large, ad the we apply Theorem 21, part b Let c > 1/2 be a costat Let ν = c 1/2 Choose τ such that 34 fτν e 8 4 > 1 Let β > 0 be a costat such that τ + β < 1/2 Choose η such that 35 4ηfτ + β < 1 Let 0 = c 0 < c 1 < c 2 < < c = τ + β be costats such that c i+1 c i η for i = 0,, 1 ow we defie a sequece of times i our process For i = 0,, let T i be the smallest value of k for which 36 XG k f c i iβ We show that whp we have i G Ti has a K i, α i compoet tail where K i ad α i are costats, ad ii T i c i for i = 1,, We go by iductio o i as i the previous subsectio Suppose T i satisfies coditios i ad ii ote that coditio i ad the defiitio of T i give i 36 imply X G Ti = f c i iβ + o1 We cosider the 2K = c i+1 c i turs that follow tur T i As there are less tha 2η radom edges geerated durig these turs, it follows from 35 ad part a of Theorem 21 that whp G Ti +2K has a K i+1, α i+1 compoet tail for some costats K i+1, α i+1 For k = 0,,K set i + 1β Y k = X G Ti +2k f c i + 2k ote that Y 0 > γ for a costat γ > 0 Let the stoppig time T be the smallest idex k such that G Ti +2k has a compoet with more tha α log vertices or Y k < γ/2

9 PRODUCT RULE WIS 9 This yields a submartigale with bouded differeces By a applicatio of the Azuma-Hoeffdig iequality, whp we have i + 1β X G Ti +2K > f c i+1 Coditios i ad ii o T i+1 follow ow suppose T satisfies coditios i ad ii The graph G T has a K, α compoet tail ad T / < c = τ + β We cosider the ν turs after tur T For Creator s tur k durig this iterval, the edge chose is a purely radom edge wheever e k is isolated This assertio would follow from the assumptio that Creator s product rule imposes the tie-breakig covetio that f k is chose whe e k is isolated Alterately, we could focus oly o rouds where oe edge is isolated ad the other is ot For the sake of brevity we omit this detail The umber of vertices that do ot appear i the set of edges {e k : 1 k } {f k : 1 k } is cocetrated aroud e 4 We coclude that durig the ν turs after tur T whp at least e 8 ν/4 purely radom edges are added to the graph G It follows from Theorem 21 part b ad 34 that G T +ν has a compoet with Ω vertices whp 4 Product Rule Wis Proof of Theorem 12 I Sectio 31 we saw that, uder the assumptio that Destroyer uses Destroyer s product rule, the susceptibility is bouded above by the trajectory f that describes the evolutio of the susceptibility for the Erdős-Réyi radom graph If we add the assumptio that Creator makes may mistakes relative to the product rule the we ca adapt our argumet to show that there are costats c < 1/2 ad δ > 0 such that whp the susceptibility of G c is bouded by fc δ Sice the argumet that bouds the progress of the susceptibility does ot deped o the umber of edges i the graph, ay groud that Creator loses relative to f caot be regaied This gives the proof of part a of Theorem 12 We ow sketch the details The proof of part b of Theorem 12 follows alog similar lies Let δ, β > 0 be costats such that f 1 2 ǫ 2 δ Let η > 0 be chose such that 4ηf = f 1 2 ǫ 2 + β ǫ ǫ 2 + β < 1 Let 0 = c 0 < c 1 < c 2 < < c = 1/2 ǫ/2 be costats that satisfy c i+1 c i < η for i = 0,, 1 We follow the argumet of Sectio 31 to show that the susceptibility is bouded at each of the ladmarks However, istead of workig with X k = X G ci +2k, we cosider X k = X G ci +2k + A c i +2k If we cosider two turs of the game ad coditio o the evet that Creator makes a mistake relative to the product rule the the expected chage i the susceptibility

10 10 BEVERIDGE, BOHMA, FRIEZE, AD PIKHURKO is bouded above by the expressio give i Lemma 31 mius 2/ It follows that we ca apply the argumet i Sectio 31 directly to X to show that we have X G ci < f c i + βi for i = 1,, So, we have X G /2 ǫ/2 = X Gc = X G c A c f c + β ǫ 2 1 = f 2 ǫ 2 + β ǫ2 12 = f ǫ2 δ ow let η > 0 satisfy 1 4η f 2 δ < 1 4 We cosider ladmarks c < c +1 < c +2 < < c M = 1/2 + δ/4 which satisfy c i+1 c i < η for i =,, M 1 Followig the machiery of Sectio 31 we see that G ci has a K i, α i compoet tail ad δi X G ci < f c i δ + 2M for i = + 1,,M This completes our sketch of the proof of part a of Theorem 12 Refereces [1] T Bohma ad A Frieze, Avoidig a giat compoet Radom Structures ad Algorithms , [2] T Bohma, A Frieze ad Wormald, Avoidace of a giat compoet i half the edge set of a radom graph Radom Structures ad Algorithms [3] T Bohma ad J H Kim, A phase trasitio for avoidig a giat compoet Radom Structures ad Algorithms [4] T Bohma ad D Kravitz, Creatig a giat compoet Combiatorics, Probability ad Computig [5] A Flaxma, D Gamarik ad G Sorki, Embracig the giat compoet pp i Lati 2004: Theoretical Iformatics, Farach ad Colti eds Spriger 2004 [6] J Specer, Percolatig thoughts Upublished mauscript dated Jauary 31, 2001 [7] J Specer ad Wormald, Birth Cotrol for Giats Combiatorica, to appear [8] Wormald, the differetial equatios method for radom graph processes ad greedy algorithms pp i Lectures o Approximatio ad Radomized Algorithms, Karoński ad Prömel eds PW, Warsaw 1999 Departmet of Mathematical Scieces, Caregie Mello Uiversity, Pittsburgh, PA Departmet of Mathematical Scieces, Caregie Mello Uiversity, Pittsburgh, PA Departmet of Mathematical Scieces, Caregie Mello Uiversity, Pittsburgh, PA Departmet of Mathematical Scieces, Caregie Mello Uiversity, Pittsburgh, PA 15213