Refined Runtime Analysis of a Basic Ant Colony Optimization Algorithm

Transcription

1 Refied Rutime Aalysis of a Basic At Coloy Optimizatio Algorithm Bejami Doerr ad Daiel Johase Max-Plack-Istitut für Iformatik Campus E1 4, Saarbrücke, Germay Abstract Neuma ad Witt (2006) aalyzed the rutime of the basic at coloy optimizatio (ACO) algorithm 1-At o pseudoboolea optimizatio problems For the problem OeMax they showed how the rutime depeds o the evaporatio factor I particular, they proved a phase trasitio from expoetial to polyomial rutime I this work, we simplify the view o this problem by a appropriate traslatio of the pheromoe model This results i a profoud simplificatio of the pheromoe update rule ad, by that, a refiemet of the results of Neuma ad Witt I particular, we show how the expoetial rutime boud gradually chages to a polyomial boud iside the phase of trasitio 1 Itroductio I 1991 Dorigo, Maiezzo ad Colori [7] itroduced the cocept of at coloy optimizatio (ACO) Sice the ACO algorithms have prove to be successful heuristics i practice ad have bee applied to a wide rage of combiatorial optimizatio problems, several of which are NP-hard The techique of ACO is based o a atural optimizatio process, amely the search of a at coloy for a shortest path to a source of food The ats commute o radom paths betwee their coloy ad the food source, leavig behid traces of pheromoes The ats preferably choose those paths with high pheromoe desity This behavior reiforces existig pheromoe trails oce they are sigificatly stroger tha the average trail Essetial for the covergece of this procedure to a optimal path is the fact that pheromoes evaporate over time Sice a at eeds more time to traverse a log path the a short oe, a larger amout of pheromoes evaporates durig the traversal of such a log path Together with the mechaism of reiforcemet this leads to a high cocetratio of pheromoes o short paths which evetually are chose almost exclusively Applied to combiatorial optimizatio problems we iterpret the radom paths chose by the ats as radom walks o a give costructio graph Such a walk represets a solutio to the optimizatio problem We assig a pheromoe value to every edge of the costructio graph This value determies the probability that a radom walk uses the edge This is the author s versio of the work It is posted here for your persoal use Not for redistributio The defiitive versio was published i Proceedigs of the 2007 IEEE Cogress o Evolutioary Computatio (CEC 2007), pages , IEEE It is available at alljsp?arumber= I each step of the algorithm a radom walk is geerated ad afterwards the pheromoe values are updated accordig to the quality of the walk as solutio to the give problem I additio to the problem-specific costructio graph we defie the evaporatio factor ρ as the mai parameter of the algorithm Durig the pheromoe update the pheromoe values of all edges are decreased by their ρ-th part (evaporatio) The, the pheromoe values of the edges i the ew walk are icreased by ρ (stregtheig) Optioally, the pheromoe values are ormalized afterwards For a evaporatio factor of zero, this meas the ew walk does ot cotribute to the updated pheromoe values at all, leavig them at their iitial assigmet throughout all steps of the algorithm For a evaporatio factor of oe, the ew walk completely determies the updated pheromoe values I both of these extreme cases the algorithm looses all iformatio obtaied i previous steps, thus the evaporatio factor is chose to be strictly betwee zero ad oe Neuma ad Witt showed i [18] that the choice of the evaporatio factor ρ ca be critical to the optimizatio 1 time of a ACO algorithm They ivestigated the basic ACO algorithm 1-At o the pseudo-boolea optimizatio problem OeMax ad proved that with high probability 2 the optimizatio time of 1-At o OeMax is expoetial for ρ 1/ 1+ɛ ad polyomial for ρ 1/ 1 ɛ where is the umber of variables i OeMax ad ɛ > 0 We cotiue this study o how the optimizatio time of 1-At o OeMax depeds o the evaporatio factor I particular, we focus o the phase trasitio from expoetial to polyomial optimizatio time, i e we ivestigate the optimizatio time as ρ assume the critical values betwee 1/ 1+ɛ ad 1/ 1 ɛ Our first achievemet is that we traslate the pheromoe model i [18] to a form that is much simpler to aalyze but is fully equivalet This ew model was idepedetly proposed i [13], but without showig its equivalece to the existig model Buildig o the techiques of Neuma ad Witt, we give bouds o the optimizatio time of 1-At o Oe- Max withig the critical widow (1/ 1+ɛ, 1/ 1 ɛ ) I particular, i Theorem 4 we show that with high probability the optimizatio time is already super-polyomial for ρ = o(1/ l ) ad show how this lower boud ca be cotiuously stregtheed as ρ takes values closer to 1/ 1+ɛ 1 We use the otios of rutime ad optimizatio time of a ACO algorithm syoymously I both cases we mea the umber of solutios costructed before a optimal solutio is produced 2 A evet occurs with high probability if, for ay α 1, the evet occurs with probability at least 1 c α/ α, where c α depeds oly o α 1

2 2 Outlie I the ext sectio we put our work ito the cotext of related research As idicated already i the title of this paper we focus o theoretical rutime aalyses of evolutioary search heuristics i geeral ad ACO algorithms i particular I Sectio 4 we itroduce the basic ACO algorithm 1-At ad discuss geeral otios like costructio graph, radom walk geeratio, ad pheromoe values Note that the importat cocepts of pheromoe updates ad evaporatio factor are postpoed to the followig two sectios The applicatio of 1-At to geeral pseudo-boolea fuctios is discussed i Sectio 5 ad Sectio 6 First, we preset the model used by Neuma ad Witt, i particular, their defiitios of the costructio graph, pheromoe updates ad evaporatio factor I Sectio 6 we traslate this model to a simplified versio ad prove their equivalece Besides a ew costructio graph we itroduce a ew pheromoe model which leads to a rescaled evaporatio factor ad a strogly simplified update rule I Sectio 7 we itroduce the pseudo-boolea optimizatio problem OeMax i the cotext of the simplified model ad show basic properties Fially, i Sectio 8, we use the simplified model to prove refied lower bouds o the optimizatio time of OeMax o 1-At 3 Kow Results Sice 1991, ACO algorithms have bee applied to a large umber of optimizatio problems, see [8] for a overview of problems i combiatorial optimizatio I cotrast to the large amout of practical ad experimetal work o ACO, theoretical results o the rutime behavior of ACO algorithms are still scarce Radomized search heuristics i geeral have bee aalyzed for twety years Examples are the rather geeral aalyses of Radomized Local Search i 1990 [19] ad of the Metropolis Algorithm i 1993 [14] I 2002 the rutime of basic evolutioary algorithms (EAs) o pseudo-boolea optimizatio problems [9] was studied The pseudo-boolea fuctios aalyzed were example fuctios such as OeMax, LeadigOes, ad BiVal Due to the well-kow structure of these fuctio it was possible to develop stochastic methods for the aalysis of EAs, some of which were highly o-trivial I the followig years, the rutime of EAs o several problems i combiatorial optimizatio was studied Some examples are the partitio problem [20] ad the problems of fidig maximum matchigs [10], miimum spaig trees [16], ad Euler tours [15, 2, 4, 3] i graphs Although models ad dyamics of ACO algorithms were studied before, util recetly oly covergece results o the rutime of such algorithms existed I [6] Dorigo ad Blum explicitly formulate the demad for the theoretic ivestigatio of the rutime of simple optimizatio problems similar to which was doe for geeral EAs I 2006 Neuma ad Witt proved the first rigorous bouds o the rutime of a ACO algorithm [18] They aalyzed the optimizatio time of the basic ACO algorithm 1-At o the pseudo-boolea optimizatio problem Oe- Max I [5] other pseudo-boolea fuctios were ivestigated ad i [17] the combiatorial optimizatio problem of fidig a miimum spaig tree was addressed We cotiue this work of theoretic rutime aalysis ad refie the results preseted i [18] 4 The 1 At Algorithm We revisit the basic ACO algorithm 1-At aalyzed by Neuma ad Witt i [18] This algorithm is derived from the graph-based at system proposed i [11] Like other radomized search heuristics, 1-At successively geerates potetial solutios to the give optimizatio problem util a optimal solutio is foud A solutio geerated by 1-At is the outcome of a radom walk of a sigle at o a give costructio graph G More precisely, let G = (V, E) be a directed graph with edge weights τ : E [0, 1] The weights τ are called the pheromoe values of the edges Give a startig vertex s V, the radom walk of a sigle at is costructed as follows I every step of the walk the at radomly chooses a edge to a uvisited eighbor This is doe with probability proportioal to the pheromoe value of that edge The walk eds if all eighbors of the curret vertex are visited I the procedure AtWalk the radom walks are geerates The iput of AtWalk cosists of the costructio graph G, the pheromoe values τ, ad the startig vertex s AtWalk the returs a walk W = (V W, E W ) costructed radomly as described above AtWalk(G, s, τ) 1 V W := s 2 E W := 3 v := s 4 while (N G (v) \ V W ) 5 do Choose w N G (v) \ V W radomly with probability 6 V W := V W w} 7 E W := E W (v, w)} 8 v := w 9 retur W = (V W, E W ) P τ(v,w) w N G (v)\v τ(v,w ) W Let W be the set of maximal o-self-itersectig walks o G startig i s, i e, the set of walks that ca be retured by AtWalk The algorithm 1-At tries to optimize a objective fuctio f : W R o all such walks This is doe by successively geeratig such walks ad storig the curretly best walk W Wheever W is chaged, 1-At also updates the pheromoe values τ accordig to the edges used i W The iput of 1-At is the costructio graph G, the startig vertex s ad the iitial pheromoe values τ iit Iitially, all edges are assiged the same pheromoe value First, 1-At geerates a iitial solutio W usig the procedure AtWalk The, the pheromoe values τ are updated accordig to W This update is performed by the procedure Update which is discussed i the ext two sectios 2

3 w 1 w 2 w u 0 u 1 u 2 u 1 u w 1 w 2 w Figure 1: The classical chai G bool Next, 1-At repeatedly geerates solutios W If the value of the ew solutio f(w ) is at least as large as the objective value of the curretly best solutio f(w ) the W is replaced by W ad the pheromoe values τ are agai updated 1-At(G, s, τ iit ) 1 τ := τ iit 2 W := AtWalk(G, s, τ) 3 τ := Update(G, τ, W ) 4 repeat 5 W := AtWalk(G, s, τ) 6 if ( f(w ) f(w ) ) 7 the W := W 8 τ := Update(G, τ, W ) 9 forever The optimizatio time of a ACO is the umber of solutios geerated util a optimal solutio is foud Clearly, the optimizatio time is a radom variable depedig o the choice of G, s, τ iit ad the defiitio of the procedure Update For the theoretical ivestigatio of the optimizatio time it is commo to assume that the algorithm ever termiates 5 The Classical Model for At Coloy Optimizatio o Pseudo-Boolea Fuctios I this sectio we preset the ACO model for pseudoboolea optimizatio problems as used i [12] ad [18] We discuss a suitable costructio graph ad pheromoe values for such problems ad defie a correspodig Update procedure Neuma ad Witt use the chai graph for the represetatio of pseudo-boolea fuctios f : 0, 1} R Let N The chai graph G bool as depicted i Figure 1 is the directed graph (V, E) with distict vertices u 0,, u, w 1,, w, w 1,, w ad edges (u i 1, w i ), (w i, u i ), (u i 1, w i ), ad (w i, u i ) for all i 1,, } The vertex u 0 serves as startig vertex s A radom walk of the at has to pass the vertices u 0,, u i their give order, havig the choice betwee w i ad w i whe passig from u i 1 to u i Thus, a walk W = (V W, E W ) of the at defies a vector x 0, 1} such that x i = 1 if w i V W ad x i = 0 if w i V W The pheromoe values τ : E [0, 1] are chose such that i every step of the algorithm 1/2 2 τ(e) ( 1)/2 2, e E τ(e) = 1, ad τ(u i 1, w i ) + τ(u i 1, w i ) = 1/2 hold for all i 1,, } Iitially, all pheromoe values are set to τ iit (e) = 1/4 for all e E The reaso to restrict the values of τ to the iterval [1/2 2, ( 1)/2 2 ] is to maitai the flexibility of the algorithm The mai idea behid ACO is to stregthe the pheromoe values of successful solutios More precisely, if the algorithm geerates a walk that has a objective value of at least the curret optimum, the the pheromoe values of all edges used by the walk are icreased (stregtheig) At the same time the pheromoe values of uused edges are decreased (evaporatio) These chages are determied by the evaporatio factor ρ [0, 1] I 1-At, the pheromoe values are chaged by the procedure Update I the model proposed i [18] this procedure is defied as follows Update(G, τ, E W ) 1 for (e E) 2 do if (e E W ) 3 the τ(e) := mi (1 ρ)τ(e)+ρ 1 ρ+2ρ, } 4 else τ(e) := max (1 ρ)τ(e) 1 ρ+2ρ, } 5 retur τ Note that wheever a at uses the edge (u i 1, w i ), the it also uses the edge (w i, u i ) Respectively, the edge (u i 1, w i ) is alway used simultaeously with the edge (w i, u i ) Thus, τ(u i 1, w i ) = τ(w i, u i ) ad τ(u i 1, w i ) = τ(w i, u i ) I order to maitai the two properties e E τ(e) = 1 ad τ(u i 1, w i ) + τ(u i 1, w i ) = 1/2 for all i 1,, } the ormalizatio of τ by 1 ρ + 2ρ is ecessary Because of the secod property the probability for a at to choose the edge e E is p e = 2τ(e) 6 The Simplified Model I order to ease the study of the radom optimizatio time of 1-At for pseudo-boolea fuctios we propose a simplified model We itroduce three modificatios to the model of the previous sectio: We simplify the costructio graph by removig uecessary vertices ad edges We idetify the pheromoe value of a edge with the probability that the edge is chose by the at Most importat, as a cosequece of the previous modificatio we fid a appropriate scalig 3

4 e 1 e 2 e u 0 u 1 u 2 u 1 u e 1 e 2 e Figure 2: The simplified chai H bool of the evaporatio factor which supports the uderstadig of the model ad eases a improvemet of the give rutime bouds First, we adapt the graph G bool from the previous sectio to our specific eeds At the ed of the previous sectio we observed that for all i 1,, }, wheever the at uses the edge (u i 1, w i ) it also uses the edge (w i, u i ) ad that the same holds for the edges (u i 1, w i ) ad (w i, u i ) Thus, as ew costructio graph we use a multi-graph istead, i e, a directed graph with multiple edges We replace the paths u i 1, w i, u i ad u i 1, w i, u i by two parallel edges e i = (u i 1, u i ) ad e i = (u i 1, u i ), see Figure 2 By this, the graph we cosider becomes a sequece of vertices such that successive vertices are joied by two parallel edges For N let H bool = (U, E) be a multi-graph with vertices u 0,, u ad distict edges e 1,, e, e 1,, e, such that for all i 1,, }, both e i ad e i start i u i 1 ad ed i u i A walk W = (V W, E W ) o H ow defies the boolea vector x 0, 1} by settig x i to oe if e i E W ad x i to zero if e i E W This adaptatio, while mior, seems atural ad obviously simplifies the graph First, the artificial couplig of the pheromoe values of the edges (u i 1, w i ) ad (w i, u i ) i G bool disappears ad reduces the umber of edges ad pheromoe values by oe half Secod, the etries of the boolea vector are defied by the edges of the correspodig radom walk rather the by its vertices This fact better reflects the paradigm of ACO algorithms to have a edgebased represetatio of the solutios The secod modificatio of the previous model is agai mior i detail but major with respect to uderstadig the behavior of the algorithm I the previous sectio, the pheromoe values were requested to sum up to oe We ow drop this costrait Istead, we directly assig the probability to chose a edge i the radom walks as its pheromoe value We therefore deote the pheromoe value of the edge e i by p i ad that of the edge e i by p i = 1 p i The value p i ow marks both, the pheromoe value of the edge e i ad the probability of its occurrece i a walk of the at As i the previous model, we restrict the p i to the iterval [1/, 1 1/] ad set the iitial pheromoe values to τ iit := p (0) = (1/2,, 1/2) I cosequece to the modificatio of the pheromoe values we propose the third, most importat simplificatio I the old model, the probability to choose a edge e is p e = 2τ(e) Thus, after a update mi 1 ρ 1 ρ+2ρ p e := p e + 2ρ 1 ρ+2ρ, 1 } if e E W, 1 ρ max 1 ρ+2ρ p e, 1 } else We redefie the evaporatio factor by ρ := this, the procedure Update chages to Update(H, p, E W ) 2ρ 1 ρ+2ρ 1 for (i 1,, }) 2 do if (e i E W ) 3 the p i := mi(1 ρ)p i + ρ, 1 1 } 4 else p i := max(1 ρ)p i, 1 } 5 retur p Comparig this ew Update procedure to the previous oe clearly demostrates the simplificatio itroduced by ρ the ew model Sice ρ := 2 (2 1) ρ all results i the ew model ca be traslated to the previous model ad vice versa To ease readig, we computed some useful values i Table 1 The ifluece of the evaporatio factor ρ o the pheromoe values ow becomes directly clear For pheromoe values ot close to 1/ or 1 1/ we ca iterpret a update as the evaporatio of a ρ-th of the existig pheromoes ad the subsequet stregtheig of the used edges by ρ Alteratively, we ca rewrite (1 ρ)p i + ρ as p i + ρ(1 p i ) ad (1 ρ)p i as p i ρp i Thus, if e i is used by the at, the its pheromoe value is raised by ρ times the probability that the edge is ot chose Respectively, if e i is ot used the p i is decreased by ρ times its curret value Because of this, pheromoe values are less icreased if close to oe ad less decreased if close to zero By ρ 0 1/ 1+ɛ 1/(2 l ) 1/ 1 ɛ 1 ρ 0 2/ ɛ 1/ l 1 1/2 ɛ 1 Table 1: The relatio betwee ρ ad ρ 7 Optimizatio of OeMax The mai advatage of the simplified model becomes clear if we ivestigate optimizatio problems with liear objective fuctios like OeMax The objective fuctio to be maximized i OeMax is the size x = i=1 x i of the boolea vector x 0, 1} Obviously, this maximum is uique ad attaied by the vector (1,, 1) of size 4

5 We cosider the vector x 0, 1} correspodig to a radom walk of a at to be a sample of a radom 0-1-vector = ( 1,, ) The distributio of is depeds o the ew pheromoe values p 1,, p, sice p i is the probability that i = 1 for all i 1,, } Note that all i are mutually idepedet The value of the objective fuctio is the size x of x ad represeted by the radom variable S = Observe that by liearity of expectatio E [ ] S = i=1 p i A typical ru of 1-At o OeMax first produces a sample x 0, 1} of the radom variable (0) distributed accordig to p (0) = (1/2,, 1/2) Let (0) = x, the (0) is a radom vector distributed like (0) Next, 1-At updates p (0) to p (1) Suppose ρ 1 2/, the ρ 2 if (0) i = 1, p (1) i = 1 2 ρ 2 if (0) i = 0 We ca ow idetify phases t N i the ru of 1-At The iitial triple p (0), (0), ad (0) marks the 0-th phase I geeral, a ew phase starts after the pheromoe vector has bee updated Thus, there is a ew pheromoe vector p (t) for each phase t This pheromoe vector defies the radom vector (t) ad its size S (t) = i=1 (t) i Durig phase t 1-At repeatedly geerates samples x 0, 1} of (t) The phase eds whe a sample x with size at least the size of all the x from previous phases has bee geerated The radom vector (t) is ow defied to be this last x of phase t Accordigly, S (t) = i=1 (t) is defied to be its size Sice (t) has size at least as the size of all its predecessors, S (t) S(t 1) holds for all t > 0 I particular, phase t eds exactly whe a geerated x has size at least S (t 1) Betwee phase t ad phase t+1 the pheromoe vector p (t) is updated: p (t+1) i = mi(1 ρ)p (t) i + ρ, 1 1 (t) } if max(1 ρ)p (t) i, 1 i = 1, (t) } if i = 0 For early phases t, the values of p (t) are bouded as follows Lemma 1 Let ρ 1/2, t N ad i 1,, } The 1 e 2 ρt 2 p (t) i 1 1e 2 ρt 2 Proof By iductio 1 2 (1 ρ)t p (t) i (1 ρ)t holds for all i 1,, } The claim the follows as 1 ρ e 2 ρ holds for all 0 ρ 1/2 As a corollary we obtai for t 1/ ρ ad 2e 2 (1 ρ)p (t 1) i + ρ if (t 1) i = 1, p (t) i = (1 ρ)p (t 1) i if (t 1) i = 0 Note that for t 1 the distributios of (t) ad (t) differ This ca be easily see, for example S (t) ca be arbitrarily small while S (t) is at least S(t) However, (t) ad (t) have the same distributio uder the coditio (t) (t 1) Formally, for t > 0 ad x 0, 1} Pr [ (t) = x ] = Pr [ (t) = x S (t) ] S(t 1) As we see, the distributio of (t) depeds ot oly o p (t) like that of (t) but also o (t 1) The two radom variables S (t) ad S(t) tur out to be the key to the uderstadig of how the optimizatio time of 1-At behaves While S (t) describes the sigle solutios geerated i the curret phase, S (t) deotes the best solutio 1-At has geerated at the ed of phase t The distributios of S (t) ad S(t) are related like those of (t) ad (t) For t > 0 ad k 0,, } Pr [ S (t) = k] = Pr [ S (t) = k S(t) ] S(t 1) = E[ S (t) ] ca ow be iterpreted meaigfully Due to the choice of our model, the values ad σ (t) = Var[S(t) ] 1 2 Namely, i phase t with derivatio σ (t) deotes the average value of a sigle solutio For example, the size of = /2 first vector almost surely differs by at most σ (0) from µ (0) = /2 We will see that we ca keep track o the behavior of µ (0) To moitor ad σ(t) = E [ S (t) ] is more difficult Obviously, = /2 sice (0) is distributed like (0) But for t > 0 we ca express oly as µ(t) = E[ S (t) S(t) ] S(t 1) We use the same stochastic techiques as [18] to approximate the probability that S (t) exceeds S(t 1) Nevertheless, our scalig of the pheromoe values allow us to directly traslate the results to a uderstadig of the algorithm s behavior ad results i more cocise proofs The followig lemma is cetral i the study of 1-At o OeMax ad a good example of this claim It states that the average size of a vector geerated i early phases depeds oly o the size of the so far accepted solutios, but ot o how these solutios actually look like Lemma 2 Let ρ 1/2 ad 0 < t 1/ ρ, the Proof = µ(t 1) = i : (t 1) i =1 ( = i=1 + ρ(s (t 1) p (t) i + (1 ρ)p (t 1) i µ (t 1) ) i : (t 1) i =0 p (t) i ) + ρs (t 1) ad the statemet follows from the defiitio of µ (t 1) We coclude this sectio with a geeral lower ad upper bouds o the stadard deviatio σ (t) = Var[S(t) ] 1 2 Lemma 3 Let ρ 1/2 ad t N The σ (t) 1 2 e ρt 1/2 Proof By Lemma 1 ad (σ (t) )2 = i=1 p(t) i (1 p (t) i ) 5

6 8 A New Lower Boud for the Optimizatio Time of 1 At o OeMax I this sectio we study how the optimizatio time of 1-At o OeMax depeds o the evaporatio factor ρ I [18] it was show that for all fixed ɛ > 0 ad ρ = 1/ ɛ (equivaletly, ρ = 1/ 1+ɛ ) ad ɛ > 0 with high probability the optimizatio time is expoetial i The followig theorem refies this result ad shows that with high probability already for ρ = o(1/ l ) the optimizatio time is super-polyomial It particular, we derive a explicit correspodece betwee a boud o the optimizatio time ad the probability that this boud holds Theorem 4 There exists a c (0, 1) ad a N N such that for N ad ρ c the optimizatio time of 1-At o OeMax with variables ad evaporatio factor ρ is at most e c ρ c with probability less the 1/e ρ We first motivate this theorem, the full proof is give at the ed of this sectio Suppose that ρ 1/ l ad let us cosider a typical ru of 1-At We are particularly iterested i the first T = 1/ ρ l phases Throughout these T phases, the solutios S (t) are more or less biomially distributed I particular, the deviatio σ (t) stays approximately σ = /2 With high probability, the first solutio S (0) is close to its expected value, hece we suppose S (0) = /2 Wheever a solutio is accepted as ew S (t), the with costat probability the icrease S (t) S(t 1) is at least σ/ l Thus, by ap=plyig the Cheroff bouds, we see that with high probability S (T ) /2 + / ρ O the other had, Lemma 2 tells us, that after T phases (T +1) µ /2 + 1/2 / ρ Applyig the Cheroff bouds agai, we ca boud the (T +1) probability that the ext solutio S ca bridge the gap of σ (T +1) 1/ ρ betwee its expected value µ ad the curret optimal solutio S (T ) For ρ = 1/ l this probability is roughly polyomial i 1/, for ρ = o(1/ l ) it decreases rapidly The remaider of this sectio is devoted to the proof of Theorem 4 First, ote that while S (t) is clearly icreasig i t, this is ot ecessarily true for This is due to the fact that we may accept solutios of size less the /2 i early phases We do, however, have the followig lower boud o Lemma 5 For all t 0, Proof Sice p (0) i µ (0) mi 2, S(0) } = i=0 p(0) i = /2 Furthermore, S (t) = 1/2 for all i 1,, }, we have that S(0) ad by = (1 ρ)µ(t 1) + ρs (t 1) Thus, the i-, S (t 1) } holds by a simple iduc- Lemma 2 also equality tio o t miµ(t 1) From this lower boud o upper boud of Lemma 6 For all t 0, 2 we obtai the followig + ρt maxs(t 1) 2, S(t 1) S (0) } Proof Agai ivokig Lemma 2, we compute t = µ(0) + = 2 + t s=1 2 + ρ t = 2 (µ (s) s=1 ρ(s (s 1) (S (t 1) s=0 µ(s 1) ) µ (s 1) ) mi 2, S(0) }) + ρt maxs(t 1) 2, S(t 1) S (0) } We eed two tools from probability theory The followig lemma was prove by Neuma ad Witt ( [18], Lemma 3) Lemma 7 Let S = i=0 i be the sum of mutually idepedet radom 0-1-variables 1,,, µ = E [ ] S ad σ 2 = Var[S ] Let 0 a σ, γ = max a µ σ, 2} ad σ The there exists a p 0 (0, 1) ad a N N such that Pr [ S a + σ γ S a ] p 0 for all N The secod tool we use are the well-kow Cheroff bouds (see for example[1] ) Theorem 8 (Cheroff) Let S = i=0 i be the sum of mutually idepedet radom 0-1-variables 1,, ad let µ = E [ S ] The for all a > 0 ad δ [0, 1], (i) Pr [ S µ + a ] e 2a2, (ii) Pr [ S µ + a ] e 2a2, ad (iii) Pr [ S (1 δ)µ ] e δ2 µ 2 We are ow ready to prove Theorem 4 Proof Let N N ad p 0 (0, 1) as i Lemma 7 Also, let c = p0 5120, ρ c, = 8c/ ρ, ad T = 1 4 ρ The, it holds that c (0, 1),, ad 1 5 ρ T 1 4 ρ Let E 1 be the evet that 2 < S(0) < 2 + ad E 2 the evet that S (T ) > 2 + We first show that Pr [ ] E 1 1 2e 16c (0) ρ Sice S has the same distributio as S (0), we apply the first two Cheroff bouds ad obtai ad Pr [ S (0) Pr [ S (0) 2 + ] e ] e 2 2 6

7 Hece, E 1 occurs at least with probability 1 2e 2 2 = 1 2e 16c/ ρ Next, we show that Pr [ ] E 2 E 1 1 e 128c ρ Suppose, that E 1 holds, i particular S (0) 2 Suppose further, that S (T ) 2 + We show that for this to happe, a umber of very ulikely evets have to occur For t T, let γ t = max S(t 1) σ (t), 2} Sice T 1 4 ρ, we have S(t 1) 2 by Lemma 6 ad σ (t) by Lemma 3 It also holds that , ad thus we get γ t Hece, by Lemma 7 we obtai Pr [ S (t) S(t 1) + 32 S(t) ] S(t 1) E 1 p0 Thus, Pr [ S (t) S(t 1) 32 E 1] p0 For t 0 we defie the auxiliary radom 0-1-variables Z (t) 1 if S (t) = S(t 1) 32, 0 else Let S Z = T t=1 Z(t) ad µ Z = E [ ] S Z E 1 p0 T The µ Z 1024c ρ, as T 1 5 ρ ad p 0 = 5120c By the third Cheroff boud Hece, sice 2 Pr [ S Z 1 2 µ Z E 1 ] e 128c ρ Pr [ S Z > 64 2 = 8c ρ Now, S (T ) = S (0) + T t=1 S (t) E 1 ] 1 e 128c ρ, S(t 1) S Z Thus, Pr [ S (T ) > 2 + E 1] 1 e 128c/ ρ Suppose, 2 S(0) < 2 + < S(T ), that is, E 1 ad E 2 hold The there is a t T such that S (t 1) 2 +, but ) S(t > 2 + Let µ = E [ S (t ) E ] 1 E 2 The by Lemma 6 it holds that µ ρt Thus, by the Cheroff boud Pr [ S (t ) µ + 2 E ] 1 E 2 e 2 2 Hece, Pr [ S (t ) > 2 + E ] 1 E 2 e 4c/ ρ Uder the coditio E 1 E 2, the probability that the optimizatio time of 1-At o OeMax is at most e c ρ [ is bouded by e c ρ (t Pr S ) > 2 + E ] 1 E 2 e 3c/ ρ But the the ucoditioal probability that the optimizatio time of 1-At o OeMax is at most e c ρ is bouded by 2e 16c/ ρ + e 128c/ ρ + e 3c/ ρ e c/ ρ 9 Coclusio I this work we proposed a ovel pheromoe model for pseudo-boolea optimizatio problems We idetified the pheromoe values with their correspodig probabilities This lead to a redefied evaporatio factor that supports a better uderstadig of pheromoe updates ad liks the mathematical otios of expectacy ad variace to coceptual ivariats of the optimizatio process We were able to refie the rutime bouds i [18] ad to exteded them to values of ρ iside the critical widow of the phase trasitio We are cofidet that our refied model will allow further isight i the rutime behavior withi the phase trasitio ad prove superior for the future aalysis of ACO algorithms o other pseudo-boolea optimizatio problems Refereces [1] N Alo ad J Specer The Probabilistic Method Itersciece Series i Discrete Mathematics ad Optimizatio Wiley, 2000 [2] B Doerr, N Hebbighaus, ad F Neuma Speedig up evolutioary algorithms through restricted mutatio operators I Proceedigs of the 9th Iteratioal Coferece o Parallel Problem Solvig From Nature (PPSN), volume 4193 of Lecture Notes i Computer Sciece, pages Spriger, 2006 [3] B Doerr ad D Johase Adjacecy list matchigs a ideal geotype for cycle covers I Geetic ad Evolutioary Computatio Coferece (GECCO- 2007), pages ACM, 2007 [4] B Doerr, C Klei, ad T Storch Faster evolutioary algorithms by superior graph represetatio I Proceedigs of the First IEEE Symposium o Foudatios of Computatioal Itelligece (FOCI), pages IEEE Press, 2007 [5] B Doerr, F Neuma, D Sudholt, ad C Witt O the rutime aalysis of the 1-ANT ACO algorithm I D Thieres, editor, Geetic ad Evolutioary Computatio Coferece (GECCO-2007), pages ACM, 2007 [6] M Dorigo ad C Blum At coloy optimizatio theory: A survey Theor Comput Sci, 344: , 2005 [7] M Dorigo, V Maiezzo, ad A Colori The at system: A autocatalytic optimizig process Techical Report Revised, Politecico di Milao, 1991 [8] M Dorigo ad T Stützle At Coloy Optimizatio MIT Press, 2004 [9] S Droste, T Jase, ad I Wegeer O the aalysis of the (1+1) evolutioary algorithm Theor Comput Sci, 276:51 81,

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