Maximum Matchings via Glauber Dynamics

Maxim um Matc hings via Glau b er Dynamics Anant Jindal ∗ Gazal Ko char † Manjish P al ‡ Abstract In this pap er w e study the classic problem o f computing a maxim um cardinalit y matc hing in general graph s G = ( V , E ). This prob lem h as b een stud ied extensiv ely more than four decades. The b est kno wn algorithm for this problem till date runs in O ( m √ n ) time due to Micali and V azirani [24]. Ev en for general bipartite graphs this is the b est kno wn runn ing time (the algo rithm of Karp and Hop croft [16] also ac hiev es this b ound ). F or r egular bipartite graphs one can ac h iev e an O ( m ) time algorithm whic h, follo wing a series of pap ers, has b een recen tly improv ed to O ( n log n ) by Go el, Kapralo v and Khanna (ST OC 2010) [15]. In this pap er we pr esen t a randomized algorithm based on the Mark ov Chain Mon te Carlo paradigm whic h runs in O ( m log 2 n ) time, th ereb y obtaining a significant im p ro v ement o v er [24]. W e use a Mark o v chain similar to the har d-c or e mo del for Glaub er Dynamics with fugacity parameter λ , whic h is used to sample ind ep endent set s in a graph from the Gibb s Distribution [31], to design a faster algorithm for finding maxim um matc hings in general graphs. Mot iv ated b y r esults which show that in the hard -core m o del one can prov e fast mixin g times (for e.g. it is kno wn that for λ less than a critical thresh old the mixing time of the hard -core mo del is O ( n log n ) [27], we defi ne an analogous Marko v c h ain (dep end ing up on a parameter λ ) on the space of all p ossible partial m atc hings of a giv en graph G , f or whic h the probabilit y of a particular matc hing M in the stationary follo ws the Gibbs distribution wh ic h is: π ( M ) = λ | M | P x ∈ Ω λ | x | where Ω is th e set of all p ossible matc hings in G . W e pr o ve u pp er and low er b oun ds on the mixing time of this Marko v c h ain. Although our Mark ov c hain is essen tially a simple modification of the one used for sampling ind ep endent sets from the Gibbs distribution, their p rop erties are qu ite d ifferen t. Our result cru cially relies on the fact that the mixin g time of our Mark o v Ch ain is indep enden t of λ , a s ignifican t deviation from the recen t series of w orks [11, 26, 28, 29, 30] whic h ac h iev e computational trans ition (for estimating the partition function) on a thresh old v alue of λ . As a r esu lt w e are able to design a randomized algorithm w hic h ru n s in O ( m log 2 n ) time that provides a ma j or impro v ement ov er the runn ing time of the algorithm d ue to Micali and V azirani. Using the conductance b ound, w e also prov e that mixing tak es Ω( m k ) time w h ere k is the size of th e maxim u m matc hing. ∗ Laxmi Niwas Mittal Institute of Infor mation T echnology , India . anant jinda l1@gma il.com † Laxmi Niwas Mittal Institute of Infor mation T echnology , India . gkoch ar@gm ail.co m ‡ Indian Institute o f T ec hnology Gandhinagar , India. ma njish pal@ii tgn.a c.in 1 1 In tro ducti on Giv en an unw eigh ted u ndirected graph G = ( V , E ) with | E | = m and | V | = n , a matc h ing M is a set of edges b elonging to E su ch th at n o t wo edges in M are inciden t on a verte x. If there is a matc hing of size n / 2 (for n ev en), then it is called a p erfe ct matching . The Maxim um Matc hing problem is to fin d the maxim um sized matc hing in a given graph. T he computational complexit y of this prob lem has b een studied extensively for more than four d ecades starting with an algorithm of Edmond s. 1.1 General Graphs Edmonds’s celebrated pap er ‘P aths, T rees and Flo w ers’ [9] w as the first to giv e an efficien t algorithm (also called th e blossom shrinking algorithm ) for fin ding maxim um matc h ing in general graph s. This algorithm can b e implemen ted in O ( n 4 ) time. The ru nning time w as subsequently improv ed in a n um b er of pap ers [10, 20, 22]. All these pap ers we re v arian ts of Edmonds algorithm. Eve n and Kariv [12] obtained an improv emen t to O ( n 2 . 5 ) which w as impro v ed by Micali and V azirani [24] who ga ve an O ( m √ n ) time algorithm for the problem b y a careful handling of blossoms. This is the b est kno wn algorithm for fi nding maximum matc hings in general bipartite graph s. 1.2 Bipartite Graphs F or bipartite graphs, the problem can easily b e solv ed using the max-flow algorithm by F ord and F ulke rson, an algorithm us ually taught in an undergradu ate algorithms course [21], whic h has a r unnin g time of O ( mn ). The fi rst algorithm for this problem was giv en by Konig [19 ]. Hop croft and Karp [16] ga ve an algorithm that ru ns in O ( m √ n ) time. This algorithm is an exact and d eterministic algorithm. The problem b ecomes significan tly simpler for regular bi- partite graph s. In a d-r e gular bip artite graph eve ry v ertex has degree d . When d is a p o wer of 2, Gab or and Kariv w ere able to ac hiev e an O ( m ) algorithm. After significan t efforts, the ideas used ther e w ere u sed by Cole, O st and Schirra [4] to obtain a get an O ( m ) algorithm for general d . In a recent line of attac k b y Go el, Kapralo v and Khann a [13, 14], the authors were able to use sampling b ased metho ds to get impr ov ed run ning time. In the most recent pap er th ey w ere able to ac hiev e a r unning time of O ( n log n ) for d -regular graph s [15]. Their algorithm p erforms an app ropriately truncated r an d om-w alk on a mo dified graph to su ccessiv ely fi nd augmen tin g path. 2 Our Results In this pap er w e giv e a Mark ov C hain Monte Carlo algorithm f or fin ding a maximum matc hing in general bipartite graphs. Our algorithm is in th e spirit similar to [15] whic h also is a ‘truncated random walk’ based algorithm, ho wev er the stationary distribu tion of the underlying Mark ov Chains in their case is different from ours. Inspired fr om the hard-core mo del with fugacit y parameter λ of sampling in dep end en t sets from graphs w e d efi ne a similar Marko v Chain o v er the space of all p ossible partial matc hing suc h that its stationary distribu tion π ( · ) 2 is the Gibb s distribution, ie. given a matc h ing M its probabilit y π is π ( M ) = λ | M | P σ ∈ Ω λ | σ | where Ω is th e set of all p ossible matc hings in G . Notice that π ( M ) is maxim um for maxim um matc hings and if λ is a s ignifi can tly large num b er , π ( M ) tend s to 1 for maximum matc hin gs. Our algorithm is extremely simple (b eing a standard in the MCMC paradigm). S tarting from a fixed matc hing w e start a random wa lk in Ω according to the underlying graph ˜ G of the Mark ov c hain. After T mix (mixing time) steps the d istribution reac hed by the algorithm is roughly the same as the Gibb s distribu tion. More formally , the v ariation distance of D t (the distribution after t steps) from the Gibbs distrib ution is less than 1 2 e . This ju st lea v es the task of p ro v in g an upp er-b ound on the mixing time of th e Marko v Ch ain, for wh ic h we r esort to the Coup ling Metho d introd uced by Bubley and Dye r [2]. W e define a metric Φ( · , · ) and app ly the b ound from [2], to sho w that the Mark ov Ch ain mixes in O ( m log n ) time. W e also us e the conductance metho d to sho w that the mixing time will b e at least Ω( m k ) wh ere k is the s ize of the maximum matc hin g. Thus up to logarithmic factors the b ounds are s ame when k is small (at most p olylog arithmic). The main result of our pap er can b e concisely written as follo ws: Theorem 1 ( Main ) . Ther e exists a r andomize d algorithm which g iven a gr aph G = ( V , E ) with | V | = n and | E | = m finds a maximum matching in O ( m log 2 n ) time with high pr ob ability. Imp ortant remark: It has b een p oin t ed to us indep endently b y Y uv al P eres, Jonah Sherman, Piyush Sriv asta v a and other anonymo us review ers t ha t the cou- pling used in this pap er do esn’t hav e t he righ t marginals b ecause of w hic h the mixing time b ound do e sn’t hold, and also t he main result presented in the pap er. W e thank them for reading t he pap er w it h in t erest and promptly p ointing out this mistake. 2.1 Organization The pap er is organized as f ollo ws: in S ection 3 we giv e a brief description of th e basic idea and tec hniqu e th at are underlyin g our algorithm. Subsequent ly in Section 4 we giv e an o verview of the MCMC paradigm wh ic h includ es b asic preliminaries and defin itions r egarding Marko v Chain and Mixing. Section 7 is dev oted to the details of the chain b eing used b y us and pro ving that in d eed it has the desired pr op erties. W e then prov e u pp er and lo wer b ounds on its mixing time usin g the coupling metho d and conductance argument in Section 8 and Section 9 resp ectiv ely . W e end the pap er with a conclusion and some op en pr oblems. 3 The Idea: Maxim um Matc hings vi a Glaub er Dy- namics Our ideas are insp ired mainly f rom the results in the Glaub er dyn amics of the hard -core mo d el of fugacit y parameter λ , for sampling in dep endent sets from th e Gibbs distribution. Acco rding to the Gibb s d istribution the probabilit y of an ind ep endent set I is give n by G ( I ) = λ | I | P ρ ∈ Ω λ | ρ | 3 where Ω is the set of all p ossible indep enden t sets in G and Z = P ρ ∈ Ω λ | ρ | is also called the p artition function . Clearly for λ = 1 the partition function v alue is same as the num b er of indep end en t sets in the graph (computing whic h is a #P-Hard problem). In the hard -core mod el giv en a particular configu r ation σ of an indep end en t set ( σ can b e though t of an n -dimensional 0/1 v ector whic h is 1 for all th e v ertices whic h are in the inde- p endent set and 0 otherwise), we choose a v ertex ran d omly uniformly , if this vertex is already present in the in d ep endent set we k eep it with probabilit y λ 1+ λ and discard it with pr obabilit y 1 1+ λ otherwise the verte x is not in the indep end en t set and if this ve rtex can b e added to the indep end en t set (i.e. none of its neigh b ors are already p resen t in the ind ep endent set) then again it is added with with probability λ 1+ λ and rejected with pr obabilit y 1 1+ λ . The b eauty of this Mark o v C h ain is that the stationary distribution is the Gibbs d istribution. The details of this c hain can b e found in [31]. The ke y d ifference b et we en the mixing time of Glaub er Dyn amics for in dep end ent sets and our case is that one can ac hieve fast mixing time in the former case only for small v alues of λ . In fact in tuitiv ely one should not b e able to obtain f ast mixin g times for large v alues of λ b ecause s u c h a result w ould imply th at we can design a ran d omized p olynomial time algorithm for fi nding maximum indep enden t set in a graph , wh ic h is an NP-Hard problem. Th is int uition has led to a series of pap ers [11 , 26, 28, 29, 30] which ultimately h as b een successful in pro ving that there exists a threshold v alue of λ = λ c suc h that if λ > λ c then estimating the partition function is hard (the exact tec hnical condition is that unless N P = R P no FPRAS exists for estimating Z ) and for λ < λ c one can obtain an FPT AS for the same pr oblem. Previous to this result, the computational complexit y of estimating the partition fu nction (and counting the num b er of ind ep endent sets) wa s only understo o d for sp ecial graph s [5, 32]. Apart from these results su bstan tial attent ion has b een giv en to obtain go o d b ounds on the mixing time of this chain for trees [28]. W e define a Mark ov Chain wh ic h is tuned to our n eed. In our case σ is a set of edges which form a partial matc hing. W e m ak e a simple mo d ification to the ab ov e c h ain, w herein instead of pic kin g a random v ertex w e p ic k a r andom ed ge e r ∈ E and p erform the same exp erim ent with th e parameter λ as in the case of ind ep endent sets (notice th at the c hosen edge won’t b e added if in the pr esen t m atching there is an edge sharin g an end p oin t with e r ). W e then sho w that this c hain is ap erio dic and irredu cible with stationary as the Gibbs distr ib ution o ver the s pace of all p ossib le partial matc hin gs with parameter λ . W e then use the tec hniques of b ound ing the mixing time to ac h iev e a λ indep enden t up p er b ound. This remark able p r op ert y allo ws us to exploit the nature of Gibbs distribution (which we obtain for v ery large v alues of λ ) without getting an o verhead on the mixing time. 3.1 Mark o v Chain M on te Carlo Mark ov C hain Mont e Carlo algorithms hav e p la yed a significant role in statistics, econometrics, physic s and computing s cience ov er the last t w o d ecades. F or some h igh-dimensional problems in geometry , such as computing the volume of a conv ex b o d y in d dimensions, MCMC simula- tion is the only known general appr oac h for p ro vid ing a solution w ithin time p olynomial in d [6]. F or a num b er of other hard p roblems like approxima ting the p erm anen t [18], appr oximate coun ting [17], the only kn o wn FPRASs ( F ul ly Polynomial time R andomize d Appr oximation Schemes ) rely on the MCMC paradigm. In th is pap er, we use this metho d to obtain a faster algorithm for the classical pr oblem of findin g m aximum matc h ings in general graphs, a p rob- lem which is kn o wn to b e solv able in p olynomial time. 4 The Marko v Chain Mon te C arlo (MCMC) metho d is a simple and frequently used approac h for sampling from the Gibb s distr ibution of a s tatistica l mec hanical system. The idea go es lik e th is, w e d esign a Mark o v c hain whose state sp ace is Ω wh ose stationary distribu tion is the desired Gibbs d istribution. Starting at an arbitrary state, we sim ulate the Mark ov chain on Ω unt il it is s u fficien tly close to its stationary d istribution. W e then ou tp ut the fin al s tate whic h is a sample from (close to) th e desired d istribution. T h e required length of the simula tion, in order to get close to the stationary distribution, is traditionally r eferr ed to as the mixing time τ or T mix and the aim is to b oun d th e mixin g time to ensu re that the simulati on is efficien t. F or a detailed u nderstanding of the theory of Marko v Chains we would r ecommend the recent excellen t b o ok by Levin, P eres and Wilmer [23]. 4 Preliminaries 4.1 Mark o v c hains Consider a sto c hastic pr o cess ( X t ) ∞ t =0 on a finite state space Ω. Let P d enote a non-negativ e matrix of size | Ω | × | Ω | which satisfies X j ǫ Ω P ij = 1 ∀ i ∈ Ω The pr o cess is called a Mark o v c hain if for all times t and i, j ∈ Ω prob ab ility of going f rom i th state to j th state is indep enden t of th e path by which i th state is reac hed i.e. if X t is the state of the pro cess at time t then P[ X t +1 | X t = x t , X t − 1 = x t − 1 . . . X 0 = x 0 ] = P[ X t +1 | X t = x t ] A distribution π is called a stationary distribution if it satisfies π P = π . A necessary and sufficien t condition f or a c hain to ha ve a uniqu e stationary distribu tion is that the c hain is 1. Irr e ducible : for all i, j ∈ Ω there exists a time t such that P t ij > 0; and 2. Ap erio dic : for all i ∈ Ω, GCD { t : P t ii > 0 } = 1. A Mark o v Chain which has b oth of th e ab o ve p rop erties is called er go dic . F or an ergo dic Mark ov c hain, if a distribution π satisfies the detailed balance equations π i P ij = π j P j i for all i, j ∈ Ω then π is the (uniqu e) stationary distrib u tion and such a c hain is called r eversible . 4.2 Mixing T ime The notion of mixing time is defined as a wa y to measur e the closeness of the distribution after t steps w.r.t. the s tationary distr ibution. The total v ariation distance b et wee n t w o discrete probabilit y d istributions o ver a finite space Ω is defined as the half of th e l 1 norm of the corresp ondin g probability vecto rs. d T V ( µ, ν ) = 1 2 X ω ∈ Ω | µ ( ω ) − ν ( ω ) | 5 If P t is the pr ob ab ility distribution after t steps in the rand om w alk then T mix is the minim um t for which, d T V ( P t , π ) ≤ 1 2 e where π is the statio nary d istribution. Th erefore, if w e in tend to get close to a stationary distribution we jus t h a ve to trun cate the random walk on th e state s pace after τ steps . 4.3 Conductance The conductance of a Mark o v c hain is defin ed as the follo win g qu an tity , φ ( G ) = min S ⊂ V P i ∈ S,j ∈ ¯ S π i p ij ( P i ∈ S π i )( P i ∈ ¯ S π i ) Another qu an tit y of our inte rest here is T r elax , the r elaxation time of the Marko v chain. T r elax is defined as the inv erse of conductance i.e. T r elax = 1 φ ( G ) . It is kno w n that T mix and T r elax ob ey the f ollo wing inequalit y [1]. T r elax + 1 ≤ T mix . Also it is kn o wn [28] that T r elax = Ω  1 φ  , a b ou n d usu ally used to p r o ve low er b oun ds on th e Mixing time of Mark ov Chains. 5 The Ch ain In this section, we d escrib e the chain considered by our algorithm whic h is essentia lly a mo d i- fication of th e hard-core mo d el of Glaub er Dynamics. Recall that our ob jectiv e is to come up with a c hain w hose stationary distribution ensures th at the probabilit y of b eing at a maxim um matc hing is the largest. Recall that first w e n eed to ensu re that our chain is ap erio d ic and irreducible. W e use the f ollo wing natural m o dification of the Marko v chain for the hard-core mo del of Glaub er dyn amics. • Cho ose an edge e r uniformly at rand om from E . • Let σ ′ = ( σ S { e r } , with probability λ 1+ λ σ \{ e r } , w ith probabilit y 1 1+ λ • If σ ′ is a v alid matc hing, mov e to state σ ′ otherwise remain at state σ . W e are no w prepared to pr o ve that this is a v alid Mark o v Chain with stati onary as th e Gibbs d istribution, where we defin e Gibbs distribution as th e follo wing distribu tion o v er the set of all p ossible matc hings Ω as G ( M ) = λ | M | P x ∈ Ω λ | x | In the rest, w e will call Z = P x ∈ Ω λ | x | . 6 Lemma 1. X t is er go dic with stationary distribution as G . Pr o of. Sin ce for eve ry state there is some p robabilit y b y wh ic h the w alk can remain in the same s tate, the chain is ap erio dic. Also the underlyin g grap h is connected b ecause giv en an y matc hing there is a at least one p ath to reac h any other matc hin g (consider the path that fir s t drops all the edges of the initial matc hing an d adds the edges of the new matc h in g on e b y one). T o s ho w that the stationary of this distribution is G , w e would sh ow that the c hain is rev ersib le w.r.t. G . Consider t wo distinct states i and j in the Marko v Chain. Assume w.l.o.g that ther e is an edge from i to j (if there is n o edge from i to j then the balance equations corresp ond in g to i and j are trivially satisfied). Let the configuration i has t edges in it then according to our construction, j will ha ve either t + 1 or t − 1 states (it w on’t b e t b ecause w e h a ve assu med th at the t w o s tates are distinct). W e will just lo ok at the case when j h as t + 1 ed ges, the other case is analogous. Since we are interested in sh o wing rev ersibilit y w.r.t. G , π i = λ t Z and π j = λ t +1 Z . Therefore, π i P ij = λ t Z · λ m (1+ λ ) , and π j P j i = λ t +1 Z · 1 m (1+ λ ) . Th us π i P ij = π j P j i = λ t +1 mZ (1+ λ ) 6 Upp er Boun d using Coupl ing In this section we pr o ve an upp er b ound on T mix of the c h ain defin ed in the previous section. Our b ound is b ased on the coupling argument in tro duced by Bubley and Dy er [2]. W e fir st giv e a d escription of the coupling metho d. 6.1 Coupling M etho d A coupling of a Mark o v c hain on state space is a sto c h astic pro cess ( σ t , η t ) on | Ω | × | Ω | suc h that: • σ t and η t are copies of the original Marko v c h ain and • if σ t = η t , then σ t +1 = η t +1 Th us the chains follo w eac h other after the first instant when they hit eac h other. In order to measure the distance b et w een th e tw o copies of th e c hain, one in tro duces a distance function Φ on the pro du ct state s p ace Ω × Ω so that Φ = Φ ( σ t , η t ) = 0 ⇐ ⇒ σ t = η t . F or t w o states σ and η let ρ ( σ, η ) b e the set of all p aths fr om σ to η in the Mark o v Chain. The f ollo wing theorem du e to to Bubley and Dy er is us ed to prov e mixing time on Marko v c h ain. Theorem 2. L et Φ b e an inte ger-value d metric define d on Ω × Ω which takes values in { 0,1...D } such that, for al l σ, η ∈ Ω ther e exists a p ath ξ ∈ ρ ( σ, η ) with Φ( σ, η ) = X i Φ( ξ i , ξ i +1 ) Supp ose ther e exists a c onstant β < 1 and a c oupling ( σ t , η t ) of the Markov chain su c h that, for al l σ t , η t , E [Φ( σ t +1 , η t +1 )] ≤ β Φ( σ t , η t ) 7 Then the mixing time is b ounde d by τ ≤ log(2 eD ) 1 − β Pr o of. Can b e found in [2, 31]. In order to b oun d the m ixing time, w e will d efi ne a coupling so as to minimize the time unt il b oth copies of the Mark ov c h ain reac h the same state, and we will d o that b y defining the coupling in suc h a wa y that on ev ery step b oth marko v chains reac h to w ards same state. Th e aim is to pro v e a go o d u p p er b ound on E [Φ( σ t +1 , η t +1 )] in term s of Φ ( σ t , η t ). In the follo wing subsection we defin e the coupling: 6.1.1 Coupling Consider the follo wing p ro cess ( σ t , η t ) on | Ω |× | Ω | where Ω is the space of all p ossible matc hings. Definition 1. Cho ose an e dge u niformly at r andom, 1. If insertion is p ossible in b oth σ and η then add it pr ob ability λ 1+ λ and r emove it with pr ob ability 1 1+ λ . 2. If i nsertion is p ossible in one and not p ossible in the other then r emove that e dge if it is alr e ady pr e sent in one matching. Notice that here we are relying on the fact that one can ins er t an edge if it is already present in it. It is easy to v er if y that this indeed is a couplin g for the Mark ov c hain defined in the previous section. W e use th e f ollo wing distance f u nction for th e aforementioned coupling. L et σ ⊕ η = n e ∈ E | e ∈  ( σ \ η ) [ ( η \ σ ) o where σ and η ∈ Ω define the distance function Φ( σ t , η t ) = | σ ⊕ η | whic h is the n um b er of edges present in one bu t not in the other. Our ob jectiv e is to upp er-b ound E [Φ ( σ t +1 , η t +1 )] in terms of d t = Φ ( σ t , η t ). Based on the definition of our coup ling the follo w ing cases may arise once w e pic k an edge e r uniformly randomly: 1. e r can b e added to bot h σ t and η t : The sub ev ents are (a) e r w as present in b oth of them, in this case Φ( σ t +1 , η t +1 ) = d t , (b) e r is not present in b oth of them, in wh ich case again Φ( σ t +1 , η t +1 ) = d t and (c) e r is p r esen t in one b ut n ot in other, in which case Φ( σ t +1 , η t +1 ) = d t − 1. 2. e r can b e added in exactly one of σ t and η t : The sub ev en ts for this case are (a) e r is not present in b oth, wh ic h giv es Φ( σ t +1 , η t +1 ) = d t (b) e r is present in one matc hing and not in other, in whic h case Φ( σ t +1 , η t +1 ) = d t − 1 3. e r can’t b e added t o an y one of σ t and η t : In this case Φ( σ t +1 , η t +1 ) = d t . Using the ab o v e even ts we can pr ov e the follo win g: Lemma 2. E [Φ( σ t +1 , η t +1 )] = Φ( σ t , η t )  1 − 1 m  . 8 Pr o of. Sin ce Φ ( σ t +1 , η t +1 ) can only tak e tw o v alues either d or d − 1 we only need to calculate the the probabilit y of th e happ ening of one of th ese cases. This h app ens when either the even t 1(c) or the ev ent 2(b) take s p lace (as mentioned ab o v e). Th us, Pr [Φ( σ t +1 , η t +1 ) = d t − 1] = Pr [1( c ) ∪ 2( b )] clearly the d istance b eco mes d t − 1 when the c hosen edge is one of the edges in σ ⊕ η . W e can divide σ ⊕ η in to t w o sets U and ¯ U . U is the set of edges which can b e add ed to th e matc hing in wh ic h it is n ot present, and ¯ U is the set of edge which can’t b e added to the matc h ing in whic h it is n ot present. Using this notation we can write the d esired probability as Pr [1( c ) ∪ 2( b )] = X e ∈ U 1 m  λ 1 + λ + 1 1 + λ  + X e ∈ ¯ U 1 m = | σ ⊕ η | m = Φ( σ t , η t ) m Therefore, E [Φ( σ t +1 , η t +1 )] = ( d t − 1) Φ( σ t , η t ) m + d t  1 − Φ( σ t , η t ) m  = Φ( σ t , η t )  1 − 1 m  ( since d t = Φ( σ t , η t )) W e can now pro v e the follo wing: Lemma 3. T mix = O ( m log n ) . Pr o of. Give n any σ and η w e defi n e a d = Φ( σ, η ) length p ath as σ = ξ 1 , ξ 2 . . . , ξ d = η suc h that ξ i +1 is the state obtained by remo ving exactly one edge from ξ i ∈ σ T ( σ ⊕ η ) for i = 1 , 2 . . . j where ξ j consists only of edges wh ic h do not b elong to σ ⊕ η and for all k ≥ j , ξ k +1 is obtained b y add in g one edge to ξ k whic h b elongs to η T ( σ ⊕ η ). S ince ξ i , ξ i +1 = 1 f or all i = 1 , 2 . . . d − 1, w e ha v e Φ( σ, η ) = X i Φ( ξ i , ξ i +1 ) Also we can write E [Φ( σ t +1 , η t +1 )] = β Φ( σ t , η t )( with β =  1 − 1 m  ) this allo ws u s to apply th e r esult from Theorem 2 whic h giv es th e follo win g result T mix ≤ log 2 eD 1 −  1 − 1 m  ≤ m log (4 en ) = O ( m log n ) where we hav e us ed β = (1 − 1 /m ) and D = 2 n . 9 Our algorithm is concisely pr esen ted as follo w s: Input : A Graph G = ( V , E ) with | V | = n and | E | = m σ 0 ← any matching ; λ = 2 m ; for t = 0 to 10 m log n do c h o ose an edge e r uniformly r an d omly ; σ ′ = ( σ t S { e r } , with probability λ 1+ λ σ t \{ e r } , w ith probabilit y 1 1+ λ ; if σ ′ is a v alid matc hing; σ t +1 = σ ′ ; else; σ t +1 = σ t ; end return σ t Algorithm 1: R andMatc hing Theorem 3. The pr ob ability that the r andom walk in Algorithm 1 ends on a maximum match- ing is at le ast 20 189 . Pr o of. Let M i b e the set all of matc hings of size i (and S i b e its cardinalit y) in the giv en graph. Also, let k b e the s ize of th e maxim u m matc hing. W e need to fin d the p r obabilit y th at the rand om wa lk lands u p on a maxim um matc hin g after T = 10 m log n ≥ T mix steps. Let X T b e the s tate after T steps, and π T b e the p robabilit y distr ibution after T steps then b y definition of mixing time and tr iangle inequalit y ,       X ω ∈M k π T ( ω ) − X ω ∈M k G ( ω )       ≤ X ω ∈M k | π T ( ω ) − G ( ω ) | ≤ X ω ∈ Ω | π T ( ω ) − G ( ω ) | ≤ 1 e where P ω ∈M k π T ( ω ) := Pr k ( π T ) is th e probability of reac hing a maxim um m atc hing after T steps (the success pr obabilit y of th e algorithm) and P ω ∈M k G ( ω ) := Pr k ( G ) is the probabilit y of finding a maxim um matc hing according to the Gibb s distribution. W e n eed to find the probabilit y that X T is a maxim u m matc h ing. T hus we ha v e, Pr k ( π T ) ∈  Pr k ( G ) − 1 e , Pr k ( G ) + 1 e  Also, Pr k ( G ) = λ k S k k X i =0 λ i S i = λ k S k λ k S k + λ k − 1 S k − 1 + λ k − 2 S k − 2 + · · · + S 0 dividing by λ k b oth numerator and denominator 10 = S k S k + S k − 1 λ + S k − 2 λ 2 + ... + S 0 λ k W e put λ = S m where S m = k max i S i = S k S k + S k − 1 S m + S k − 2 S 2 m + · · · + S 0 S k m = 1 1 + S k − 1 S k S m + S k − 2 S k S 2 m + · · · + S 0 S k S k m By defin ition of S m , S i S m is alw ays ≤ 1, h ence we ha v e 1 1 + S k − 1 S k S m + S k − 2 S k S 2 m + · · · + S 0 S k S k m ≥ 1 1 + 1 S k + 1 S k S m + 1 S k S 2 m ... + 1 S k S k − 1 m ≥ 1 1 + 1 S k  1 + 1 S m + 1 S 2 m · · · + 1 S k − 1 m  ≥ 1 1 + 1 S k    1 −  1 S m  k 1 − 1 S m    ≥ 1 1 + 1 S k  S k m − 1 S k − 1 m ( S m − 1)  Since  S k m − 1 S k − 1 m ( S m − 1)  =  1 + S k − 1 m − 1 S k − 1 m ( S m − 1)  = 1 + Θ  1 S m  ≤ 11 10 Pr k ( G ) ≥ 1 1 + 1 11 S k 10 ≥ 10 21 This giv es us , Pr k ( π T ) ≥ 10 21 − 1 e ≥ 20 189 Notice that our pro of still go es thr ough ev en if we c h o ose a λ th at is larger than S m (w e can tak e λ = 2 m ). Therefore λ can b e r ep resen ted using m b its. W e can now prov e th e main theorem, 11 Theorem 4. Given a gr aph G = ( V , E ) with | V | = n and | E | = m , ther e exists a r andomize d algorithm that runs in O ( m log 2 n ) time and finds a maximum matching with high pr ob ability. Pr o of. Eac h step of the algorithm runs in O (1) time (w e just need to mainta in an array whic h indicates whether the i th v er tex is o ccupied in the matc hing or not), thus one call of Algorithm 1 r uns in O ( m log n ) time whic h by Th eorem 3 lands on a maximum matching w ith probabilit y 20 169 . Thus calling it 10 log n times ind ep endently , ens u res that we land on a maxim um matc hin g in one call is at least 1 −  169 189  10 log n = 1 − 1 n Ω(1) . 7 Lo wer Bound via C onductanc e The condu ctance m etho d as defined in the Section is used to obtain lo wer b ounds on the mixing time of the a Mark o v c hain. T o get a low er b ound on T r elax w e n eed an up p er-b ound on φ , and b y definition of φ , for an y cut ( S, ¯ S ). φ ≤  P i ∈ S,j ∈ ¯ S π i P ij ( P i ∈ S π i )( P i ∈ ¯ S π i )  This allo w s us to observe : Lemma 4. F or any gr aph G , the c onductanc e of our Markov Chain satisfies φ ≤ O  k m  wher e k is the size of maximum matching. Pr o of. T o giv e an upp er b ou n d on conductance we need to construct a cut ( S, ¯ S ) for whic h w e can estimate the ab o ve quant it y . Let S b e consisting of exactly one matc h ing whic h is the maxim u m matc hing m k where k is the size of the maximum matc hing. T h us the n um b er of edges going out of S is k . P i ∈ S,j ∈ ¯ S π i P ij ( P i ∈ S π i )( P i ∈ ¯ S π i ) = λ k Z · k · 1 m ( λ +1) λ k Z  1 − λ k Z  = k (1 + λ ) m · 1  1 − λ k Z  Also using the termin ology of T h eorem 3 λ k Z = λ k P k i =0 S i λ i = 1 P k i =0 S k − i 1 λ i ≤ 1 P k i =0 1 λ i = λ k ( λ − 1) λ k +1 − 1 = 1 − Θ  1 λ  Therefore, P i ∈ S,j ∈ ¯ S π i P ij ( P i ∈ S π i )( P i ∈ ¯ S π i ) = k (1 + λ ) m · 1 Θ  1 λ  = Θ  k m  Th us the resu lt f ollo ws. As a result of the p revious lemma we hav e th e follo w in g result. Lemma 5. F or any gr aph G , the mixing time of our Markov chain satisfies, T mix ≥ Ω  m k  12 Pr o of. F ollo ws from results in Section 4.3. Th us the lo wer b ound is sharp (u pto logarithmic factors) if the size of the matc h ing in the giv en graph is sm all (sa y at most O (p oly log n )). A Note on Other metho ds to pro ve Lo w er Bound : There are other metho ds, apart from conductance, which can b e used to pr o ve lo wer b ound s on mixing time. Although p o we rful and usefu l in many con texts, it is not clear wh ether such metho ds could b e applied to our chain. F or eg. the Wilson’s method [1] exp ects the k n o wledge of one eigen vecto r of the transition v ector that is differen t from the all 1’s v ector and the corresp ond ing eige n v alue lies in the range  0 , 1 2  . Sin ce the matrix P for our case is an exp onen tial sized matrix w ith apparently no usefu l pattern in the en tries, it is not clear ho w to come up with suc h an eigen vecto r. In fact, we made several edu cated guesses f or coming up w ith suc h a ve ctor all of whic h failed to serv e our pu rp ose. 8 Conclus ion In this pap er, we ga v e a new randomized algorithm for fin ding maxim u m matc hings in general bipartite graphs that runs in O ( m log 2 n ) time that imp ro ves up on the run ning time of Micali and V azirani. Our algorithm wa s b ased on the MCMC paradigm wh ic h p erforms a trun cated random walk on the Mark ov C hain defin ed by the Glaub er Dynamics with parameter λ . Ap art from th e b en efi t of b eing ve ry sim p le (b oth in analysis and implementat ion) our algorithm is the first near linear time complexit y algorithm for the maxim um matc h ing pr oblem for general graphs. Moreo ver, u nlik e [15 ] the r u nning time of our algorithm is n ot a random v ariable. T o our knowledge this is for the fi rst time Glaub er d ynamics and the nature of Gibbs distribution has b een exploited to d esign an faster algorithm for a pr oblem for w hic h efficien t solutions are already known, an d w e hop e this idea can b e of use in other p roblems as w ell. The obvio us op en problem will b e to impro v e b oth the upp er-b ounds and lo wer b ounds on the mixing time. Is it p ossible to impro v e u p on the present b ou n d to get an O ( m ) time algorithm? 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