An analysis of a random algorithm for estimating all the matchings

An analysis of a random algorithm for estimating all the matc hings Jinshan Zhang, ∗ Y an Huo, † and F engshan Bai ‡ Dep artment of Mathematic al Scienc es, Tsinghua Uni versity, 10008 4, Beijing, PRC. Count in g the n u m b er of all the matc hings on a bipartite graph has b een trans- formed in to calculating the p ermanen t of a matrix obtained from t h e extended bi- partite graph b y Y an Huo, and Rasmussen presen ts a simple app roac h (RM) to appro xim ate the p ermanen t, whic h just yields a critical ratio O( nω ( n )) for almost all the 0-1 matrices, pro vided it’s a simp le pr omising practica l w ay t o compute this #P-complete problem. In this pap er, the p erformance of this m etho d will b e sho wn when it’s applied to compute all the matc hings based on that transformation. T he critical r atio will b e pro ved to b e v ery large w ith a certain probab ility , owning an increasing f actor larger than any p olynomial of n eve n in the sense f or almost all the 0-1 matrices. Hence, RM fails to work well when coun tin g all the matc h ings via computing the p er m anen t of the matrix. In other w ords, we must carefully utilize the kn o wn method s of estimating the p ermanen t to coun t all the matc hings through that transf ormation. Keywords: matc hing; permanent; critical ratio; bipar tite gr a ph; determinant; Monte-Carlo algorithm;r a ndom algor ithm; RM;fpras ∗ Electronic address: zjs0 2 @mails.tsinghua.edu.cn † Electronic a ddr ess: huo y03@ mails.tsinghua.edu.cn ‡ Electronic a ddr ess: fbai@math.tsinghua.edu.cn 2 I. INTR O DUCTION Let G = ( V , E ) b e a bipartite graph, where V = V 1 ∪ V 2 is the set of v ertices and E ⊂ V 1 × V 2 is the set of edges. In the followin g sections w e supp ose # V 1 = # V 2 = n if there’s no special illustration. A set of edges S ⊂ E is called a matc hing if no t w o distinct edges e 1 , e 2 ∈ S con tain a common vertex . S is called a k-match ing if # S = k . In sp ecial case, S is called a perfect matc hing if k = n . Let S k b e the set of k-matching in G and A ( G ) b e the set of all t he k-matc hing, k = 0 , 1 , . . . , n . F or the con venie nce of discussion, let # S 0 = 1, then the n umber of a ll the matc hings in G is # A ( G ) = n P i =0 # S k . The p ermanent of a 0-1 A = a ij , 1 ≤ i, j ≤ n is define d as P er ( A ) = X π n Y i =1 a i,π ( i ) (1) where the sum is o ve r all the p erm utations π of [ n ] = { 1 , . . . , n } . It’s w ell kno wn that the p ermanen t of an adjacen t matr ix of bipartite graph equals the num b er of its p erfect matc hing. Let AM(G) denote the n um b er o f all the matc hings in G , and A be adjacent matrix of G . [8] has pro ved that AM ( G ) = 1 n ! per   A I n × n 1 n × n 1 n × n   (2) where I n × n is n × n unit matrix, 1 n × n denotes n × n matrix with a ll the elemen ts 1. This means in order to coun t the n um b er o f all the matc hings of a bipartite graph w ith 2 n v ertices w e only need to compute the p ermanen t of a 2 n × 2 n correspo nding matr ix transformed from adjacen t matrix. The computation of p ermanen t has a lo ng history and w as sho wn to b e #P-complete in [2]. Th us, in t he past 20 y ears or so, many random alg o rithms ha ve b een deve lo p ed to approximate the p ermanen t, whic h can b een divided at least four categories[3]: elemen tary recursiv e algorithms(the original one is Rasm ussen metho d(RM)) [4]; reductions to determinan ts [5, 7, 9, 11]; iterative balancing [12]; a nd Mark o v c hain Mon te Carlo [13, 1 6, 19]. All these metho ds try to find a fully-p olynomial randomized appro ximation sc heme fpr as for computing the p ermanen t. fpr as is suc h a sche me whic h, when giv en ε and inputs mat r ix A , outputs a estimator(usually a un biased estimator) Y of the p ermanen t suc h that P r ((1 − ε ) per ( A ) ≤ Y ≤ (1 + ε ) per ( A )) ≥ 3 4 (3) 3 and runs in p olynomial time in n and ε − 1 , here 3 / 4 may b e b o osted to 1 − δ fo r an y desired δ > 0 b y running t he algo r ithm O ( l og ( δ − 1 )) and taking the median of the trials [10]. Then a straigh tf orw ard a pplication of Cheb yc hev’s inequalit y shows that r unning the algorithm O ( E ( Y 2 ) E 2 ( Y ) ε − 2 ) times and ta king the mean of the results can mak e the pro babilit y more than 3 / 4(e.g. running 4 E ( Y 2 ) E 2 ( Y ) ε − 2 times). Hence, if the critical ra tio E ( Y 2 ) E 2 ( Y ) is b ounded b y a p olynomial of inputs A , we’ll get an fpr as for the p ermanent of A . Another mo dified sc heme called fpr as for almost all inputs means: c ho ose a matrix fr o m A ( n, 1 / 2)( A ( n, 1 / 2 ) denotes a proba bilit y space of n × n 0-1 matrices where each entry is c ho sen to b e 1 or 0 with the same probabilit y 1 / 2), or equiv a lently c ho ose a matrix u.a.r. from A ( n ) ( A ( n ) represen ts the set of n × n 0-1 matrices), and the f ollo wing Pr(critical ra tio of A is b ounded by a p olynomial o f the input A )= 1 − o (1) as n − → ∞ holds.(Note that this is a muc h w eak er requiremen t than that of an fpr as ). If a prop osition P relating to n satisfies Pr(P is true)= 1 − o ( n ), w e sa y P holds whp ( whp is the abbreviation of ”with high probabilit y”). Thus, tha t there is an fpr as for almost all the mat rix means the critical ratio of A is b ounded b y a p olynomial of the input A whp . A exciting result, that Mark ov Chain approach led to the first fp r as for the p ermanen t of an y 0- 1 matrix(actually of an y matrix with nonnegativ e en try) w as sho wn by [1 6]. Ho we v er, it s high exp onen t of p olynomial running time mak es it difficult to b e a practical metho d to approximate the p ermanen t. RM and reductions to determinan t s seem to b e t w o practical a ppro ac hes esti- mating p ermanent due to their simply feasibility , and b oth of t hem hav e b een prov ed to b e an fpr as for almost all t he 0-1 matr ices. b eside s, [3] promises a go o d prosp ect on computing p ermanen t via clifford algebra if some difficulties can b e conquered. RM also has dev elop ed to b e a kind o f approa ches called sequen tial imp ortance sampling wa y , whic h is widely used in statistical ph ysics, see[14]. In this pap er, we ’ll, b y RM, compute the num ber of all the matchings based o n the ab ov e t r a nsformation and give its p erforma nce t heoretically , say , an analysis of critical ratio in the sense ”for almost all the 0-1 matrix” of t ha t matrix with a sp ecial structure. In section II, A new alternative estimator opera t ing directly on the adjacent matrix without an y transformation will b e presen ted and pro v ed to b e equiv alent to approximation p erforming on the transformed matrix b y RM. In section I I I, a low b ound of the critical ratio for almost all the matrices will b e presen ted, whic h is lar g er than any polynomial of n with a certain probabilit y . Hence, RM do es not p erform we ll in computing the n umber of all the matc hings 4 as in computing the n umber of p erfect matc hing. In section IV we ’ll prop ose some analytic results w.r.t. t he exp ectation and v ariance of the num ber o f all the matc hings of a matrix selected u.a.r from G ( m, n )( G ( m, n ) denotes the set of bipartite graph with # V 1 = # V 2 = n as its v ertices and exact m edges). These results seem likely to con tribute to the upp er b ound of critical ratio for almost all matrices, but the calculations are more arduous and will b e left for latter pap er. I I. AN EQ UIV ALENT ESTIMA TOR All the notations ha v e the same meanings as those in the previous section without sp ecial illustration. Let A an n × n 0-1 matrix b e an adjacen t matrix of a bipartit e graph G = ( V , E ), ( V = V 1 S V 2 ). Set Y A a random v ariable. Then RM can b e stated as follo ws: inputs: A an n × n 0-1 matrix; outputs: Y A the estimator of p ermanent A; if n=0; then Y A = 1 else W = { j : a 1 j = 1 } if W = ∅ then Y A = 0 else Cho ose J u.a.r. from W Y A = | W | Y 1 J Y 1 j denotes the submatrix obtained from A b y remo ving the 1st ro w a nd the jth column. Note t his heuristic idea comes from the Laplace’s expansion. Our follo wing algorithm(for easy discussion, call it AMM) is a lso inspired by another expans io n. we first presen ts our algorithm fo r the n umber of all the matc hings, and then giv e the explanation and pro of of equiv alence betw een AMM and R M on the transformed matrix: inputs: A an n × n 0-1 adjacen t matrix of G ; 5 outputs: Y A the estimator of the n um b er of all the matc hings o f G ; if n=0; then Y A = 1 else W = { j : a 1 j = 1 } S { 0 } Cho ose J u.a.r. from W Y A = | W | Y 1 J Y 10 denotes a submatrix of A by remo ving the 1 st row (of course, it’s not necessarily a square matrix). Define a new terminology AM on t he matrix. let B = { b ij , 1 ≤ i ≤ m, 1 ≤ j ≤ n } an m × n matrix, m ≤ n . let AM ( ∅ ) = 1, by induction on m . AM ( B ) := AM ( B 10 ) + n X j =1 b 1 ,j B 1 j (4) Then we hav e the followin g theorem. Theorem 1. Let A b e an n × n adjacen t mat rix of a bipartite graph G, Then AM(A)is t he n um b er of all the matc hings of G . Pro of: It’s easy to ch ec k, when k ≥ 1, the num b er of k-matc hing of G equals P i 1 , ··· ,i k P π a i 1 ,π ( i 1 ) · · · a i k ,π ( i k ) , where i 1 < i 2 · · · < i k c hosen f r o m { 1 , 2 , · · · , n } , π de- notes the p ermu tation of { i 1 , i 2 , · · · , i k } . Th us, the n umber of all the matc hings is n P k =1 P i 1 < ··· 0, a nd in our results p = 1 2 − ε where ε is no more than 0.02. T o pro ve the theorem, w e need the follo wing lemma, whic h will be pro v ed in section IV. Lemma2 Let B ( m, n ) denote the set of all n × n 0-1 matrices with exact m 1’s, m ≫ n . Cho ose B u.a.r. f r om B ( m, n ). Then E ( AM ( B )) = n X k =0 ( C k n ) 2 k ! C m − k n 2 − k C m n 2 and E ( AM 2 ( B )) E 2 ( AM ( B )) = 1 + o (1) , n → ∞ Theorem7 Cho o se A n,n u.a.r. from A ( n, n ), and let X A n,n b e the output b y AMM. Then P r ( E σ ( X 2 A n,n ) E 2 σ ( X A n,n ) ≥ n √ n/ 2 − 1 ) ≥ n 2 P i =(1 / 2+ ε ) n 2 C k n 2 2 n 2 where c is a constan t no more 10 , and ε ≤ 0 . 02. Pro of: F rom lemma2 w e kno w if w e set m = (1 / 2 + ε ) n 2 and q = C m − k n 2 − k C m n 2 . When n go es to infinit y , noting k ≤ n ≪ m, n 2 , there holds q = C m − k n 2 − k C m n 2 = m ( m − 1) · · · ( m − k ) n 2 ( n 2 − 1) · · · ( n 2 − k ) and ln ( q ) = k − 1 X i =0 [ ln ( m − i ) − ln ( n 2 − i )] = k l n ( m n 2 ) + k − 1 X i =0 [ ln (1 − i m ) − l n (1 − i n 2 )] = k l n ( m n 2 ) − k − 1 X i =0 [ i m − i n 2 + O ( i 2 m 2 )] = k l n ( m n 2 ) − k ( k − 1) 2 ( 1 m − 1 n 2 ) + O ( k 3 m 2 ) 13 Th us, noting tha t k m − 1 ≤ 2 nm − 1 = O ( n 3 m − 2 ) q = ( m n 2 ) k exp [ − k 2 2 ( 1 m − 1 n 2 ) + O ( n 3 m 2 )] = ( (1 / 2 + ε ) n 2 n 2 ) k exp [ − k 2 2 ( 1 (1 / 2 + ε ) n 2 − 1 n 2 ) + O ( n 3 ((1 / 2 + ε ) n 2 ) 2 )] ≤ e − 1 (1 / 2 + ε ) k Let B sele cted u.a.r. fro m B ( m, n ) Since E ( AM 2 ( B )) E 2 ( AM ( B )) = 1 + o (1), as n → ∞ then P r ( AM ( B ) < 5 6 E ( AM ( B ))) → 0 , a s n → ∞ . So, if m ≥ (1 / 2 + ε ) n 2 and ε ≤ 0 . 0 2, w e ha ve whp E σ ( X 2 B ) ≥ E 2 σ ( X B ) = AM 2 ( B ) ≥ ( 5 6 E ( AM ( B ))) 2 = ( 5 6 n X k =0 ( C k n ) 2 k ! C m − k n 2 − k C m n 2 ) 2 ≥ ( n X k =0 ( C k n ) 2 k ! 5 e − 1 6 (1 / 2 + ε ) k ) 2 ≥ n X k =0 P k n P k n +3 2 n + k X s 0 + s 1 + ··· + s k = n − k s 0 , ··· s k ≥ 0 ( n + 2) s 0 ( n + 2 − 1) s 1 · · · ( n + 2 − k ) s k = E A ( E σ ( X A n,n )) . Noting P r ( A ∈ S m ≥ (1 / 2+ ε ) n 2 B ( m, n )) = n 2 P i =(1 / 2+ ε ) n 2 C k n 2 2 n 2 , th us P r ( E σ ( X 2 A n,n ) ≥ E A ( E σ ( X A n,n ))) ≥ n 2 P i =(1 / 2+ ε ) n 2 C k n 2 2 n 2 . Using Marko v’s inequalit y , P r ( E σ ( X 2 A n,n ) ≥ nE A ( E σ ( X A n,n ))) ≤ 1 n → 0 then whp E σ ( X 2 A n,n ) ≤ nE A ( E σ ( X A n,n )). F inally , we ha ve P r ( E σ ( X 2 A n,n ) E 2 σ ( X A n,n ) ≥ 1 n E A ( E σ ( X 2 A n,n )) E A ( E σ ( X A n,n )) ) ≥ n 2 P i =(1 / 2+ ε ) n 2 C k n 2 2 n 2 14 Apply theorem6 to the ab o v e form ula, w e ha v e P r ( E σ ( X 2 A n,n ) E 2 σ ( X A n,n ) ≥ n √ n/ 2 − 1 ) ≥ n 2 P i =(1 / 2+ ε ) n 2 C k n 2 2 n 2 IV. THE NUMBER O F ALL THE MA TCHINGS ON RANDOM GRAPH. In this section, w e consider the exp ectation and v ariance of the n um b er o f all the matc hings on G sele cted u.a.r. from G ( m, n ). W e hav e the follow ing theorem. Theorem8 Cho ose G u.a.r. from G ( m, n ), where G ( m, n ) denotes the set of bipar- tite graph with # V 1 = # V 2 = n as its v ertices and exact m edges, m ≫ n , and let AM(G ) denotes the n um b er of all the matc hings in G . Then w e ha v e E ( AM ( G )) = n X k =0 ( C k n ) 2 k ! E ( X M ( k ) ) and E ( AM 2 ( G )) = n X k =0 k X i =0 ( C k n ) 2 k ! min ( i,n − k ) X p =0 C p n − k C i − p k P p n − i + p i − p X j =0 C j i − p [ F n − j ( i − p − j )] E ( X M ( k + i − j ) ) + n X k =1 k − 1 X i =0 ( C k n ) 2 k ! min ( i,n − k ) X p =0 C p n − k C i − p k P p n − i + p i − p X j =0 C j i − p [ F n − j ( i − p − j )] E ( X M ( k + i − j ) ) where E ( X M ( k ) ) = C m − k n 2 − k /C m n 2 and F n ( p ) = p P r =0 ( − 1) r C r p P p − r n − r Pro of: w e’ll use the metho dolo gy in [6]; Let M ( k ) b e a k-mat c hing on V 1 + V 2 , F or G ∈ G ( m, n ), define the random v ariable X M ( G ) to b e 1 if M ( k ) is contained in G, and otherwise 0. The expectation and second momen t of AM ( G ) is as follo ws. E ( AM ( G )) = E ( n X k =0 X M ( k ) X M ( k ) ) = n X k =0 X M ( k ) E ( X M ( k ) ) and E ( AM 2 ( G )) = E (( n X k =0 X M ( k ) X M ( k ) ) 2 ) = n X k =0 n X i =0 X M ( k ) ,M ′ ( i ) E ( X M ( k ) X ′ M ( i ) ) where ∀ 0 ≤ k ≤ n , M ( k ) and M ′ ( k ) ra ng e o ve r all ( C k n ) 2 k ! k-matc hing’s on V 1 + V 2 . Not e that E ( X M ( k ) ) = C m − k n 2 − k C m n 2 15 The first equation follo ws quic kly . F o r the second, in order t o compute E ( X M ( k ) X ′ M ( i ) ), w e hav e to calculate the nu m b er of pa irs o f M ( k ) and M ′ ( i ) as a function of the ov erlap j = | M ( k ) T M ′ ( i ) | . F or any fixed k , suppose i ≤ k , we need to compute the num b er of the pairs o f M ( k ) and M ′ ( i ), where i = 0 , · · · , k , and M ′ ( i ) r a nges ov er all ( C i n ) 2 i ! i -mat c hing’s on V 1 + V 2 . The problem can b e equiv alen tly stated as follows : There’re n differen t letters and n differen t en v elop es. Among these letters, t here’re exact k (0 ≤ k ≤ n ) la b eled letters, eac h of whic h has only one ’mother en v elop e’ a mong en v elop es. Different lab eled letters ha ve differen t mot her env elop es. W e call a j -fit if there’re exact j lab eled letters put into its own mother en v elop e. Now c ho ose i (0 ≤ i ≤ k )letters from these n letters, then put them into i env elopes, and each letter can only b e put in to one en v elop e. ∀ p ossible j , ho w many circumstances o f j -fit are there? W e can solv e this problem like this: Supp ose there’re p letters unlab eled and i − p lab eled letters a mong the selected letters, ob viously , 0 ≤ p ≤ min ( n − k , i ), the num b er of w ay s of c ho osing letters is C p n − k C i − p k . If the lab eled letters has b een laid, then the num b er of the w ay s of putting p unlab eled letters is P p n − ( i − p ) . F or any j (0 ≤ j ≤ i − p ), there’re C j i − p w ay s putting exact j lab eled letters in its o wn mother en v elop e. The last one w e need to deal with is ho w man y w a ys to put i − p − j lab eled letters in to n − j env elop es whic h contain all these i − p − j letters’ mother en ve lop es, satisfying 0-fit. By the principle of inclus ion-exclusion see[1], w e can easily obtain the n um b er of the w ay s is F n − j ( i − p − j ), where F n ( p ) = p P r =0 ( − 1) r C r p P p − r n − r . Noting that p ranges ov er 0 to min ( i, n − k ), and j ranges ov er 0 to i − p , for eac h k and i ≤ k . Then X M ′ ( i ) E ( X M ( k ) X ′ M ( i ) ) = min ( i,n − k ) X p =0 C p n − k C i − p k P p n − i + p i − p X j =0 C j i − p [ F n − j ( i − p − j )] E ( X M ( k + i − j ) ) 16 where E ( X M ( k ) ) = C m − k n 2 − k /C m n 2 and F n ( p ) = p P r =0 ( − 1) r C r p P p − r n − r . Consider, n X k =0 n X i =0 X M ( k ) ,M ′ ( i ) E ( X M ( k ) X ′ M ( i ) ) = ( n X k =0 k X i =0 + n − 1 X k =0 n X i = k +1 ) X M ( k ) ,M ′ ( i ) E ( X M ( k ) X ′ M ( i ) ) = ( n X k =0 k X i =0 + n − 1 X k =0 n X i = k +1 ) X M ( k ) ,M ′ ( i ) E ( X M ( k ) X ′ M ( i ) ) = ( n X k =0 k X i =0 + n X i =1 i − 1 X k =0 ) X M ( k ) ,M ′ ( i ) E ( X M ( k ) X ′ M ( i ) ) = ( n X k =0 k X i =0 + n X k =1 k − 1 X i =0 ) X M ( k ) ,M ′ ( i ) E ( X M ( k ) X ′ M ( i ) ) = ( n X k =0 k X i =0 + n X k =1 k − 1 X i =0 ) X M ( k ) X M ′ ( i ) E ( X M ( k ) X ′ M ( i ) ) = ( n X k =0 ( C k n ) 2 k ! k X i =0 + n X k =1 ( C k n ) 2 k ! k − 1 X i =0 ) X M ′ ( i ) E ( X M ( k ) X ′ M ( i ) ) Replace P M ′ ( i ) E ( X M ( k ) X ′ M ( i ) ) b y min ( i,n − k ) P p =0 C p n − k C i − p k P p n − i + p i − p P j =0 C j i − p [ F n − j ( i − p − j )] E ( X M ( k + i − j ) ), then the second equation is ac hiev ed.  Remark: T o complete t he pro of of t heorem7, w e also need to know whether the ratio E ( AM 2 ( G )) E 2 ( AM ( G )) go es to 1 as n g o es to infinit y , a dding the condition suc h as m 2 n − 3 → ∞ as n → ∞ . 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