On the graph isomorphism problem

On the graph isomorphism problem Shm uel F riedland ∗ † Jan uary 9, 2008 Abstract W e relate the graph isomor phism problem to the solv a bility of certain sys- tems of linear equations and linear inequalities. The num b er of these equations and inequa lities is related to the complexity of the gra phs isomorphism and subgraph isomorphim pr o blems. 2000 Mathematics Sub ject Classification: 03D15, 05C50, 0 5C60, 15A48, 1 5A51, 15A69, 90C05. Keywords and phra s es: graph isomorphism, subgraph isomorphism, tensor pro ducts, doubly s to chastic matrices, ellipso ida l a lgorithm. 1 In tro duction Let G 1 = ( V , E 1 ) , G 2 = ( V , E 2 ) b e t w o simple und irected graphs, where V is the set of v ertices of cardinalit y n and E 1 , E 2 ⊂ V × V the set of edges. G 1 and G 2 are called isomorp hic if there exists a bijectio n σ : V → V whic h ind uces the corresp onding bijection ˜ σ : E 1 → E 2 . T he graph isomorphism problem, abbr eviated h ere as GIP , is the problem of determination if G 1 and G 2 are isomorphic. Clearly the GIP in the class NP . It is one of a very small n umb er of p roblems whose complexit y is un kno wn [4, 6 ]. F or certai n graphs it is kno wn that the complexit y of GIP is p olynomial [1, 2, 3, 5, 9, 10]. Let G 3 = ( W , E 3 ), where # W = m ≤ n . G 3 is cal led isomorphic to a subgraph of G 2 if there exits an injection τ : V 3 → V 2 whic h induces an injection ˜ τ : E 3 → E 2 . The subgraph isomorphism , abbreviated here as SGIP , is the problem of determi- nation if G 3 is isomorphic to a sub graph of G 2 . It is wel l kno wn that SGIP is NP-Complete [4]. In the previous ve rsions of th is pap er we related the graph iso morph ism problem to the solv abilit y of certain systems of linear equations and linear inequalities. It w as p oin ted out to me by N. Alon and L . Babai, that m y approac h relates in a similar w a y the SGIP to the solv abilit y of certain systems of linea r equations and linear inequalities. Hence f ( n ), the num b er of these linear equalitie s and inequ alities for V = n , is probably exp onen tial in n . Th us, the suggested approac h in this pap er do es not seem to b e the righ t approac h to d etermine the complexit y of the ∗ Department of Mathematics, Statistics, and Computer Science, Universit y of I llinois at Chicago Chicago, Illinois 60607-704 5, USA , E-mai l: friedlan@u ic.edu † Visiting Professor, Berlin Mathematical Sc ho ol, Institut f ¨ ur Mathematik, T echnisc he Univer- sit¨ at Berlin, Strasse des 17. Juni 136, D-10623 Berlin, Germany 1 GIP . Nev ertheless, in this pap er w e summarize the main ideas and results of this approac h. It seems that our approac h is r elated to the ideas and results discussed in [11]. Let Ω n ⊂ R n × n + b e the conv ex set of n × n doubly sto c hastic m atrices. In this pap er w e relate the complexit y of the GIP to the min imal num b er of su p p orting h yp erplanes determining a ce rtain con v ex p olytope Ψ n,n ⊂ Ω n 2 . More precisely , t wo graph are isomorphic if ce rtain system of n 2 h yp erplanes in tersect Ψ n,n . More gen- eral, if the corresp onding system n 2 half spaces intersect Ψ n,n then G 3 is isomorphic to a s u bgraph of G 2 . Hence th e minimal num b er of supp orting hyp erplanes defining Ψ n,n , denoted b y f ( n ), is clo sely relat ed to the complexit y o f S GI P . W e giv e a larger p olytope Φ n,n , charact erized b y (4 n − 1) n 2 linear equations in n 4 nonnegativ e v ariables satisfying Ψ n,n ⊂ Φ n,n ⊂ Ω n 2 . (1.1) In the first version of this p ap er we err one ously claime d that Φ n,n = Ψ n,n . T he error in m y pro of w as p oin ted out to me b y Babai, Melk eb eek, Rosen b erg and V a v asis. The inequalit y Ψ n,n $ Φ n,n for n ≥ 4 is implied b y the example of J. Rosen b erg. Th us if t wo graphs are isomorphic then certain system of n 2 h yp erplanes in tersect Φ n,n . This of co ur se yields a n ecessary conditions for GIP and SGIP . W e now outline the main ideas of the pap er. Let A, B b e n × n adjacency matrices of G 1 , G 2 . So A, B are 0 − 1 symmetric matrices with zero diagonal. It is enough to consider the case where A and B h a v e the same num b er of 1’s. Let P n b e the set of n × n p erm utation matrices. Then G 1 and G 2 are isomorphic if and only if P AP ⊤ = B for some P ∈ P n . It is easy to see that this condition is equiv alent P ( A + 2 I n ) Q ⊤ = B + 2 I n for some P , Q ∈ P n , (1.2) where I n is the n × n iden tit y matrix. F or C, D ∈ R n × n denote by C ⊗ D ∈ R n 2 × n 2 the Kronec ker pro du ct, see § 2. Let P n ⊗ P n := { P ⊗ Q, P , Q ∈ P n } . Denote by Ψ n,n ⊂ R n 2 × n 2 + the conv ex set spanned by P n ⊗ P n . Ψ n,n is a su bset of n 2 × n 2 doubly stoc hastic matrices. Then th e condition (1.2) implies the solv abilit y of th e system of n 2 equations of the form Z ( \ A + 2 I n ) = \ B + 2 I n for some Z ∈ Ψ n,n . Here \ B + 2 I n ∈ R n 2 is a column vect or comp osed of the columns of B + 2 I n . Vice v ersa, the solv abilit y of Z ( \ A + 2 I n ) = \ B + 2 I n for some Z ∈ Ψ n,n implies (1.2). The ellipsoi d algo rithm in linear p rogramming [8, 7] yields that the existence a solution to this system of equati ons is determined in p olynomial time in max( f ( n ) , n ). S imilarly , for the SGIP one needs to consider the the solv ab ilit y of Z ( \ C + 2 n 2 I n ) ≤ \ B + 2 n 2 I n for some Z ∈ Ψ n,n , where C is the adjacency matrix of the graph ˜ G 3 = ( V , E 3 ) obtained from G 3 b y app ending n − m isolate d vertic es. W e now survey br iefly the con ten ts of this paper. In § 2 w e in tro duce the needed concepts from linear algebra to giv e the c haracterizatio n of Φ n,n in terms of (4 n − 2) n 2 linear equations in n 4 nonnegativ e v ariables. This is done for the general set Φ m,n , whic h con tains Ψ m,n , the con v ex h ull of P m ⊗ P n . § 3 d iscus s es the p ermutati onal similarit y of A, B ∈ R n × n and p erm utational equiv alence of A, B ∈ R n × m . W e sho w the second main result that the p ermuta tional similarit y and equiv alence is equiv alent to solv abilit y of the corresp onding system of equations discuss ed ab ov e. In § 4 w e deduce the complexit y results claimed in this pap er. 2 This pap er generated a lot of interest. I would lik e to thank a ll the p eople who sen t their commen ts to me. 2 T ensor pro ducts of doubly sto c hastic matrices F or m ∈ N denote h m i := { 1 , . . . , m } . F or A ⊂ R denote b y A m × n the set of m × n matrices A = [ a ij ] m,n i,j =1 suc h that eac h a ij ∈ A . Recall th at A = [ a ij ] ∈ R m × m + is called doubly sto c hastic if m X j =1 a ij = m X j =1 a j i = 1 , i = 1 , . . . , m . (2.1) Since th e su m of all rows of A is equal to the sum of all columns of A it follo ws that at most 2 m − 1 of ab o v e equations are linearly indep enden t. It is well kno wn that an y 2 m − 1 of the ab o ve equ ation are linearly indep endent. L et 1 := (1 , . . . , 1) ⊤ ∈ R m + . Note th at A = [ a ij ] ∈ R m × m satisfies (2.1) if and only if and A 1 = A ⊤ 1 = 1 . Denote b y Ω m the set of doub ly sto c hastic matrices. Clearly , Ω m is a conv ex compact set. Birkhoff theorem claims that the set of the extreme p oints of Ω m is the set of p erm utations matrices P m ⊂ { 0 , 1 } m × m . Lemma 2.1 De note by Λ m ⊂ R m × m + the set of nonne gative matric es satisfying the c onditions A 1 = A ⊤ 1 = a 1 for some a ≥ 0 dep ending on A . Then Λ m is a multiplic ative c one: Λ m + Λ m = Λ m , a Λ m ⊂ Λ m for al l a ≥ 0 , Λ m · Λ m = Λ m . F urthermo r e, A = [ a ij ] ∈ R m × m + is in Λ m if and only if the fol lowing 2( m − 1) e q u alities hold. m X j =1 a ij = m X j =1 a j i = m X j =1 a 1 j for i = 2 , . . . , m. (2.2) Pro of. Th e fact that Λ m is a cone is straigh tforw ard. Since I m ∈ Λ m w e deduce the equalit y Λ m · Λ m = Λ m . Observe next that the conditions (2.2) imply that A 1 = a 1 , where a is the sum of the elemen ts in the fir st row. Also the su m of the elemen ts in eac h column except the fir st is equal to a . Since the su m of all elemen ts of A is ma it follo ws that the su m of th e elements in the fir st column is also a , i.e A ⊤ 1 = a 1 . ✷ F or A = [ a ij ] ∈ R m × m , B = [ b k l ] ∈ R n × n denote by A ⊗ B ∈ R mn × mn the tensor pro duct of A and B . The ro ws and columns of A ⊗ B are indexed by double ind ices ( i, k ) and ( j, l ), where i, j = 1 , . . . m, k, l = 1 , . . . , n . T h us A ⊗ B = [ c ( i,k )( j,l ) ] ∈ R mn × mn , (2.3) where c ( i,k )( j,l ) = a ij b k l for i, j = 1 , . . . , m, k, l = 1 , . . . , n. If w e arrange the in d ices ( i, k ) in the lexicographic order then A ⊗ B h as the follo wing blo c k matrix form called the Kr one cker pr o duct 3 A ⊗ B =      a 11 B a 12 B . . . a 1 m B a 21 B a 22 B . . . a 2 m B . . . . . . . . . . . . a m 1 B a m 2 B . . . a mm B      . (2.4) F or simp licit y of the exp osition w e will iden tify A ⊗ B with the blo ck matrix (2. 4 ) unless stated otherwise. N ote that an y other orderin g of h m i × h n i ind uces a dif- feren t represent ation of A ⊗ B as C ∈ R mn × mn , where C = P ( A ⊗ B ) P ⊤ for some p ermuta tion matrix P ∈ P mn . Recall that A ⊗ B is bilinear in A and B . F u r thermore ( A ⊗ B )( C ⊗ D ) = ( AC ) ⊗ ( B D ) for all A, C ∈ R m × m , B , D ∈ R n × n . (2.5) Prop osition 2.2 L et A ∈ Ω m , B ∈ Ω n . Then A ⊗ B ∈ Ω mn . Pro of. Clearly A ⊗ B is a nonn egativ e matrix. Assu me the r epresen tation (2.3). Then m,n X j,l =1 c ( i,k )( j,l ) = m,n X j,l =1 a ij b k l = ( m X j =1 a ij )( n X l =1 b k l ) = 1 · 1 = 1 , m,n X j,l =1 c ( j,l )( i,k ) = m,n X j,l =1 a j i b lk = ( m X j =1 a j i )( n X l =1 b lk ) = 1 · 1 = 1 . ✷ Lemma 2.3 De note by Ψ m,n ⊂ Ω mn the c onvex hul l sp anne d by Ω m ⊗ Ω n , i.e. al l doubly sto c hastic matric es of the form A ⊗ B , wher e A ∈ Ω m , B ∈ Ω n . Th en the extr eme p oints of Ψ m,n is the set P m ⊗ P n , i.e. e ach extr eme p oint is of the form P ⊗ Q , wher e P ∈ P m , Q ∈ P n . Pro of. Use Birkhoff ’s theorem and the bilinearit y of A ⊗ B to deduce that Ψ m,n is sp anned by P m ⊗ P n . Clearly P m ⊗ P n ⊂ P mn . Since Birkhoff ’s theorem implies that P mn are extreme p oin ts of Ω mn it follo ws that P m ⊗ P n ⊂ P mn are con v exly indep end en t. ✷ Theorem 2.4 L et Φ m,n b e the c onvex set of mn × mn non ne gative matric es char acterize d by 2 mn + (2 n − 2) m 2 + (2 m − 2) n 2 line ar e quations of the fol lowing form. View C ∈ R mn × mn as a matrix with entries c ( i,k )( j,l ) wher e i, j = 1 , . . . , m, k , l = 1 , . . . , n . Then C ∈ R mn × mn + b e longs to Φ m,n if the fol lowing e qualities hold. 4 m,n X j,l =1 c ( i,k ) , ( j,l ) = m,n X j,l =1 c ( j,l )( i,k ) = 1 , i = 1 , . . . , m, k = 1 , . . . , n, (2.6) m X j =1 c ( i,k )( j,l ) = m X j =1 c (1 ,k )( j,l ) , m X j =1 c ( j,k )( i,l ) = m X j =1 c (1 ,k )( j,l ) (2.7) wher e i = 2 , . . . , m and k , l = 1 , . . . , n, n X l =1 c ( i,k )( j,l ) = n X l =1 c ( i, 1)( j,l ) , n X l =1 c ( i,l )( j,k ) = n X l =1 c ( i, 1)( j,l ) (2.8) wher e k = 2 , . . . , n and i, j = 1 , . . . , m. F urthermo r e Ψ m,n ⊂ Φ m,n ⊂ Ω mn (2.9) Pro of. The conditions (2.6) state that C ∈ Ω mn . W e no w sh o w the condi- tions Ψ m,n ⊆ Φ m,n . Let A ∈ Ω m , B ∈ Ω n and consider the Kronec k er pro du ct (2.4). Then for i, j ∈ h m i , the ( i, j ) b lo c k of A ⊗ B is a ij B ∈ Λ n . Since Λ n is a cone, it follo ws that for an y C ∈ Ψ m,n , ha ving the blo c k form C = [ C ij ] , C ij ∈ R n × n + , i, j ∈ h m i , ea c h C ij ∈ Λ n . Lemma 2.1 yields the conditions for eac h i, j ∈ h m i . Since A ⊗ B = P ( B ⊗ A ) P ⊤ w e also deduce the conditions (2 .7 ) for eac h k, l ∈ h n i . ✷ Lemma 2.5 Ψ 2 , 2 = Φ 2 , 2 . Pro of. Let D = [ d pq ] 4 p,q =1 ∈ Φ 2 , 2 . S ince F 11 :=  d 11 d 12 d 21 d 22  , F 12 :=  d 13 d 14 d 23 d 24  ∈ Λ 2 it follo ws that d 11 = d 22 = a, d 12 = d 21 = b, d 13 = d 24 = c, d 14 = d 23 = d. Since G 11 :=  d 11 d 13 d 31 d 33  , G 12 :=  d 12 d 14 d 32 d 34  ∈ Λ 2 it follo ws that d 31 = c, d 32 = d, d 33 = a, d 34 = b. Since F 21 :=  d 31 d 32 d 41 d 42  , F 22 :=  d 33 d 34 d 43 d 44  ∈ Λ 2 it follo ws that d 41 = d, d 42 = c, d 43 = b, d 44 = b. So a, b, c, d ≥ 0 and a + b + c + d = 1. This set has 4 extreme p oint s which form the set P 2 ⊗ P 2 . ✷ The follo wing result was comm unicated to me by J. Rosen b erg. Recall that P ∈ P n is called a cyclic p erm utation if P n i =1 P i is a matrix whose all en tries are equal to 1. 5 Lemma 2.6 L et P , Q ∈ P n b e c yclic p ermutations. Then the blo ck matrix D = 1 n [ P i Q j ] n i,j =1 b e longs to Φ n,n . If P 6 = Q i for i = 1 , . . . , n − 1 then D 6∈ Ψ n,n . In p articular Ψ n,n $ Φ n,n for n ≥ 4 . F or n = 3 e ach D of the ab ove form is in Ψ 3 , 3 . Pro of. Since P i , Q j ∈ Ω n it follo ws that P i Q j ∈ Ω n for i, j = 1 , . . . , n . Hence the conditions (2.8) and (2.6 ) are s atisfied. It is left to s h o w the conditions (2.7). Denot e A i = [ a ( i ) k p ] n k ,p =1 , B j = [ b ( j ) pl ] n p,l =1 ∈ Ω n . View D = [ c ( i,k )( j,l ) ]. Then c ( i,k )( j,l ) = 1 n n X p =1 a ( i ) k p b ( j ) pl , i, j, k , l = 1 , . . . , n. (2.10) Since P n j =1 b ( j ) pl = 1 for p, l = 1 , . . . , n and A i ∈ Ω n w e obtain 1 n P j =1 c ( i,k )( j,l ) = 1 n P n p =1 a ( i ) k p = 1 n . In a similar wa y we d educe that P n j =1 c ( j,k )( i,l ) = 1 n . S o D ∈ Φ n,n . Supp ose that D ∈ Ψ n,n . Obs erve that P n Q n = I n I n = I n . Assume D as a con v ex com bination of some extreme p oin ts U ⊗ V ∈ P n ⊗ P n with p ositiv e co efficients. Express U ⊗ V as a blo c k matrix [( U ⊗ V ) ij ] n i,j =1 . Sup p ose fur thermore that ( U ⊗ V ) nn 6 = 0 n × n . Then V = I n . Hence th ere exists j ∈ h n − 1 i suc h P Q j = I , i.e P = Q n − j . If P is not a p o we r of Q w e d educe that D 6∈ Ψ n,n . F or n ≥ 4 it is easy to construct suc h t w o p erm utations. F or example, let P and Q are represen ted b y the cycles 1 → 3 → 2 → 4 → . . . → n → 1 , 1 → 2 → 3 → 4 → . . . → n → 1 . If n = 3 then one has only tw o cycles R and R 2 . A straigh tforward calculat ion sho w that if P , Q ∈ { R, R 2 } the D ∈ Ψ 3 , 3 . ✷ Note that the system (2.6) h as 2 mn − 1 linear in dep endent equ ations. Since any p ermuta tion matrix is an extreme p oin t in Ω mn w e deduce. Corollary 2.7 The c onvex set Φ m,n ⊂ R mn × mn + is given b y at most 2(( n − 1) m 2 + ( m − 1) n 2 + mn ) − 1 line ar e quations. It c ontains al l the extr eme p oints P m ⊗ P n of Ψ m,n . It is in teresting to un derstand the structure of the set Φ m,n and to c haracterize it extreme p oin ts. It is easy to c haracterize the follo wing larger set. Lemma 2.8 L et Θ m,n b e the c onvex set of mn × mn nonne gative matric es char acterize d by 2 mn + (2 n − 2) m 2 line ar e quations o f the fol lowing form. View C ∈ R mn × mn as a matrix with entries c ( i,k )( j,l ) wher e i, j = 1 , . . . , m, k, l = 1 , . . . , n . Then C ∈ R mn × mn + b e longs to Θ m,n if the e qualities (2.6) and (2.8) hol d. Then Φ m,n ⊂ Θ m,n ⊂ Ω mn . F urthermo r e, a ny C = [ C ij ] m i,j =1 ∈ Θ m,n is of the fol lowing form C ij = a ij D ij , D ij ∈ Ω n , i, j = 1 , . . . , m, A = [ a ij ] m i,j =1 ∈ Ω m . (2.11) In p articular, the extr eme p oints of Θ m,n ar e of the the ab ove form wher e A ∈ P m , D ij ∈ P n for i, j = 1 , . . . , n . 6 Pro of. Observe first that C in the blo ck from C = [ C ij ] , C ij ∈ R n × n where C ij ∈ R n × n + . Conditions (2.8) equiv alen t to the assum ptions that C ij ∈ Λ n . Hence C ij = f ij D ij for some D ij ∈ Ω n and f ij ≥ 0. If f ij = 0 w e can c ho ose an y D ij ∈ Ω n . If f ij > 0 then D ij is a unique doubly sto c hastic matrix. Let F = [ f ij ] ∈ R m × m . Then the cond itions (2.6) are equiv alen t to the condition that F ∈ Ω m . Th us the conditions (2.8) and (2.6 ) are equiv alen t to the statemen t that C = [ f ij D ij ] where eac h D ij ∈ Ω n and F = [ f ij ] ∈ Ω m . Since the extreme p oin ts of Ω n are P n w e deduce that an y extreme p oint of Θ m,n is of the blo c k form C = [ f ij P ij ] where eac h P ij ∈ P n . Since the extreme p oin ts of Ω m are P m it follo w s that the extreme p oin ts of Θ m,n are of the form E = [ E ij ] satisfying the follo w ing cond itions. There exists a p ermutat ion σ : h m i → h m i such that E iσ ( i ) ∈ P n for i = 1 , . . . , m and E ij = 0 m × m otherwise. ✷ 3 P erm utatio nal similarit y and equiv alence of matrice s F or A ∈ R n × n denote b y tr A the tr ac e of A . Recall that h A, B i , the standard inner pro duct on R n × n , is giv en b y tr AB ⊤ . W e say that A, B ∈ R n × n are p erm utationally similar, and denote it b y A ∼ B if B = P AP ⊤ . Clearly , if A ∼ B then A and B h a v e the s ame c haracteristic p olynomial, i.e. det( xI n − A ) = det( xI n − B ). In what follo ws w e need the follo win g three lemmas. The pro of of the first t w o straigh tforw ard and is left to the reader. Lemma 3.1 L et A = [ a ij ] , B = [ b ij ] ∈ R n × n . Assume A ∼ B . Then the fol lowing c onditio ns hold. P ( a 11 , . . . , a nn ) ⊤ = ( b 11 , . . . , b nn ) ⊤ for some P ∈ P n , ( 3.1) R ( a 12 , . . . , a 1 n , a 21 , a 23 , . . . , a 2 n , . . . , a n 1 , . . . , a n ( n − 1) ) ⊤ = (3.2) ( b 12 , . . . , b 1 n , b 21 , b 23 , . . . , b 2 n , . . . , b n 1 , . . . , b n ( n − 1) ) ⊤ for some R ∈ P n 2 − n . Lemma 3.2 Assume that A, B ∈ R n × n satisfy the c onditions (3.1) and (3.2). Then tr( A + tI n )( A + tI n ) ⊤ = tr( B + tI n )( B + tI n ) ⊤ for e ach t ∈ R . Lemma 3.3 L et A = [ a ij ] , B = [ b ij ] ∈ R n × n satisfy th e c onditions (3.1) and (3.2). Fix t ∈ R such that t 6 = a ij − a k k for e ach i, j, k ∈ h n i such that i 6 = j . Then the fol lowing c onditions ar e e quivalent. 1. A ∼ B . 2. B + tI n = P ( A + tI n ) Q ⊤ for some P , Q ∈ P n . Pro of. Supp ose that 2 holds. Hence there exists t wo p ermuta tions σ , η : h n i → h n i suc h that b ij + tδ ij = a σ ( i ) η ( j ) + tδ σ ( i ) η ( j ) for all i, j ∈ h n i . Assume that σ 6 = η . Then there exists i 6 = j ∈ h n i suc h that σ ( i ) = η ( j ) = k . Hence b ij = a k k + t . The condition (3. 2) imp lies that b ij = a i 1 j 1 for s ome i 1 6 = j 1 ∈ h n i . 7 So t = a i 1 j 1 − a k k , wh ic h con tradicts the assump tions of the lemma. Hence σ = η whic h is equiv alen t to P = Q . Th us B + tI n = P ( A + tI n ) P ⊤ = P AP ⊤ + tI n ⇒ B = P AP ⊤ . Rev erse the implication in the abov e stat ement to deduce 2 f r om 1 . ✷ W e recall standard facts fr om linear algebra. Lemma 3.4 L et X = [ x lj ] n,m l,j =1 = [ x 1 x 2 . . . x m ] ∈ R n × m , wher e x 1 , . . . , x m ∈ R n ar e the m c olumns of X . Denote by ˆ X ∈ R mn the c olumn ve c tor c omp ose d of the c olumns of X , i.e. ( ˆ X ) ⊤ = ( x ⊤ 1 , x ⊤ 2 , . . . , x ⊤ m ) . L et A = [ a ij ] ∈ R m × m , B = [ b k l ] ∈ R n × n . Co nsider the line ar tr ansform ation of R m × n to itself giv en by X 7→ B X A ⊤ = [( B X A ⊤ ) k i ] n,m k ,i =1 : ( B X A ⊤ ) k i = m,n X j,l =1 a ij b k l x lj , k = 1 , . . . , n, i = 1 , . . . , m. (3.3) Then this line ar tr ansformat ion is r epr esente d by the Kr one cker pr o duct A ⊗ B . Tha t is, \ B X A ⊤ = ( A ⊗ B ) ˆ X fo r al l X ∈ R n × m . (3.4) Pro of. O bserv e fi rst that B X = [ B x 1 B x 2 . . . B x m ]. This shows (3.4) in the case A = I m . Consider now the case B = I n . A straigh tforward calculation sh o ws that ( A ⊗ I n ) ˆ X = [ X A ⊤ . Sin ce A ⊗ B = ( A ⊗ I n )( I m ⊗ B ) w e deduce the equalit y (3.4). ✷ Theorem 3.5 L et A = [ a ij ] , B = [ b ij ] ∈ R n × n . The fol lowing c onditions a r e e q u ivalent. 1. A ∼ B . 2. The fol lowing c onditions hold. (a) The c onditions (3.1) and (3.2) hold . (b) Fix t ∈ R such that t 6 = a ij − a k k for e ach i, j, k ∈ h n i such that i 6 = j . Then ther e exists Z ∈ Ψ n,n satisfying Z \ ( A + tI n ) = \ B + tI n . Pro of. Assume 1 . S o B + tI n = P ( A + tI n ) P ⊤ for some P ∈ P n and eac h t ∈ R . Use Lemma 3. 4 to deduce that ( P ⊗ P ) \ ( A + tI n ) = \ B + tI n . Hence the condition 2b holds. Lemma 3.1 yiel ds the co nd itions (3.1) and (3.2). Assume 2 . Use Lemma 3.2 yields that tr( A + tI n )( A + tI n ) ⊤ = tr( B + tI n )( B + tI n ) ⊤ . W e claim that max P ,Q ∈P n tr P ( A + tI n ) Q ⊤ ( B + tI n ) ⊤ = max Y ∈ Ψ n,n ( \ B + tI n ) ⊤ Y ( \ A + tI n ) . (3.5) T o fin d the maximum on the r igh t-hand side it is enou gh to r estrict the maximum on the right- hand side to the extreme p oin ts of Ψ n,n . Lemma 2.3 yields that the extreme p oin ts of Ψ n,n are P n ⊗ P n . Let Y = Q ⊗ P ∈ P n ⊗ P n . (3. 4) yields that ( \ B + tI n ) ⊤ Y ( \ A + tI n ) = tr P ( A + tI n ) Q ⊤ ( B + tI n ) ⊤ . 8 Compare th e ab ov e expression with the left-hand side of (3.5 ) to deduce the equalit y in (3.5 ). Assume that the maxim um in the left-hand side of (3.5) is ac hiev ed for P ∗ , Q ∗ ∈ P n . Use Cauc hy-Sc h wa rz inequalit y to dedu ce that tr P ∗ ( A + tI n ) Q ⊤ ∗ ( B + tI n ) ⊤ ≤ ((tr P ∗ ( A + tI n )( A + tI n ) ⊤ P ⊤ ∗ ) tr( B + tI n )( B + tI n ) ⊤ ) 1 2 = tr( B + tI n )( B + tI n ) ⊤ . Equalit y h olds if and only if B + tI n = P ∗ ( A + tI n ) Q ⊤ ∗ . The assu mption 2b yields the opp osite inequalit y tr( B + tI n )( B + tI n ) ⊤ = ( \ B + tI n ) ⊤ Z ( \ A + tI n ) ≤ max Y ∈ Ψ n,n ( \ B + tI n ) ⊤ Y ( \ A + tI n ) = tr P ∗ ( A + tI n ) Q ⊤ ∗ ( B + tI n ) ⊤ . Hence B + t I n = P ∗ ( A + tI n ) Q ⊤ ∗ . Lemma 3.3 implies that A ∼ B . ✷ The p ro of of the ab ov e theorem yields. Corollary 3.6 Assume that the c onditions 2 of The or em 3.5 holds. L et Ψ n,n ( A, B ) b e the set of al l Z ∈ Ψ n,n satisfying the c ondition Z ( \ A + tI n ) = \ B + tI n . Then al l the extr eme p oints of this c omp act c onvex set ar e of the form P ⊗ P ∈ P n ⊗ P n wher e P AP ⊤ = B . A, B ∈ R n × n are called p ermutational ly e quivalent , denoted as A ≈ B , if B = P AQ ⊤ for some P ∈ P n , Q ∈ P m . The arguments of the pr o of of Theorem 3.5 yield. Theorem 3.7 L et A, B ∈ R n × m . The fol lowing c onditio ns ar e e quivalent. 1. A ≈ B . 2. tr AA ⊤ = tr B B ⊤ and ther e exists Z ∈ Ψ m,n satisfying Z b A = b B . That is, view the entries of Z as z ( i,k ) , ( j,l ) wher e i, j ∈ h m i , k , l ∈ h n i . Then these m 2 n 2 nonne gative variables satisfy 2(( n − 1) m 2 + ( m − 1) n 2 + mn ) c onditions (2.6-2.8) and the mn c onditions: m,n X j,l =1 z ( i,k )( j,l ) a lj = b k i for k = 1 , . . . , n, i = 1 , . . . , m. (3.6) Corollary 3.8 Assume that the c onditions 2 of The or em 3.7 holds. L et Ψ m,n ( A, B ) b e the set of al l Z ∈ Ψ m,n satisfying the c ondition Z b A = b B . Then al l the extr eme p oints of this c omp act c onvex set ar e of the form Q ⊗ P ∈ P m ⊗ P n wher e P AQ ⊤ = B . 4 GIP and SGIP 4.1 Graph isomorphisms Theorem 4.1 Assume that Ψ n,n is char acterize d by f ( n ) numb er of line ar e qual- ities and ine qualities. Then isomorphism of two simple undir e cte d gr aphs G 1 = ( V , E 1 ) , G 2 = ( V , E 2 ) wher e # V = n is de cidable in p olynom ial time in max( f ( n ) , n ) . 9 Pro of. Let A, B ∈ { 0 , 1 } n × n b e the adjace ncy matrices of G 1 , G 2 resp ec- tiv ely . Recall that A, B are symmetric and ha v e zero diagonal. G 1 and G 2 are isomorphic if and only if A ∼ B . It is left to show that the conditions 2 of Theorem 3.5 can b e v erified in p olynomial time in max( f ( n ) , n ). 2a means that G 1 and G 2 ha v e the same d egree sequence. Th is requires at most 4 n 2 computations. Assume that 2a holds. Note that t = 2 satisfies the fi rst part of the condition 2b . The existence of Z ∈ Ψ n,n satisfying Z ( \ A + 2 I n ) = \ B + 2 I n is equiv alen t to the solv abil- it y of f ( n ) + n 2 linear equations and inequalities in n 4 nonnegativ e v ariables. The ellipsoid metho d [8, 7] yields that the existence or nonexistence of suc h X ∈ Ψ n,n is decidable in p olynomial time in max( f ( n ) , n ). ✷ Theorem 4.2 Assume that Ψ n,n is char acterize d by f ( n ) numb er of line ar e qual- ities a nd ine qualities. Then th e isomorphism of two simple dir e cte d gr aphs G 1 = ( V , E 1 ) , G 2 = ( V , E 2 ) , (self-lo ops al lowe d), w her e # V = n is de cidable in p olyno- mial time in max( f ( n ) , n ) . Pro of. Let A, B ∈ { 0 , 1 } n × n b e the adjacency matrices of G 1 , G 2 resp ectiv ely . Apply p art 2 of Theorem 3.5 with t = 2 to d educe the theorem. ✷ The app licatio n of part 2 of Theorem 3.5 yields. Theorem 4.3 Assume that Ψ n,n is char acterize d by f ( n ) numb er of line ar e qual- ities and ine qualities. L et A, B ∈ R n × n . Then p e rmutationa l similarity of A and B is de cidable in p olynomial time in max( f ( n ) , n ) and the entries of A and B . Let G = ( V 1 ∪ V 2 , E ) b e an u ndirected simple bipartite graph with the set of v ertices divided to t wo classes V 1 , V 2 suc h that E ⊂ V 1 × V 2 . Assume that # V 1 = n, # V 2 = m and iden tify V 1 , V 2 with h n i , h m i resp ectiv ely . Then G is represen ted by the incidence matrix A = [ a ij ] ∈ { 0 , 1 } n × m where a ij = 1 if and only if the v ertices i ∈ h n i , j ∈ h m i are connected b y an edge in E . L et H = ( V 1 ∪ V 2 , F ) b e another bipartite graph with the incidence matrix B ∈ { 0 , 1 } n × m . If m 6 = n then G and H are isomorp h ic if and only if A ≈ B . If m = n G and H are isomorphic if and only if either A ≈ B or A ≈ B ⊤ . T h eorem 3.7 yields. Theorem 4.4 Assume that Ψ m,n is char acterize d by a g ( m, n ) numb er of lin- e ar e qualities and ine q u alities. The isomorph ism of two simple undir e cte d bip artite gr aphs G 1 = ( V 1 ∪ V 2 , E 1 ) , G 2 = ( V 1 ∪ V 2 , E 2 ) wher e # V 1 = n, V 2 = m is de cidable in p olyno mial time in max ( g ( m, n ) , n + m ) . Theorem 4.5 Assume that Ψ m,n is char acterize d by g ( m, n ) numb er of line ar e q u alities and ine qualities. L et A, B ∈ R n × m . Then p ermutationa l e quivalenc e of A and B is de cidable in p olynomial time in max( g ( m, n ) , n + m ) and the entries of A and B . W e n o w remark that if we r eplace in Theorem 3.5 and Theorem 3.7 the sets Ψ n,n and Ψ m,n b y the sets Φ n,n and Φ m,n resp ectiv ely , w e will obtain necessary conditions for p erm utational similarit y and equiv alence, which can b e v erified in p olynomial time. 10 4.2 Subgraph isomorphism Theorem 4.6 Assume that Ψ n,n is char acterize d by f ( n ) numb er of line ar e qual- ities and ine q u alities. L et G 3 = ( W , E 3 ) , G 2 = ( V , E 2 ) b e two simple undir e cte d gr aphs, wher e # W = m ≤ # V = n . Then the pr oblem of determining if G 3 is isomorp hic to a sub g r aph of G 2 is de cidable in p olynom ial time in max( f ( n ) , n ) . Pro of. Add n − m isolated v ertices to G 3 to obtain th e graph ˜ G 3 on n v ertices. Let C, B ∈ { 0 , 1 } n × n b e the adjacency matrices of ˆ G 3 , G 2 resp ectiv ely . W e claim that G 3 is isomorphic to a subgraph of G 2 if and only if Z ( \ C + 2 n 2 I n ) ≤ \ B + 2 n 2 I n for some Z ∈ Ψ n,n . (4.1) Assume first that G 3 is isomorph ic to a subgraph of G 2 . T his is equiv alent to the statemen t t hat P C P ⊤ ≤ B for some P ∈ P n . (That is in eac h place where P C P ⊤ has en try 1, then B has entry 1 at the same p lace.) As P P ⊤ = I we d educe that (4.1) holds for Z = P ⊗ P . Assume that (4.1) is satisfied. Let Z = X P ,Q ∈P n w ( P , Q ) P ⊗ Q, w ( P , Q ) ≥ 0 for eac h P , Q ∈ P n and X P ,Q ∈P n w ( P , Q ) = 1 . Hence there exists P ∗ , Q ∗ ∈ P n suc h that w ( P ∗ , Q ∗ ) ≥ 1 ( n !) 2 . (4.1 ) yields that 1 ( n !) 2 Q ∗ ( C + 2 n 2 I n ) P ⊤ ∗ ≤ B + 2 n 2 I n . Since n = 2 n − 1 for n = 1 , 2 and n < 2 n − 1 for 2 < n it follo ws that n ! < 2 n ( n − 1) 2 for n > 2. Hence ( n !) 2 < 2 n 2 for n ≥ 1. Since all offdiagonal ele ments of B are at most 1 it follo ws that P ∗ = Q ∗ . Hence P ∗ C P ⊤ ∗ ≤ ( n !) 2 B . T h us if P ∗ C P ⊤ ∗ has 1 in the place ( i , j ) then B can not ha v e zero in the place ( i , j ). That is B has 1 in the place ( i, j ). T herefore G 3 is isomorphic to a subgraph of G 2 . ✷ References [1] L. Babai, D.Y u. Grigory ev and D.M. Moun t, Isomorph ism of graphs with b ounded eigen v alue m ultiplicit y , Pr o c e e dings of the 14th Annual ACM Sym- p osium on The ory of Computing , 198 2, pp. 310- 324, . [2] H. Bo dlaender, Pol ynomial algorithms for graphs isomorphism and c hro- momatic in dex on partial k -trees, J. Algo rithms 11 (19 90), 631-643 . [3] I.S. 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