A Family of Counter Examples to an Approach to Graph Isomorphism

A F amily o f Coun ter Exa mples to an Approac h to Graph Iso morphism Jin-Yi Cai ∗ Pin y an Lu † Ming ji Xia ‡ Jan uary 10,2008 Abstract W e give a family of co un ter examples showing that the t wo s e- quences of polytop es Φ n,n and Ψ n,n are different. These p olyto pes were defined recently by S. F riedland in an attempt at a polyno mial time algorithm for graph isomorphis m. 1 In tro duc tion In a recent p osting at arXiv (arXiv:0801 .0398 v1 [cs.CC] 2 J an 2008 and arXiv:0801 .0398 v2 [cs.CC] 4 Jan 2008), S. F riedland defined t wo s equ ences of p olytop es Φ n,n and Ψ n,n . Let Ω n ⊂ R n × n + denote the n × n doubly sto chastic m atrices. Then Ψ n,n ⊂ Ω n 2 is the con v ex hull of the tensor pro ducts A ⊗ B , where A, B ∈ Ω n . Mean while Φ n,n is defined to b e the subset of Ω n 2 defined b y the follo wing set of linear constrain ts. n,n X j,l =1 c ( i,k ) , ( j,l ) = n,n X j,l =1 c ( j,l ) , ( i,k ) = 1 , i = 1 , . . . , n , k = 1 , . . . , n , n X j =1 c ( i,k ) , ( j,l ) = n X j =1 c (1 ,k ) , ( j, l ) , n X j =1 c ( j,k ) , ( i,l ) = n X j =1 c (1 ,k ) , ( j, l ) , where i = 2 , . . . , n, and k, l = 1 , . . . , n, ∗ Universit y of Wisconsin-Madison, jyc@cs.wisc.e du † Tsingh ua U niversit y lpy@mails. tsinghua.edu.cn ‡ Institute of Softw are, Chinese Academy of Sciences, xmjljx@gma il.com 1 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 ) , where i = 2 , . . . , n, and k, l = 1 , . . . , n. It w as shown that Ψ n,n ⊆ Φ n,n . (In the earlier v ersion it was claimed that Ψ n,n = Φ n,n . If this w ere the case, then graph isomorphism w ould b e in P , as one can reduce th e problem to linear programming. In th e Jan 4th v ersion F riedland stated that the equalit y Ψ n,n = Φ n,n “is p r obably wrong”.) In this note w e giv e an explicit family of coun ter examples sho wing Ψ n,n 6 = Φ n,n . F or ev ery n ≥ 4, ou r examples consist of an exp onen tial num b er of matricies whic h are v ertices of Φ n,n , b ut do not b elong to Ψ n,n . 2 Coun ter Examples Let ρ ∈ S n b e the cyclic p ermutatio n (1 2 3 . . . n ). Let σ ∈ S n b e any p ermutati on. Lemma 2.1. Ther e ar e exactly n ! − nφ ( n ) many p ermutations σ ∈ S n , such that σ ρσ − 1 do es not b elong to the sub gr oup gener ate d by ρ . Pr o of. A conjugate σ ρσ − 1 of ρ is also an n -cycle. T o b e in the su bgroup generated by ρ , iff it is a p o we r ρ i for some i relativ ely p rime to n . T o b e of this form, iff σ is of the form σ ( i + 1) − σ ( i ) (in a cyclic sense) is a constan t relativ ely prime to n , which means there are exactly nφ ( n ) m any . Let A b e the matrix whose fi rst ro w is ( x 1 , x 2 , . . . x n ), and its i -th r o w is obtained b y applying ( i − 1) times the cyclic p erm utation ρ . Let B b e the matrix whose first ro w is ( x 1 , x 2 , . . . x n ) p ermuted by σ , and its i -th ro w is obtained by fur ther applyin g ( i − 1) times the cyclic p ermuta tion ρ . Lemma 2.2. Whenever σ ∈ S n satisfies L emma 1, ther e do es not exist a p air of p ermutation matric e s P and Q , such that A = P B Q . Pr o of. The first t wo ro ws of B are σ ( x 1 , x 2 , . . . x n ) and ρσ ( x 1 , x 2 , . . . x n ). Assume for con tradiction that there do es exist a pair of p ermutati on m a- trices P a nd Q , su ch that A = P B Q . The fir st tw o ro ws of B Q are q σ ( x 1 , x 2 , . . . x n ) and q ρσ ( x 1 , x 2 , . . . x n ), where q is the p ermutation cor- resp ondin g to Q . They must b e t w o rows of A , so there exist i and j ( i 6 = j ) su c h that q σ ( x 1 , x 2 , . . . x n ) = ρ i ( x 1 , x 2 , . . . x n ) and q ρσ ( x 1 , x 2 , . . . x n ) = ρ j ( x 1 , x 2 , . . . x n ). W e get σ − 1 ρσ = ρ j − i , con tradicting with lemma 1. 2 Supp ose A = ( a ij ) is an n × n matrix. we use b A to denotes th e column v ector ( a 11 , . . . , a 1 n , a 21 , . . . , a 2 ,n , a 3 , 1 , . . . , a nn ) T of length n 2 . Giv en A and B , d efine T to b e the n 2 × n 2 matrix comp osed of 0 and 1 /n such that b A = T b B . An example of this is sho wn as follo ws, for n = 4 and σ = (3 4): A =     x 1 x 2 x 3 x 4 x 2 x 3 x 4 x 1 x 3 x 4 x 1 x 2 x 4 x 1 x 2 x 3     , B =     x 1 x 2 x 4 x 3 x 2 x 4 x 3 x 1 x 4 x 3 x 1 x 2 x 3 x 1 x 2 x 4     , T = 1 / 4                              1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0                              Theorem 2.1. F or any σ ∈ S n satisfying L emma 1.1, the matrix T is an extr eme p oint of Φ n,n . However, T 6∈ Ψ n,n . Pr o of. By the definition of A , B and T = ( t ( i,k ) , ( j,l ) ), for eac h fixed pair i, j , ( t ( i,k ) , ( j,l ) ) (resp ective ly , for eac h fi xed k , l , ( t ( i,k ) , ( j,l ) )) is a p erm utation matrix multiplied by 1 /n . Obviously , T ∈ Φ n,n . F or eac h double row ind ex ( i, k ), either fix i , or fix k , and v arying the other index, and for eac h doub le column ind ex ( j, l ), either fix j , or fix l , and v arying the other ind ex, w e alw a ys get an n by n p erm utation matrix. Supp ose T = P s w s T s , where T s ∈ Φ n,n , w s > 0, and P s w s = 1. So within eac h b lo c k (fixed i, j , v arying k and l , ) the non-zero en tries of T s are a subset of non -zero entries of T w ithin th at b lo c k, which form a p erm utation matrix. then by the equations for T s within th e b lo ck, it m ust b e either 3 totally zero or a p ositiv e m u ltiple of the same p erm utation matrix made up of n on-zero entries of T within that blo c k. F or eac h block, the p ermutation matrix is the same for eve ry T s . The m ultipliers form a doubly stochasti c matrix M s ∈ Ω n , by the global su m P n,n j,l =1 = 1. Therefore T s is as follo ws: its ( i, j ) blo ck is obtained by multiplying eac h entry of a d oubly sto chastic matrix M s ∈ Ω n with the p ermuta tion matrix of T for eac h blo c k. No w if we consider the sum P n j =1 c ( i,k ) , ( j,l ) = P n j =1 c (1 ,k ) , ( j, l ) , by the prop erty of T eac h row of M s is a constan t. (Similarly eac h column of M s is a constan t.) Thus M s is jus t th e all 1 / n matrix 1 /nJ . This implies that there is exactly one term in the su m T = P s w s T s , and T is an extreme p oin t. Assume for a con tradiction that T ∈ Ψ n,n and T = P s w s P s ⊗ Q s , wh ere P s , Q s are p ermutation matrices, w s > 0, and P s w s = 1. W e get T ≥ w 1 P 1 ⊗ Q 1 (Here the relation of ≥ is entry-wise). F or an y x 1 , x 2 , . . . , x n ≥ 0, T b B ≥ w 1 P 1 ⊗ Q 1 b B , that is, A ≥ w 1 P 1 B Q 1 . By lemma 1.2, P 1 B Q 1 is differen t fr om A , so there must b e an en try ( i, j ) suc h that they are differen t at that en try . Notic e th at eac h entry of A or P 1 B Q 1 is a single v ariable from { x 1 , . . . , x n } . W.l.o.g, we ca n assume the ( i, j )-th en try of A and P 1 B Q 1 are x 1 and x 2 . W e can set x 1 = 0 and x 2 = 1 suc h that A ij < ( w 1 P 1 B Q 1 ) ij , whic h is a con tradiction. S o T 6∈ Ψ n,n . Before we p osted this note, we note th at Babai (h ttp://p eople.cs.uc hicago.edu/ ∼ laci/polytop e.p df ) and Onn (arXiv:08 01.1410 ) ha ve b oth p oin ted out that the linear optimization problem ov er the p oly- top e Ψ n,n can solv e NP-complete problems, and therefore it is unlikely that Ψ n,n can b e defin ed b y a p olynomial n umb er of (in)equalities as Φ n,n can. In (h ttp://p eople.cs.uc hicago.edu/ ∼ laci/polytop e-corresp ond ence.p df ), Babai also men tion that J o el Rosen b erg already ga ve a coun ter example showing the t wo p olytop es are different, for n = 4. References [1] L. Babai, Th e d ouble p ermutatio n p olytop e is NP-hard. h ttp://p eople.cs.uc hicago.edu/ ∼ laci/polytop e.p df [2] L. Babai, Timeline of a corresp ondence. h ttp://p eople.cs.uc hicago.edu/ ∼ laci/polytop e-corr esp ondence.p d f [3] S. F riedland, Graph isomorphism is P olynomial. h cac he/arxiv/p df/0801/0 801.0398v1.p d f 4 [4] S. F riedland, On the graph isomorp hism problem. h cac he/arxiv/p df/0801/0 801.0398v2.p d f [5] S. Onn, T w o graph isomorph ism p olytop es. h ttp://arxiv.org/abs/080 1.1410 5