Two graph isomorphism polytopes

Tw o graph isomorph ism p olyto p e s Shm uel Onn Jan uary 11, 2008 Abstract The conv ex h ull ψ n,n of certain ( n !) 2 tensors w as consider ed recently in connection with graph isomorphism. W e c o nsider the co n vex hull ψ n of the n ! diagonals a mong these tenso rs. W e show: 1. The po lytop e ψ n is a face of ψ n,n . 2. Deciding if a g raph G has a subgraph isomor phic to H reduces to optimization over ψ n . 3. Optimization ov er ψ n reduces to optimization over ψ n,n . In particular, this implies that the subgraph isomor phism problem reduces to optimization o ver ψ n,n . AMS Subje ct Classific ation: 05A, 1 5A, 51 M, 52A, 5 2B, 52C, 62H, 68 Q, 68R, 68U, 68W, 9 0B, 90C 1 In tro duc tion Let P n b e the set of n × n p erm u tation matrices and consider the follo wing t wo p olytop es, ψ n := conv { P ⊗ P : P ∈ P n } , ψ n,n := conv { P ⊗ Q : P , Q ∈ P n } . The p olytop e ψ n,n w as considered recen tly in [1 ] in connection with the graph isomorphism problem. Note that ψ n and ψ n,n ha v e n ! and ( n !) 2 v ertices resp ectiv ely . In this short note w e sho w: 1. The p olytop e ψ n is a face of the p olytop e ψ n,n . 2. Dec idin g if a graph G has a su bgraph isomorph ic to a graph H reduces to optimization o v er ψ n . 3. Optimizatio n ov er ψ n reduces to optimization ov er ψ n,n . In particular, this implies a r esult of [1] that subgraph isomorphism redu ces to optimizat ion o ver ψ n,n . So if P 6 = N P then optimizatio n and separation o ver ψ n and hence o v er ψ n,n cannot b e done in p olynomial time and a compact inequalit y descrip tion of ψ n and hence of ψ n,n cannot b e determined. Deciding if G has a sub grap h that is isomorph ic to H can also b e reduced to optimizatio n o ver a related p olytop e φ n defined as follo ws. Eac h p ermutatio n σ of the v ertices of the complete graph K n naturally indu ces a p erm utation Σ of its edges b y Σ( { i, j } ) := { σ ( i ) , σ ( j ) } . Then φ n is defin ed as the con v ex hull of all  n 2  ×  n 2  p ermutatio n matrices of induced p erm utations Σ. This p olytop e and a broader class of so-calle d Y oung p olytop es h a v e b een stu d ied in [2]. I n p articular, therein it was sho wn that the graph of φ n is complete, so p iv oting algorithms cannot b e exploited for optimizat ion o ver this p olytop e. It is an in teresting question wh ether ψ n and φ n , h a ving n ! v ertices eac h, are isomorphic. 1 2 Statemen ts Define b ilinear forms on R n × n and on R n × n ⊗ R n × n (note the s h uffled indexation on the r igh t) by h A, B i := X i,j A i,j B i,j , h X, Y i := X i,j,s, t X i,s,j, t Y i,j,s, t . Let I b e the n × n identit y matrix and for a graph G let A G b e its adjacency m atrix. W e show: Theorem 2.1 The p olytop e ψ n is a fac e of ψ n,n given by ψ n = ψ n,n ∩ { X : h I ⊗ I , X i = n } . Theorem 2.2 L et G and H b e two gr aph s on n vertic es with m the numb er of e dges of H . Then max {h A G ⊗ A H , X i : X ∈ ψ n } ≤ 2 m with e quality if and only if G has a sub gr aph that is isomorphic to H . Theorem 2.3 L et W = ( W i,s,j, t ) b e any tensor and let w := 2 n 2 max | W i,s,j, t | . Then max {h W , X i : X ∈ ψ n } = max {h W + w I ⊗ I , X i : X ∈ ψ n,n } − nw . Com binin g Theorems 2.2 and 2.3 w ith W = A G ⊗ A H and w = n 2 (sufficing since W ≥ 0, as is clear from the pro of of Theorem 2.3 b elo w), we get the f ollo wing somewhat tigh ter form of a result of [1]. Corollary 2.4 L et G and H b e two gr aphs on n vertic es with m the numb er of e dges of H . Then max {h A G ⊗ A H + nI ⊗ nI , X i : X ∈ ψ n,n } ≤ 2 m + n 3 with e quality if and only if G has a sub gr aph that is isomorphic to H . 3 Pro ofs W e r ecord the follo wing statemen t that follo ws d ir ectly from the definitions of the bilinear forms ab ov e. Prop osition 3.1 F or any two simple tensors X = A ⊗ B and Y = P ⊗ Q we have h X, Y i = h A ⊗ B , P ⊗ Q i = X i,j,s, t A i,s B j,t P i,j Q s,t = h P B Q ⊺ , A i . 2 Pr o of of The or em 2 .1. F or ev ery P , Q ∈ P n , the matrix P I Q ⊺ is a p erm utation m atrix, with P I Q ⊺ = I if and only if P = Q . It follo ws that for ev ery tw o distinct P , Q ∈ P n w e ha v e h I ⊗ I , P ⊗ Q i = h P I Q ⊺ , I i ≤ n − 1 < n = h P I P ⊺ , I i = h I ⊗ I , P ⊗ P i .  (1) Pr o of o f The or em 2.2. F or any P ∈ P n , the matrix P A H P ⊺ is the adjacency matrix of th e p ermutatio n of H b y P . So h P A H P ⊺ , A G i ≤ 2 m with equ ality if and only if H is isomorph ic via P to a su b graph of G . Since the maxim um of a linear form o ve r a p olytop e is attained at a v ertex we get max {h A G ⊗ A H , X i : X ∈ ψ n } = max {h A G ⊗ A H , P ⊗ P i : P ∈ P n } = max {h P A H P ⊺ , A G i : P ∈ P n } ≤ 2 m with the last inequalit y holding with equalit y if and only if G h as a subgraph isomorph ic to H .  Pr o of of The or em 2.3. F or eve ry P , Q ∈ P n , the tensor P ⊗ Q = ( P i,j Q s,t ) h as n 2 en tries that are equal to 1 and all other ent ries equal to 0, and th erefore − 1 2 w ≤ h W , P ⊗ Q i ≤ 1 2 w . Combining this with inequalit y (1) w e see that for every tw o distinct P , Q ∈ P n w e ha ve h W + w I ⊗ I , P ⊗ Q i = h W , P ⊗ Q i + w h I ⊗ I , P ⊗ Q i ≤ 1 2 w + ( n − 1) w = − 1 2 w + nw ≤ h W , P ⊗ P i + w h I ⊗ I , P ⊗ P i = h W + w I ⊗ I , P ⊗ P i . Since th e maximum of a linear form o v er a p olytop e is attained at a vertex we obtain the inequalit y max {h W + wI ⊗ I , X i : X ∈ ψ n,n } = max {h W + wI ⊗ I , P ⊗ Q i : P, Q ∈ P n } = max {h W + wI ⊗ I , P ⊗ P i : P ∈ P n } = max {h W , P ⊗ P i : P ∈ P n } + nw = max {h W , X i : X ∈ ψ n } + n w .  References [1] S. F riedland, On the grap h isomorph ism pr oblem, e-print: arXiv:0801.03 98 [2] S. Onn, Geometry , complexit y and com binatorics of p ermuta tion p olytop es, Journal of Combina- torial The ory Series A 64:31–49 (1993) Shm uel Onn T e chnion - Isr ael Institute of T e chnolo gy, 32000 Haifa, Isr ael email: onn @ ie.te chnion.ac.il , http://ie.te chnion.a c.il/ ∼ onn 3