SDP and eigenvalue bounds for the graph partition problem

SDP and eigenvalue bounds for the graph partition problem Renata Sotirov and Edwin van Dam Tilburg University, The Netherlands

Outline... the graph partition problem

Outline... the graph partition problem matrix lifting vector lifting

Outline... the graph partition problem matrix lifting vector lifting simplify complicate

Outline... the graph partition problem matrix lifting vector lifting simplify complicate???

The Graph Partition Problem G = (V, E)... an undirected graph V... vertex set, V = n E... edge set

The Graph Partition Problem G = (V, E)... an undirected graph V... vertex set, V = n E... edge set The k-partition problem (GPP) Find a partition of V into k subsets S 1,..., S k of given sizes m 1... m k, s.t. the total weight of edges joining different S i is minimized.

The Graph Partition Problem G = (V, E)... an undirected graph V... vertex set, V = n E... edge set The k-partition problem (GPP) Find a partition of V into k subsets S 1,..., S k of given sizes m 1... m k, s.t. the total weight of edges joining different S i is minimized. m i = V k, i the graph equipartition problem k = 2 the bisection problem

The k-partition problem A... the adjacency matrix of G, m := (m 1,..., m k ) T

The k-partition problem A... the adjacency matrix of G, m := (m 1,..., m k ) T V k let X = (x ij ) R { 1, if vertex i Sj x ij := 0, if vertex i / S j

The k-partition problem A... the adjacency matrix of G, m := (m 1,..., m k ) T V k let X = (x ij ) R { 1, if vertex i Sj x ij := 0, if vertex i / S j P k := { X R n k : Xu k = u n, X T u n = m, x ij {0, 1} }, where u n... vector of all ones

The k-partition problem A... the adjacency matrix of G, m := (m 1,..., m k ) T V k let X = (x ij ) R { 1, if vertex i Sj x ij := 0, if vertex i / S j P k := { X R n k : Xu k = u n, X T u n = m, x ij {0, 1} }, where u n... vector of all ones For X P k : w(e cut ) = 1 2 tr(x T LX ) = 1 2 tr A(J n XX T ) where L := Diag(Au n ) A is the Laplacian matrix of G

The Graph Partition Problem The trace formulation: (GPP) 1 min 2 trace(x T LX ) s.t. Xu k = u n X T u n = m x ij {0, 1}

The Graph Partition Problem The trace formulation: (GPP) 1 min 2 trace(x T LX ) s.t. Xu k = u n X T u n = m x ij {0, 1} the GPP... is NP-hard (Garey and Johnson, 1976)

The Graph Partition Problem The trace formulation: (GPP) 1 min 2 trace(x T LX ) s.t. Xu k = u n X T u n = m x ij {0, 1} the GPP... is NP-hard (Garey and Johnson, 1976) applications: VLSI design, parallel computing, floor planning, telecommunications, etc.

. matrix lifting SDP for the GPP...

SDP for GPP linearize the objective: trace(lxx T ) trace(ly)

SDP for GPP linearize the objective: trace(lxx T ) trace(ly) Y conv{ỹ : X P k s.t. Ỹ = XX T } ky J n 0.

SDP for GPP linearize the objective: trace(lxx T ) trace(ly) Y conv{ỹ : X P k s.t. Ỹ = XX T } ky J n 0. S., 2013 (GPP m ) min 1 2 tr(ly ) s.t. diag(y ) = u n tr(jy ) = k mi 2 i=1 ky J n 0, Y 0

SDP for GPP linearize the objective: trace(lxx T ) trace(ly) Y conv{ỹ : X P k s.t. Ỹ = XX T } ky J n 0. S., 2013 (GPP m ) min 1 2 tr(ly ) s.t. diag(y ) = u n tr(jy ) = k mi 2 i=1 ky J n 0, Y 0 for k = 2 the nonnegativity constraints are redundant

GPP m and known relaxations GPP m... for equipartition is equivalent to the relaxation from: S.E. Karisch, F. Rendl. Semidefnite programming and graph equipartition. In: Topics in Semidefinite and Interior Point Methods, The Fields Institute for research in Math. Sc., Comm. Ser. Rhode Island, 18, 1998.

GPP m and known relaxations GPP m... for equipartition is equivalent to the relaxation from: S.E. Karisch, F. Rendl. Semidefnite programming and graph equipartition. In: Topics in Semidefinite and Interior Point Methods, The Fields Institute for research in Math. Sc., Comm. Ser. Rhode Island, 18, 1998. for bisection is equivalent to the relaxation from: S. E. Karisch, F. Rendl, J. Clausen. Solving graph bisection problems with semidefinite programming, INFORMS J. Comput., 12:177-191, 2000.

Strengthening?

Strengthening?? How to strengthen GPP m?

Strengthening?? How to strengthen GPP m? impose the linear inequalities: constraints y ab + y ac 1 + y bc, (a, b, c) independent set constraints y ab 1, W s.t. W = k + 1 a<b, a,b W

Strengthening?? How to strengthen GPP m? impose the linear inequalities: constraints y ab + y ac 1 + y bc, (a, b, c) independent set constraints y ab 1, W s.t. W = k + 1 a<b, a,b W there are 3 ( ) ( n 3, and n k+1) independent set constraints

On computational issues... in general, for graphs with 100 vertices and k = 3: the best known vector lifting relaxation is hopeless

On computational issues... in general, for graphs with 100 vertices and k = 3: the best known vector lifting relaxation is hopeless GPP m + triangle inequalities + independent set solves 3 h

On computational issues... in general, for graphs with 100 vertices and k = 3: the best known vector lifting relaxation is hopeless GPP m + triangle inequalities + independent set solves 3 h GPP m solves 14 min

On computational issues... in general, for graphs with 100 vertices and k = 3: the best known vector lifting relaxation is hopeless GPP m + triangle inequalities + independent set solves 3 h GPP m solves 14 min Can we compute GPP m with/without additional constr. more efficiently?

On computational issues... in general, for graphs with 100 vertices and k = 3: the best known vector lifting relaxation is hopeless GPP m + triangle inequalities + independent set solves 3 h GPP m solves 14 min Can we compute GPP m with/without additional constr. more efficiently? yes

Simplification highly symmetric graphs...

Simplification highly symmetric graphs... matrix *-algebra: subspace of R n n that is closed under matrix multiplication and taking transposes

Simplification highly symmetric graphs... matrix *-algebra: subspace of R n n that is closed under matrix multiplication and taking transposes Assumption: The data matrices of an SDP problem and I belong to a matrix *-algebra span{a 1,..., A r } where r n 2

Simplification highly symmetric graphs... matrix *-algebra: subspace of R n n that is closed under matrix multiplication and taking transposes Assumption: The data matrices of an SDP problem and I belong to a matrix *-algebra span{a 1,..., A r } where r n 2 Then, if the SDP relaxation has an optimal solution it has an optimal solution in the matrix *-algebra.

Simplification highly symmetric graphs... matrix *-algebra: subspace of R n n that is closed under matrix multiplication and taking transposes Assumption: The data matrices of an SDP problem and I belong to a matrix *-algebra span{a 1,..., A r } where r n 2 Then, if the SDP relaxation has an optimal solution it has an optimal solution in the matrix *-algebra. Schrijver, Goemans, Rendl, Parrilo,...

Simplification highly symmetric graphs... matrix *-algebra: subspace of R n n that is closed under matrix multiplication and taking transposes Assumption: The data matrices of an SDP problem and I belong to a matrix *-algebra span{a 1,..., A r } where r n 2 Then, if the SDP relaxation has an optimal solution it has an optimal solution in the matrix *-algebra. Schrijver, Goemans, Rendl, Parrilo,... a basis of the matrix *-algebra (coming from combinatorial or group symmetry): (i) A i {0, 1} n n, A T i {A 1,..., A r }, (i = 1,..., r) (ii) r i=1 A i = J, i I A i = I, I {1,..., r} (iii) For i, j {1,..., r}, p h ij such that A i A j = r h=1 ph ija h.

Simplification highly symmetric graphs... Y = r z i A i, z i R (r n 2 ) i=1 (GPP m ) 1 min 2 tr(aj r n) 1 2 z i tr(aa i ) i=1 s.t. z i diag(a i ) = u n i I r z i tr(ja i ) = k i=1 mi 2 i=1 k r z i A i J n 0, z i 0, i = 1,..., r. j=1

Simplification highly symmetric graphs... Y = r z i A i, z i R (r n 2 ) i=1 (GPP m ) 1 min 2 tr(aj r n) 1 2 z i tr(aa i ) i=1 s.t. z i diag(a i ) = u n i I r z i tr(ja i ) = k i=1 mi 2 i=1 k r z i A i J n 0, z i 0, i = 1,..., r. j=1 LMI may be (block-)diagonalized

Simplification highly symmetric graphs... Y = r z i A i, z i R (r n 2 ) i=1 (GPP m ) 1 min 2 tr(aj r n) 1 2 z i tr(aa i ) i=1 s.t. z i diag(a i ) = u n i I r z i tr(ja i ) = k i=1 mi 2 i=1 k r z i A i J n 0, z i 0, i = 1,..., r. j=1 LMI may be (block-)diagonalized exploit properties of A i to aggregate and indep. set const. extend the approach from: M.X. Goemans, F. Rendl. Semidefinite Programs and Association Schemes. Computing, 63(4):331 340, 1999.

On aggregating constraints... for a given (a, b, c) consider the inequality y ab + y ac 1 + y bc

On aggregating constraints... for a given (a, b, c) consider the inequality y ab + y ac 1 + y bc if (A i ) ab = 1, (A h ) ac = 1, (A j ) bc = 1 type (i, j, h) ineq.,

On aggregating constraints... for a given (a, b, c) consider the inequality y ab + y ac 1 + y bc if (A i ) ab = 1, (A h ) ac = 1, (A j ) bc = 1 type (i, j, h) ineq., by summing all ineq. of type (i, j, h), the aggregated ineq.: p i hj tr A iy + p h ij tr A h Y p i hj tr A ij + p j i h tr A jy, where p h ij : A ia j = r h=1 ph ij A h j : is the index s.t. A j = A T j

On aggregating constraints... for a given (a, b, c) consider the inequality y ab + y ac 1 + y bc if (A i ) ab = 1, (A h ) ac = 1, (A j ) bc = 1 type (i, j, h) ineq., by summing all ineq. of type (i, j, h), the aggregated ineq.: p i hj tr A iy + p h ij tr A h Y p i hj tr A ij + p j i h tr A jy, where p h ij : A ia j = r h=1 ph ij A h j : is the index s.t. A j = A T j use: Y = r j=1 z ja j

On aggregating constraints... for a given (a, b, c) consider the inequality y ab + y ac 1 + y bc if (A i ) ab = 1, (A h ) ac = 1, (A j ) bc = 1 type (i, j, h) ineq., by summing all ineq. of type (i, j, h), the aggregated ineq.: p i hj tr A iy + p h ij tr A h Y p i hj tr A ij + p j i h tr A jy, where p h ij : A ia j = r h=1 ph ij A h j : is the index s.t. A j = A T j use: Y = r j=1 z ja j of aggregated constraints is bounded by r 3

On aggregating constraints... for a given (a, b, c) consider the inequality y ab + y ac 1 + y bc if (A i ) ab = 1, (A h ) ac = 1, (A j ) bc = 1 type (i, j, h) ineq., by summing all ineq. of type (i, j, h), the aggregated ineq.: p i hj tr A iy + p h ij tr A h Y p i hj tr A ij + p j i h tr A jy, where p h ij : A ia j = r h=1 ph ij A h j : is the index s.t. A j = A T j use: Y = r j=1 z ja j of aggregated constraints is bounded by r 3 similar approach applies to independent set constr. when k = 2

Simplification highly symmetric graphs... Example. Strongly regular graph (SRG)

Simplification highly symmetric graphs... Example. Strongly regular graph (SRG) n vertices, κ the valency of the graph

Simplification highly symmetric graphs... Example. Strongly regular graph (SRG) n vertices, κ the valency of the graph A has exactly two eigenvalues r 0 and s < 0 associated with eigenvectors u n

Simplification highly symmetric graphs... Example. Strongly regular graph (SRG) n vertices, κ the valency of the graph A has exactly two eigenvalues r 0 and s < 0 associated with eigenvectors u n A belongs to the *-algebra spanned by {I, A, J A I }

Simplification highly symmetric graphs... Example. Strongly regular graph (SRG) n vertices, κ the valency of the graph A has exactly two eigenvalues r 0 and s < 0 associated with eigenvectors u n A belongs to the *-algebra spanned by {I, A, J A I } Y = I + z 1 A + z 2 (J A I )

Simplification highly symmetric graphs... Example. Strongly regular graph (SRG) n vertices, κ the valency of the graph A has exactly two eigenvalues r 0 and s < 0 associated with eigenvectors u n A belongs to the *-algebra spanned by {I, A, J A I } Y = I + z 1 A + z 2 (J A I ) (GPP m ) min 1 2 κn(1 z 1) s.t. κz 1 + (n κ 1)z 2 = 1 n 1 + rz 1 (r + 1)z 2 0 1 + sz 1 (s + 1)z 2 0 z 1, z 2 0 k mi 2 1 i=1

SRG Theorem. Let G = (V, E) be a SRG with eigenvalues κ, r, s. Let m i N, i = 1,..., k s.t. k j=1 m j = n. Then the SDP bound for the minimum k-partition is { κ r max n i<j m ( ) } 1 im j, 2 n(κ + 1) i m2 i Similarly, the SDP bound for the maximum k-partition is { } κ s min n i<j m 1 im j, 2 κn.

SRG Theorem. Let G = (V, E) be a SRG with eigenvalues κ, r, s. Let m i N, i = 1,..., k s.t. k j=1 m j = n. Then the SDP bound for the minimum k-partition is { κ r max n i<j m ( ) } 1 im j, 2 n(κ + 1) i m2 i Similarly, the SDP bound for the maximum k-partition is { } κ s min n i<j m 1 im j, 2 κn. this is an extension of the result for the equipartition: De Klerk, Pasechnik, S., Dobre: On SDP relaxations of maximum k-section, Math. Program. Ser. B, 136(2):253-278, 2012.

SRG after aggregating, 3 ( n 3) constraints remain: z 1 1 z 2 1 2z 1 z 2 1 z 1 + 2z 2 1

SRG after aggregating, 3 ( n 3) constraints remain: z 1 1 z 2 1 2z 1 z 2 1 z 1 + 2z 2 1 Prop. For SRG with n > 5 the ineq. are redundant in GPP m.

SRG after aggregating, 3 ( n 3) constraints remain: z 1 1 z 2 1 2z 1 z 2 1 z 1 + 2z 2 1 Prop. For SRG with n > 5 the ineq. are redundant in GPP m. However, the independent set constraints improve GPP m.

Simplification not a special graph...

Simplification not a special graph... closed form expression for the GPP for any graph

Simplification not a special graph... closed form expression for the GPP for any graph L = Diag(Au n ) A... the Laplacian matrix of G

Simplification not a special graph... closed form expression for the GPP for any graph L = Diag(Au n ) A... the Laplacian matrix of G L := span{f 0,..., F d } the Laplacian algebra corr. to L

Simplification not a special graph... closed form expression for the GPP for any graph L = Diag(Au n ) A... the Laplacian matrix of G L := span{f 0,..., F d } the Laplacian algebra corr. to L F i = U i Ui T, i... where U i corr. to the distinct eig. λ i d i=0 F i = I F i F j = δ ij F i for i j tr(f i ) = f i... the multiplicity of i-th eigenvalue of L

Simplification not a special graph... in GPP m : relax diag(y ) = u n tr(y ) = n remove nonnegativity constraints

Simplification not a special graph... in GPP m : relax diag(y ) = u n tr(y ) = n remove nonnegativity constraints (GPP eig ) 1 min 2 tr LY s.t. tr(y ) = n tr(jy ) = k mi 2 i=1 ky J n 0

Simplification not a special graph... in GPP m : relax diag(y ) = u n tr(y ) = n remove nonnegativity constraints (GPP eig ) 1 min 2 tr LY s.t. tr(y ) = n tr(jy ) = k ky J n 0 mi 2 i=1 Y = d z i F i, z i R (i = 0,..., d) i=0

Simplification not a special graph... in GPP m : relax diag(y ) = u n tr(y ) = n remove nonnegativity constraints (GPP eig ) 1 min 2 tr LY s.t. tr(y ) = n tr(jy ) = k ky J n 0 mi 2 i=1 Y = d z i F i, z i R (i = 0,..., d) i=0 tr(ly ) = tr( d λ j F j ( d z i F i )) = d λ i f i z i j=0 i=0 i=0 where 0 = λ 0... λ d distinct eigenvalues of L etc....

Simplification not a special graph... Theorem Let G = (V, E) be a graph, m T = (m 1,..., m k ) s.t. k j=1 m j = n. Then the GPP eig bound for the minimum k-partition of G equals m i m j, λ 1 n i<j and the bound GPP eig for the maximum k-partition of G equals m i m j. λ d n i<j

Simplification not a special graph... Theorem Let G = (V, E) be a graph, m T = (m 1,..., m k ) s.t. k j=1 m j = n. Then the GPP eig bound for the minimum k-partition of G equals m i m j, λ 1 n i<j and the bound GPP eig for the maximum k-partition of G equals m i m j. λ d n i<j for the bisection the above results coincide with: M. Juvan, B. Mohar: Optimal linear labelings and eigenvalues of graphs. Discrete Appl. Math., 36:153 168, 1992. for the min 3-partition: J. Falkner, F. Rendl, H. Wolkowicz. A computational study of graph partitioning. Math. Program., 66:211 239, 1994.

. computational results...

Quality of the presented bounds G n partition GPP eig GPP m Doob 64 8 112 160 design 90 9 360 360 grid graph 100 (50,25,25) 4 6 Higman-Sims 100 20 950 950 Table : Lower bounds for the min graph partition.

Quality of the presented bounds G n partition GPP eig GPP m Doob 64 8 112 160 design 90 9 360 360 grid graph 100 (50,25,25) 4 6 Higman-Sims 100 20 950 950 Table : Lower bounds for the min graph partition. G n m GPP m GPP m GPP m ind J(7, 2) 21 (11,10) 37 37 40 Foster 90 (45,45) 13 18 14 Biggs-Smith 102 (70,32) 10 15 10 Table : Lower bounds for the min bisection. each bound computed in a few seconds

. vector lifting for the GPP...

Vector lifting for GPP let m = (m 1,..., m k ) T, i m i = n

Vector lifting for GPP let m = (m 1,..., m k ) T, i m i = n X P k := { X R n k : Xu k = u n, X T u n = m, x ij {0, 1} }

Vector lifting for GPP let m = (m 1,..., m k ) T, i m i = n X P k := { X R n k : Xu k = u n, X T u n = m, x ij {0, 1} } define y := vec(x ), Y := yy T relax Y yy T 0

Vector lifting for GPP let m = (m 1,..., m k ) T, i m i = n X P k := { X R n k : Xu k = u n, X T u n = m, x ij {0, 1} } define y := vec(x ), Y := yy T relax Y yy T 0 (GPP v ) min 1 2 tr((j k I k ) A)Y s.t. tr((j k I k ) I n )Y = 0 tr(i k J n )Y + tr(y ) = ( k mi 2 + n) ( 1 y T y Y i=1 ) + 2y T ((m + u k ) u n ) S + nk+1, Y 0 H. Wolkowicz and Q. Zhao. Semidefinite programming relaxations for the graph partitioning problem. Discrete Appl. Math., 96 97:461 479, 1999. original Zhao-Wolkowicz relaxation does not include Y 0

Vector lifting for GPP Theorem (S., 2012) When restricted to the equipartition, GPP v and GPP m are equivalent.

Vector lifting for GPP Theorem (S., 2012) When restricted to the equipartition, GPP v and GPP m are equivalent. Theorem (S., 2013) When restricted to the bisection, GPP v dominates GPP m.

Vector lifting for GPP Theorem (S., 2012) When restricted to the equipartition, GPP v and GPP m are equivalent. Theorem (S., 2013) When restricted to the bisection, GPP v dominates GPP m. numerical experiments show: gap between GPP v and GPP m reduces for k > 5

. How to strengthen GPP v?

. How to strengthen GPP v? We demonstrate for the bisection problem.

New bound for the bisection assign a pair of vertices of G to different parts of the partition

New bound for the bisection assign a pair of vertices of G to different parts of the partition Which pair of vertices?

New bound for the bisection assign a pair of vertices of G to different parts of the partition Which pair of vertices? consider the action of aut(a) on the pair of vertices (i, j) i.e., orbital: {(Pe i, Pe j ) : P aut(a)}

New bound for the bisection assign a pair of vertices of G to different parts of the partition Which pair of vertices? consider the action of aut(a) on the pair of vertices (i, j) i.e., orbital: {(Pe i, Pe j ) : P aut(a)} orbitals represent the different kinds of pairs of vertices

New bound for the bisection assign a pair of vertices of G to different parts of the partition Which pair of vertices? consider the action of aut(a) on the pair of vertices (i, j) i.e., orbital: {(Pe i, Pe j ) : P aut(a)} orbitals represent the different kinds of pairs of vertices assume that there are t such orbitals: O h (h = 1, 2,..., t) we prove the following

New bound for the bisection Theorem. Let G be an undirected graph with adjacency matrix A, and t orbitals O h (h = 1, 2,..., t) of edges and nonedges.

New bound for the bisection Theorem. Let G be an undirected graph with adjacency matrix A, and t orbitals O h (h = 1, 2,..., t) of edges and nonedges. Let (r h1, r h2 ) be an arbitrary pair of vertices in O h (h = 1, 2,..., t).

New bound for the bisection Theorem. Let G be an undirected graph with adjacency matrix A, and t orbitals O h (h = 1, 2,..., t) of edges and nonedges. Let (r h1, r h2 ) be an arbitrary pair of vertices in O h (h = 1, 2,..., t). Then min tr Z T AZ(J 2 I 2 ) = Z P 2 min h=1,2,...,t min tr X T AX (J 2 I 2 ), X P 2(h) where P 2 (h) = {X P 2 : X rh1,1 = 1, X rh2,2 = 1} (h = 1, 2,..., t).

New bound for the bisection Theorem. Let G be an undirected graph with adjacency matrix A, and t orbitals O h (h = 1, 2,..., t) of edges and nonedges. Let (r h1, r h2 ) be an arbitrary pair of vertices in O h (h = 1, 2,..., t). Then min tr Z T AZ(J 2 I 2 ) = Z P 2 min h=1,2,...,t min tr X T AX (J 2 I 2 ), X P 2(h) where P 2 (h) = {X P 2 : X rh1,1 = 1, X rh2,2 = 1} (h = 1, 2,..., t). for each h, compute: µ h := {GPP v with two additional constraints}

New bound for the bisection Theorem. Let G be an undirected graph with adjacency matrix A, and t orbitals O h (h = 1, 2,..., t) of edges and nonedges. Let (r h1, r h2 ) be an arbitrary pair of vertices in O h (h = 1, 2,..., t). Then min tr Z T AZ(J 2 I 2 ) = Z P 2 min h=1,2,...,t min tr X T AX (J 2 I 2 ), X P 2(h) where P 2 (h) = {X P 2 : X rh1,1 = 1, X rh2,2 = 1} (h = 1, 2,..., t). for each h, compute: µ h := {GPP v with two additional constraints} the new lower bound for the bisection problem is: GPP fix := min h=1,...,t µ h

. computational results...

Comparison of bounds... in general, it is difficult to solve GPP fix

Comparison of bounds... in general, it is difficult to solve GPP fix but for graphs with symmetry... G n m T GPP m GPP v GPP m ind GPP fix J(6, 2) 15 (8,7) 23 23 26 24 Gewirtz 56 (53,3) 23 24 23 26 M 22 77 (74,3) 41 42 41 44 Higman-Sims 100 25-part. 960 960 960 964 Table : Lower bounds for the min GPP each bound computed with IPM in < 30s

Example: the bandwidth problem

Example: the bandwidth problem The Bandwidth Problem in graphs: label the vertices v i of G with distinct integers φ(v i ) s.t. max φ(v i) φ(v j ) minimal (v i,v j ) E

Example: the bandwidth problem The Bandwidth Problem in graphs: label the vertices v i of G with distinct integers φ(v i ) s.t. max φ(v i) φ(v j ) minimal (v i,v j ) E 2 3 8 1 00000000 11111111 5 4 7 6

Example: the bandwidth problem The Bandwidth Problem in graphs: label the vertices v i of G with distinct integers φ(v i ) s.t. max φ(v i) φ(v j ) minimal (v i,v j ) E 2 3 3 7 8 1 00000000 11111111 1 5 5 4 2 6 7 6 4 8

Example: the bandwidth problem The Bandwidth Problem in graphs: label the vertices v i of G with distinct integers φ(v i ) s.t. max φ(v i) φ(v j ) minimal (v i,v j ) E 2 3 3 7 8 1 00000000 11111111 1 5 5 4 2 6 7 6 0 1 0 0 0 0 1 1 1 0 1 0 1 0 0 0 0 1 0 1 0 0 0 1 0 0 1 0 0 1 0 1 0 1 0 1 0 0 1 0 0 0 0 1 0 0 1 1 1 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0 4 8 0 0 1 1 1 0 0 0 0 0 1 1 0 1 0 0 1 1 0 0 0 0 1 0 1 1 0 0 0 0 0 1 1 0 0 0 0 0 1 1 0 1 0 0 0 0 1 1 0 0 1 0 1 1 0 0 0 0 0 1 1 1 0 0

The bandwidth problem the bandwidth problem is related to the following GPP problem

The bandwidth problem the bandwidth problem is related to the following GPP problem The min-cut problem is: OPT MC := min s.t. i S 1,j S 2 a ij (S 1, S 2, S 3 ) partitions V S i = m i, i = 1, 2, 3, where A = (a ij ) is the adjacency matrix of G.

The bandwidth problem the bandwidth problem is related to the following GPP problem The min-cut problem is: OPT MC := min s.t. i S 1,j S 2 a ij (S 1, S 2, S 3 ) partitions V S i = m i, i = 1, 2, 3, where A = (a ij ) is the adjacency matrix of G. bandwidth lower bound (Povh-Rendl (2007), van Dam-S.): If for some m = (m 1, m 2, m 3 ) it holds that OPT MC ν > 0, then σ (G) m 3 + 12 + 2ν + 1 4

The bandwidth problem - SDP relaxation SDP relaxations for the min-cut: solve GPP v and GPP fix with objective 1 trace(d A)Y 2 where D = 0 1 0 1 0 0 0 0 0

Bandwidth of Hamming graphs... Hamming graph H(d, q) is the graph Cartesian product of d copies of the complete graph K q.

Bandwidth of Hamming graphs... Hamming graph H(d, q) is the graph Cartesian product of d copies of the complete graph K q. q nodes old bw v time(s) bw fix time(s) u.b. 3 27 9 10 0 12 44 13 4 64 22 22 3 25 176 31 5 125 42 43 15 47 536 60 6 216 72 74 76 78 1756 101 Table : Bounds on the bandwidth of H(3, q) bw v and bw fix obtained by use of: m 3 + 12 + 2α + 1 4 upper bounds obtained by improved rev. Cuthill-McKee algor.

More on bounds... we also compute the best known lower/upper bounds for: H(4, q) the 3-dimensional generalized Hamming graphs H q1,q2,q3 the Johnson and Kneser graphs

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