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SDP and GPP SotirovR

SDP and eigenvalue bounds for the graph partition problem

Renata Sotirov and Edwin van Dam

Tilburg University, The Netherlands

SDP and GPP SotirovR

Outline . . .

the graph partition problem

SDP and GPP SotirovR

Outline . . .

the graph partition problem ւց matrix lifting vector lifting

SDP and GPP SotirovR

Outline . . .

the graph partition problem ւց matrix lifting vector lifting ւ ց simplify complicate

SDP and GPP SotirovR

Outline . . .

the graph partition problem ւց matrix lifting vector lifting ւ ց simplify complicate ⇓ ⇓ ???

SDP and GPP SotirovR

The Graph Partition Problem

G = (V , E) . . . an undirected graph

V . . . vertex set, |V | = n E . . . edge set

SDP and GPP SotirovR

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 S1, . . . , Sk of given sizes m1 ≥ . . . ≥ mk, s.t. the total weight of edges joining different Si is minimized.

SDP and GPP SotirovR

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 S1, . . . , Sk of given sizes m1 ≥ . . . ≥ mk, s.t. the total weight of edges joining different Si is minimized. mi = |V |

k , ∀i the graph equipartition problem

k = 2 the bisection problem

SDP and GPP SotirovR

The k-partition problem

A . . . the adjacency matrix of G, m := (m1, . . . , mk)T

SDP and GPP SotirovR

The k-partition problem

A . . . the adjacency matrix of G, m := (m1, . . . , mk)T let X = (xij) ∈ R|V |×k xij := 1, if vertex i ∈ Sj 0, if vertex i / ∈ Sj

SDP and GPP SotirovR

The k-partition problem

A . . . the adjacency matrix of G, m := (m1, . . . , mk)T let X = (xij) ∈ R|V |×k xij := 1, if vertex i ∈ Sj 0, if vertex i / ∈ Sj Pk :=

• X ∈ Rn×k : Xuk = un, X Tun = m, xij ∈ {0, 1}
• ,

where un . . . vector of all ones

SDP and GPP SotirovR

The k-partition problem

A . . . the adjacency matrix of G, m := (m1, . . . , mk)T let X = (xij) ∈ R|V |×k xij := 1, if vertex i ∈ Sj 0, if vertex i / ∈ Sj Pk :=

• X ∈ Rn×k : Xuk = un, X Tun = m, xij ∈ {0, 1}
• ,

where un . . . vector of all ones For X ∈ Pk: w(Ecut) = 1 2 tr(X TLX) = 1 2 tr A(Jn − XX T) where L := Diag(Aun) − A is the Laplacian matrix of G

SDP and GPP SotirovR

The Graph Partition Problem

The trace formulation: (GPP) min

1 2trace(X TLX)

s.t. Xuk = un X Tun = m xij ∈ {0, 1}

SDP and GPP SotirovR

The Graph Partition Problem

The trace formulation: (GPP) min

1 2trace(X TLX)

s.t. Xuk = un X Tun = m xij ∈ {0, 1} the GPP . . . is NP-hard (Garey and Johnson, 1976)

SDP and GPP SotirovR

The Graph Partition Problem

The trace formulation: (GPP) min

1 2trace(X TLX)

s.t. Xuk = un X Tun = m xij ∈ {0, 1} the GPP . . . is NP-hard (Garey and Johnson, 1976) applications: VLSI design, parallel computing, floor planning, telecommunications, etc.

SDP and GPP SotirovR

.

matrix lifting SDP for the GPP . . .

SDP and GPP SotirovR

SDP for GPP

linearize the objective: trace(LXXT) trace(LY)

SDP and GPP SotirovR

SDP for GPP

linearize the objective: trace(LXXT) trace(LY) Y ∈ conv{ ˜ Y : ∃X ∈ Pk s.t. ˜ Y = XX T} ⇒ kY − Jn 0.

SDP and GPP SotirovR

SDP for GPP

linearize the objective: trace(LXXT) trace(LY) Y ∈ conv{ ˜ Y : ∃X ∈ Pk s.t. ˜ Y = XX T} ⇒ kY − Jn 0.

S., 2013

(GPPm) min

1 2 tr(LY )

s.t. diag(Y ) = un tr(JY ) =

k

• i=1

m2

i

kY − Jn 0, Y ≥ 0

SDP and GPP SotirovR

SDP for GPP

linearize the objective: trace(LXXT) trace(LY) Y ∈ conv{ ˜ Y : ∃X ∈ Pk s.t. ˜ Y = XX T} ⇒ kY − Jn 0.

S., 2013

(GPPm) min

1 2 tr(LY )

s.t. diag(Y ) = un tr(JY ) =

k

• i=1

m2

i

kY − Jn 0, Y ≥ 0 for k = 2 the nonnegativity constraints are redundant

SDP and GPP SotirovR

GPPm and known relaxations

GPPm . . . 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.

SDP and GPP SotirovR

GPPm and known relaxations

GPPm . . . 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.

SDP and GPP SotirovR

Strengthening?

SDP and GPP SotirovR

Strengthening?

? How to strengthen GPPm?

SDP and GPP SotirovR

Strengthening?

? How to strengthen GPPm? impose the linear inequalities: ∆ constraints yab + yac ≤ 1 + ybc, ∀(a, b, c) independent set constraints

• a<b, a,b∈W

yab ≥ 1, ∀ W s.t. |W| = k + 1

SDP and GPP SotirovR

Strengthening?

? How to strengthen GPPm? impose the linear inequalities: ∆ constraints yab + yac ≤ 1 + ybc, ∀(a, b, c) independent set constraints

• a<b, a,b∈W

yab ≥ 1, ∀ W s.t. |W| = k + 1 ⇒ there are 3 n

3

• ∆, and

n

k+1

• independent set constraints

SDP and GPP SotirovR

On computational issues . . .

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

SDP and GPP SotirovR

On computational issues . . .

in general, for graphs with 100 vertices and k = 3: the best known vector lifting relaxation is hopeless GPPm + triangle inequalities + independent set solves ∼ 3 h

SDP and GPP SotirovR

On computational issues . . .

in general, for graphs with 100 vertices and k = 3: the best known vector lifting relaxation is hopeless GPPm + triangle inequalities + independent set solves ∼ 3 h GPPm solves ∼ 14 min

SDP and GPP SotirovR

On computational issues . . .

in general, for graphs with 100 vertices and k = 3: the best known vector lifting relaxation is hopeless GPPm + triangle inequalities + independent set solves ∼ 3 h GPPm solves ∼ 14 min Can we compute GPPm with/without additional

• constr. more efficiently ?

SDP and GPP SotirovR

On computational issues . . .

in general, for graphs with 100 vertices and k = 3: the best known vector lifting relaxation is hopeless GPPm + triangle inequalities + independent set solves ∼ 3 h GPPm solves ∼ 14 min Can we compute GPPm with/without additional

• constr. more efficiently ?

yes

SDP and GPP SotirovR

Simplification – ‘highly symmetric’ graphs . . .

SDP and GPP SotirovR

Simplification – ‘highly symmetric’ graphs . . .

matrix *-algebra: subspace of Rn×n that is closed under matrix multiplication and taking transposes

SDP and GPP SotirovR

Simplification – ‘highly symmetric’ graphs . . .

matrix *-algebra: subspace of Rn×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{A1, . . . , Ar} where r ≪ n2

SDP and GPP SotirovR

Simplification – ‘highly symmetric’ graphs . . .

matrix *-algebra: subspace of Rn×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{A1, . . . , Ar} where r ≪ n2 Then, if the SDP relaxation has an optimal solution ⇒ it has an optimal solution in the matrix *-algebra.

SDP and GPP SotirovR

Simplification – ‘highly symmetric’ graphs . . .

matrix *-algebra: subspace of Rn×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{A1, . . . , Ar} where r ≪ n2 Then, if the SDP relaxation has an optimal solution ⇒ it has an optimal solution in the matrix *-algebra.

Schrijver, Goemans, Rendl, Parrilo, . . .

SDP and GPP SotirovR

Simplification – ‘highly symmetric’ graphs . . .

matrix *-algebra: subspace of Rn×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{A1, . . . , Ar} where r ≪ n2 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) Ai ∈ {0, 1}n×n, AT

i ∈ {A1, . . . , Ar}, (i = 1, . . . , r)

(ii) r

i=1 Ai = J,

• i∈I Ai = I, I ⊂ {1, . . . , r}

(iii) For i, j ∈ {1, . . . , r}, ∃ph

ij such that AiAj = r h=1 ph ijAh.

SDP and GPP SotirovR

Simplification – ‘highly symmetric’ graphs . . .

⇒ Y =

r

• i=1

ziAi, zi ∈ R (r ≪ n2) (GPPm) min

1 2 tr(AJn) − 1 2 r

• i=1

zi tr(AAi) s.t.

• i∈I

zi diag(Ai) = un

r

• i=1

zi tr(JAi) =

k

• i=1

m2

i

k

r

• j=1

ziAi − Jn 0, zi ≥ 0, i = 1, . . . , r.

SDP and GPP SotirovR

Simplification – ‘highly symmetric’ graphs . . .

⇒ Y =

r

• i=1

ziAi, zi ∈ R (r ≪ n2) (GPPm) min

1 2 tr(AJn) − 1 2 r

• i=1

zi tr(AAi) s.t.

• i∈I

zi diag(Ai) = un

r

• i=1

zi tr(JAi) =

k

• i=1

m2

i

k

r

• j=1

ziAi − Jn 0, zi ≥ 0, i = 1, . . . , r. LMI may be (block-)diagonalized

SDP and GPP SotirovR

Simplification – ‘highly symmetric’ graphs . . .

⇒ Y =

r

• i=1

ziAi, zi ∈ R (r ≪ n2) (GPPm) min

1 2 tr(AJn) − 1 2 r

• i=1

zi tr(AAi) s.t.

• i∈I

zi diag(Ai) = un

r

• i=1

zi tr(JAi) =

k

• i=1

m2

i

k

r

• j=1

ziAi − Jn 0, zi ≥ 0, i = 1, . . . , r. LMI may be (block-)diagonalized exploit properties of Ai 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.

SDP and GPP SotirovR

On aggregating constraints . . .

for a given (a, b, c) consider the ∆ inequality yab + yac ≤ 1 + ybc

SDP and GPP SotirovR

On aggregating constraints . . .

for a given (a, b, c) consider the ∆ inequality yab + yac ≤ 1 + ybc if (Ai)ab = 1, (Ah)ac = 1, (Aj)bc = 1 ⇒ type (i, j, h) ineq.,

SDP and GPP SotirovR

On aggregating constraints . . .

for a given (a, b, c) consider the ∆ inequality yab + yac ≤ 1 + ybc if (Ai)ab = 1, (Ah)ac = 1, (Aj)bc = 1 ⇒ type (i, j, h) ineq., by summing all ineq. of type (i, j, h), the aggregated ∆ ineq.: pi

hj′ tr AiY + ph ij tr AhY ≤ pi hj′ tr AiJ + pj i′h tr AjY ,

where ph

ij: AiAj = r h=1 ph ijAh

j′: is the index s.t. Aj′ = AT

j

SDP and GPP SotirovR

On aggregating constraints . . .

for a given (a, b, c) consider the ∆ inequality yab + yac ≤ 1 + ybc if (Ai)ab = 1, (Ah)ac = 1, (Aj)bc = 1 ⇒ type (i, j, h) ineq., by summing all ineq. of type (i, j, h), the aggregated ∆ ineq.: pi

hj′ tr AiY + ph ij tr AhY ≤ pi hj′ tr AiJ + pj i′h tr AjY ,

where ph

ij: AiAj = r h=1 ph ijAh

j′: is the index s.t. Aj′ = AT

j

use: Y = r

j=1 zjAj

SDP and GPP SotirovR

On aggregating constraints . . .

for a given (a, b, c) consider the ∆ inequality yab + yac ≤ 1 + ybc if (Ai)ab = 1, (Ah)ac = 1, (Aj)bc = 1 ⇒ type (i, j, h) ineq., by summing all ineq. of type (i, j, h), the aggregated ∆ ineq.: pi

hj′ tr AiY + ph ij tr AhY ≤ pi hj′ tr AiJ + pj i′h tr AjY ,

where ph

ij: AiAj = r h=1 ph ijAh

j′: is the index s.t. Aj′ = AT

j

use: Y = r

j=1 zjAj

♯ of aggregated ∆ constraints is bounded by r 3

SDP and GPP SotirovR

On aggregating constraints . . .

for a given (a, b, c) consider the ∆ inequality yab + yac ≤ 1 + ybc if (Ai)ab = 1, (Ah)ac = 1, (Aj)bc = 1 ⇒ type (i, j, h) ineq., by summing all ineq. of type (i, j, h), the aggregated ∆ ineq.: pi

hj′ tr AiY + ph ij tr AhY ≤ pi hj′ tr AiJ + pj i′h tr AjY ,

where ph

ij: AiAj = r h=1 ph ijAh

j′: is the index s.t. Aj′ = AT

j

use: Y = r

j=1 zjAj

♯ of aggregated ∆ constraints is bounded by r 3 similar approach applies to independent set constr. when k = 2

SDP and GPP SotirovR

Simplification – ’highly symmetric’ graphs . . .

• Example. Strongly regular graph (SRG)

SDP and GPP SotirovR

Simplification – ’highly symmetric’ graphs . . .

• Example. Strongly regular graph (SRG)

n vertices, κ the valency of the graph

SDP and GPP SotirovR

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 ⊥ un

SDP and GPP SotirovR

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 ⊥ un A belongs to the *-algebra spanned by {I, A, J − A − I}

SDP and GPP SotirovR

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 ⊥ un A belongs to the *-algebra spanned by {I, A, J − A − I} ⇒ Y = I + z1A + z2(J − A − I)

SDP and GPP SotirovR

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 ⊥ un A belongs to the *-algebra spanned by {I, A, J − A − I} ⇒ Y = I + z1A + z2(J − A − I) (GPPm) min

1 2κn(1 − z1)

s.t. κz1 + (n − κ − 1)z2 = 1

n k

• i=1

m2

i − 1

1 + rz1 − (r + 1)z2 ≥ 0 1 + sz1 − (s + 1)z2 ≥ 0 z1, z2 ≥ 0

SDP and GPP SotirovR

SRG

Theorem. Let G = (V , E) be a SRG with eigenvalues κ, r, s. Let mi ∈ N, i = 1, . . . , k s.t. k

j=1 mj = n.

Then the SDP bound for the minimum k-partition is

max

• κ−r

n

• i<j mimj,

1 2

• n(κ + 1) −

i m2 i

• Similarly, the SDP bound for the maximum k-partition is

min

• κ−s

n

• i<j mimj,

1 2κn

• .

SDP and GPP SotirovR

SRG

Theorem. Let G = (V , E) be a SRG with eigenvalues κ, r, s. Let mi ∈ N, i = 1, . . . , k s.t. k

j=1 mj = n.

Then the SDP bound for the minimum k-partition is

max

• κ−r

n

• i<j mimj,

1 2

• n(κ + 1) −

i m2 i

• Similarly, the SDP bound for the maximum k-partition is

min

• κ−s

n

• i<j mimj,

1 2κn

• .

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.

SDP and GPP SotirovR

SRG

after aggregating, 3 n

3

• ∆ constraints remain:

z1 ≤ 1 z2 ≤ 1 2z1 − z2 ≤ 1 −z1 + 2z2 ≤ 1

SDP and GPP SotirovR

SRG

after aggregating, 3 n

3

• ∆ constraints remain:

z1 ≤ 1 z2 ≤ 1 2z1 − z2 ≤ 1 −z1 + 2z2 ≤ 1

• Prop. For SRG with n > 5 the ∆ ineq. are redundant in GPPm.

SDP and GPP SotirovR

SRG

after aggregating, 3 n

3

• ∆ constraints remain:

z1 ≤ 1 z2 ≤ 1 2z1 − z2 ≤ 1 −z1 + 2z2 ≤ 1

• Prop. For SRG with n > 5 the ∆ ineq. are redundant in GPPm.

However, the independent set constraints improve GPPm.

SDP and GPP SotirovR

Simplification – ’not a special’ graph . . .

SDP and GPP SotirovR

Simplification – ’not a special’ graph . . .

closed form expression for the GPP for ’any’ graph

SDP and GPP SotirovR

Simplification – ’not a special’ graph . . .

closed form expression for the GPP for ’any’ graph L = Diag(Aun) − A . . . the Laplacian matrix of G

SDP and GPP SotirovR

Simplification – ’not a special’ graph . . .

closed form expression for the GPP for ’any’ graph L = Diag(Aun) − A . . . the Laplacian matrix of G L := span{F0, . . . , Fd} the Laplacian algebra corr. to L

SDP and GPP SotirovR

Simplification – ’not a special’ graph . . .

closed form expression for the GPP for ’any’ graph L = Diag(Aun) − A . . . the Laplacian matrix of G L := span{F0, . . . , Fd} the Laplacian algebra corr. to L Fi = UiUT

i , ∀i . . . where Ui corr. to the distinct eig. λi

d

i=0 Fi = I

FiFj = δijFi for i = j tr(Fi) = fi . . . the multiplicity of i-th eigenvalue of L

SDP and GPP SotirovR

Simplification – ’not a special’ graph . . .

in GPPm: relax diag(Y ) = un tr(Y ) = n remove nonnegativity constraints

SDP and GPP SotirovR

Simplification – ’not a special’ graph . . .

in GPPm: relax diag(Y ) = un tr(Y ) = n remove nonnegativity constraints (GPPeig) min

1 2 tr LY

s.t. tr(Y ) = n tr(JY ) =

k

• i=1

m2

i

kY − Jn 0

SDP and GPP SotirovR

Simplification – ’not a special’ graph . . .

in GPPm: relax diag(Y ) = un tr(Y ) = n remove nonnegativity constraints (GPPeig) min

1 2 tr LY

s.t. tr(Y ) = n tr(JY ) =

k

• i=1

m2

i

kY − Jn 0 Y =

d

• i=0

ziFi, zi ∈ R (i = 0, . . . , d)

SDP and GPP SotirovR

Simplification – ’not a special’ graph . . .

in GPPm: relax diag(Y ) = un tr(Y ) = n remove nonnegativity constraints (GPPeig) min

1 2 tr LY

s.t. tr(Y ) = n tr(JY ) =

k

• i=1

m2

i

kY − Jn 0 Y =

d

• i=0

ziFi, zi ∈ R (i = 0, . . . , d) tr(LY ) = tr(

d

• j=0

λjFj(

d

• i=0

ziFi)) =

d

• i=0

λifizi where 0 = λ0 ≤ . . . ≤ λd distinct eigenvalues of L

• etc. . . .

SDP and GPP SotirovR

Simplification – ’not a special’ graph . . .

Theorem Let G = (V , E) be a graph, mT = (m1, . . . , mk) s.t. k

j=1 mj = n.

Then the GPPeig bound for the minimum k-partition of G equals

λ1 n

• i<j

mimj, and the bound GPPeig for the maximum k-partition of G equals

λd n

• i<j

mimj.

SDP and GPP SotirovR

Simplification – ’not a special’ graph . . .

Theorem Let G = (V , E) be a graph, mT = (m1, . . . , mk) s.t. k

j=1 mj = n.

Then the GPPeig bound for the minimum k-partition of G equals

λ1 n

• i<j

mimj, and the bound GPPeig for the maximum k-partition of G equals

λd n

• i<j

mimj. 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.

SDP and GPP SotirovR

.

computational results . . .

SDP and GPP SotirovR

Quality of the presented bounds

G n partition GPPeig GPPm 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.

SDP and GPP SotirovR

Quality of the presented bounds

G n partition GPPeig GPPm 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 GPPm GPPm−∆ GPPm−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

SDP and GPP SotirovR

.

vector lifting for the GPP . . .

SDP and GPP SotirovR

Vector lifting for GPP

let m = (m1, . . . , mk)T,

• i mi = n

SDP and GPP SotirovR

Vector lifting for GPP

let m = (m1, . . . , mk)T,

• i mi = n

X ∈ Pk :=

• X ∈ Rn×k : Xuk = un, X Tun = m, xij ∈ {0, 1}

SDP and GPP SotirovR

Vector lifting for GPP

let m = (m1, . . . , mk)T,

• i mi = n

X ∈ Pk :=

• X ∈ Rn×k : Xuk = un, X Tun = m, xij ∈ {0, 1}
• define y := vec(X), Y := yy T relax Y − yy T 0

SDP and GPP SotirovR

Vector lifting for GPP

let m = (m1, . . . , mk)T,

• i mi = n

X ∈ Pk :=

• X ∈ Rn×k : Xuk = un, X Tun = m, xij ∈ {0, 1}
• define y := vec(X), Y := yy T relax Y − yy T 0

(GPPv) min

1 2tr((Jk − Ik) ⊗ A)Y

s.t. tr((Jk − Ik) ⊗ In)Y = 0 tr(Ik ⊗ Jn)Y + tr(Y ) = −(

k

• i=1

m2

i + n)

+ 2y T((m + uk) ⊗ un)

• 1

y T y Y

• ∈ 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.

• riginal Zhao-Wolkowicz relaxation does not include Y ≥ 0

SDP and GPP SotirovR

Vector lifting for GPP

Theorem (S., 2012) When restricted to the equipartition, GPPv and GPPm are equivalent.

SDP and GPP SotirovR

Vector lifting for GPP

Theorem (S., 2012) When restricted to the equipartition, GPPv and GPPm are equivalent. Theorem (S., 2013) When restricted to the bisection, GPPv dominates GPPm.

SDP and GPP SotirovR

Vector lifting for GPP

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

SDP and GPP SotirovR

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How to strengthen GPPv ?

SDP and GPP SotirovR

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How to strengthen GPPv ? We demonstrate for the bisection problem.

SDP and GPP SotirovR

New bound for the bisection

⇛ assign a pair of vertices of G to different parts of the partition

SDP and GPP SotirovR

New bound for the bisection

⇛ assign a pair of vertices of G to different parts of the partition Which pair of vertices?

SDP and GPP SotirovR

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: {(Pei, Pej) : P ∈ aut(A)}

SDP and GPP SotirovR

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: {(Pei, Pej) : P ∈ aut(A)}

• rbitals represent the ‘different’ kinds of pairs of vertices

SDP and GPP SotirovR

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: {(Pei, Pej) : P ∈ aut(A)}

• rbitals represent the ‘different’ kinds of pairs of vertices

assume that there are t such orbitals: Oh (h = 1, 2, . . . , t) ⇛ we prove the following

SDP and GPP SotirovR

New bound for the bisection

• Theorem. Let G be an undirected graph with adjacency matrix A,

and t orbitals Oh (h = 1, 2, . . . , t) of edges and nonedges.

SDP and GPP SotirovR

New bound for the bisection

• Theorem. Let G be an undirected graph with adjacency matrix A,

and t orbitals Oh (h = 1, 2, . . . , t) of edges and nonedges. Let (rh1, rh2) be an arbitrary pair of vertices in Oh (h = 1, 2, . . . , t).

SDP and GPP SotirovR

New bound for the bisection

• Theorem. Let G be an undirected graph with adjacency matrix A,

and t orbitals Oh (h = 1, 2, . . . , t) of edges and nonedges. Let (rh1, rh2) be an arbitrary pair of vertices in Oh (h = 1, 2, . . . , t). Then min

Z∈P2 tr Z TAZ(J2 − I2) =

min

h=1,2,...,t

min

X∈P2(h) tr X TAX(J2 − I2),

where P2(h) = {X ∈ P2 : Xrh1,1 = 1, Xrh2,2 = 1} (h = 1, 2, . . . , t).

SDP and GPP SotirovR

New bound for the bisection

• Theorem. Let G be an undirected graph with adjacency matrix A,

and t orbitals Oh (h = 1, 2, . . . , t) of edges and nonedges. Let (rh1, rh2) be an arbitrary pair of vertices in Oh (h = 1, 2, . . . , t). Then min

Z∈P2 tr Z TAZ(J2 − I2) =

min

h=1,2,...,t

min

X∈P2(h) tr X TAX(J2 − I2),

where P2(h) = {X ∈ P2 : Xrh1,1 = 1, Xrh2,2 = 1} (h = 1, 2, . . . , t). ⇒ for each h, compute:

µ∗

h := {GPPv with two additional constraints}

SDP and GPP SotirovR

New bound for the bisection

• Theorem. Let G be an undirected graph with adjacency matrix A,

and t orbitals Oh (h = 1, 2, . . . , t) of edges and nonedges. Let (rh1, rh2) be an arbitrary pair of vertices in Oh (h = 1, 2, . . . , t). Then min

Z∈P2 tr Z TAZ(J2 − I2) =

min

h=1,2,...,t

min

X∈P2(h) tr X TAX(J2 − I2),

where P2(h) = {X ∈ P2 : Xrh1,1 = 1, Xrh2,2 = 1} (h = 1, 2, . . . , t). ⇒ for each h, compute:

µ∗

h := {GPPv with two additional constraints}

⇒ the new lower bound for the bisection problem is: GPPfix := min

h=1,...,t µ∗ h

SDP and GPP SotirovR

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computational results . . .

SDP and GPP SotirovR

Comparison of bounds . . .

in general, it is difficult to solve GPPfix

SDP and GPP SotirovR

Comparison of bounds . . .

in general, it is difficult to solve GPPfix but for graphs with symmetry . . .

G n mT GPPm GPPv GPPm−ind GPPfix J(6, 2) 15 (8,7) 23 23 26 24 Gewirtz 56 (53,3) 23 24 23 26 M22 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

SDP and GPP SotirovR

Example: the bandwidth problem

SDP and GPP SotirovR

Example: the bandwidth problem

The Bandwidth Problem in graphs: label the vertices vi of G with distinct integers φ(vi) s.t. max

(vi,vj)∈E |φ(vi) − φ(vj)|

minimal

SDP and GPP SotirovR

Example: the bandwidth problem

The Bandwidth Problem in graphs: label the vertices vi of G with distinct integers φ(vi) s.t. max

(vi,vj)∈E |φ(vi) − φ(vj)|

minimal

• 1

5 2 7 6 4 3 8

SDP and GPP SotirovR

Example: the bandwidth problem

The Bandwidth Problem in graphs: label the vertices vi of G with distinct integers φ(vi) s.t. max

(vi,vj)∈E |φ(vi) − φ(vj)|

minimal

• 1

5 2 7 6 4 3 8 8 5 4 2 6 7 3 1

SDP and GPP SotirovR

Example: the bandwidth problem

The Bandwidth Problem in graphs: label the vertices vi of G with distinct integers φ(vi) s.t. max

(vi,vj)∈E |φ(vi) − φ(vj)|

minimal

• 1

5 2 7 6 4 3 8 8 5 4 2 6 7 3 1

           1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1                       1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1           

SDP and GPP SotirovR

The bandwidth problem

⇛ the bandwidth problem is related to the following GPP problem

SDP and GPP SotirovR

The bandwidth problem

⇛ the bandwidth problem is related to the following GPP problem The min-cut problem is: OPTMC := min

• i∈S1,j∈S2

aij s.t. (S1, S2, S3) partitions V |Si| = mi, i = 1, 2, 3, where A = (aij) is the adjacency matrix of G.

SDP and GPP SotirovR

The bandwidth problem

⇛ the bandwidth problem is related to the following GPP problem The min-cut problem is: OPTMC := min

• i∈S1,j∈S2

aij s.t. (S1, S2, S3) partitions V |Si| = mi, i = 1, 2, 3, where A = (aij) is the adjacency matrix of G. bandwidth lower bound (Povh-Rendl (2007), van Dam-S.): If for some m = (m1, m2, m3) it holds that OPTMC ≥ ν > 0, then σ∞(G) ≥ m3 +

• − 1

2 +

• 2ν + 1

4

SDP and GPP SotirovR

The bandwidth problem - SDP relaxation

SDP relaxations for the min-cut: solve GPPv and GPPfix with objective 1 2trace(D ⊗ A)Y where D =   1 1  

SDP and GPP SotirovR

Bandwidth of Hamming graphs . . .

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

SDP and GPP SotirovR

Bandwidth of Hamming graphs . . .

Hamming graph H(d, q) is the graph Cartesian product of d copies of the complete graph Kq. q ♯ nodes

• ld

bwv time(s) bwfix time(s) u.b. 3 27 9 10 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)

bwv and bwfix obtained by use of: m3 +

• − 1

2 +

• 2α + 1

4

• upper bounds obtained by improved rev. Cuthill-McKee algor.

SDP and GPP SotirovR

More on bounds . . .

we also compute the best known lower/upper bounds for: H(4, q) the 3-dimensional generalized Hamming graphs Hq1,q2,q3 the Johnson and Kneser graphs

SDP and GPP SotirovR

ThAnK YoU!