SDP and eigenvalue bounds for the graph partition problem Renata - PowerPoint PPT Presentation

SDP and GPP Simplification – ‘highly symmetric’ graphs . . . SotirovR r z i A i , z i ∈ R ( r ≪ n 2 ) ⇒ Y = � i =1 r 1 2 tr( AJ n ) − 1 � min z i tr( AA i ) 2 i =1 s . t . � z i diag( A i ) = u n i ∈I (GPP m ) r k m 2 � z i tr( JA i ) = � i i =1 i =1 r � z i A i − J n � 0 , z i ≥ 0 , i = 1 , . . . , r . k j =1

SDP and GPP Simplification – ‘highly symmetric’ graphs . . . SotirovR r z i A i , z i ∈ R ( r ≪ n 2 ) ⇒ Y = � i =1 r 1 2 tr( AJ n ) − 1 � min z i tr( AA i ) 2 i =1 s . t . � z i diag( A i ) = u n i ∈I (GPP m ) r k m 2 � z i tr( JA i ) = � i i =1 i =1 r � z i A i − J n � 0 , z i ≥ 0 , i = 1 , . . . , r . k j =1 LMI may be (block-)diagonalized

SDP and GPP Simplification – ‘highly symmetric’ graphs . . . SotirovR r z i A i , z i ∈ R ( r ≪ n 2 ) ⇒ Y = � i =1 r 2 tr( AJ n ) − 1 1 � min z i tr( AA i ) 2 i =1 s . t . � z i diag( A i ) = u n i ∈I (GPP m ) r k m 2 � z i tr( JA i ) = � i i =1 i =1 r � z i A i − J n � 0 , z i ≥ 0 , i = 1 , . . . , r . k 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.

SDP and GPP On aggregating constraints . . . SotirovR for a given ( a , b , c ) consider the ∆ inequality y ab + y ac ≤ 1 + y bc

SDP and GPP On aggregating constraints . . . SotirovR 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.,

SDP and GPP On aggregating constraints . . . SotirovR 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.: hj ′ tr A i J + p j p i hj ′ tr A i Y + p h ij tr A h Y ≤ p i i ′ h tr A j Y , ij : A i A j = � r where p h h =1 p h ij A h j ′ : is the index s.t. A j ′ = A T j

SDP and GPP On aggregating constraints . . . SotirovR 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.: hj ′ tr A i J + p j p i hj ′ tr A i Y + p h ij tr A h Y ≤ p i i ′ h tr A j Y , ij : A i A j = � r where p h h =1 p h ij A h j ′ : is the index s.t. A j ′ = A T j use: Y = � r j =1 z j A j

SDP and GPP On aggregating constraints . . . SotirovR 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.: hj ′ tr A i J + p j p i hj ′ tr A i Y + p h ij tr A h Y ≤ p i i ′ h tr A j Y , ij : A i A j = � r where p h h =1 p h ij A h j ′ : is the index s.t. A j ′ = A T j use: Y = � r j =1 z j A j ♯ of aggregated ∆ constraints is bounded by r 3

SDP and GPP On aggregating constraints . . . SotirovR 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.: hj ′ tr A i J + p j p i hj ′ tr A i Y + p h ij tr A h Y ≤ p i i ′ h tr A j Y , ij : A i A j = � r where p h h =1 p h ij A h j ′ : is the index s.t. A j ′ = A T j use: Y = � r j =1 z j A j ♯ of aggregated ∆ constraints is bounded by r 3 similar approach applies to independent set constr. when k = 2

SDP and GPP Simplification – ’highly symmetric’ graphs . . . SotirovR Example. Strongly regular graph (SRG)

SDP and GPP Simplification – ’highly symmetric’ graphs . . . SotirovR Example. Strongly regular graph (SRG) n vertices, κ the valency of the graph

SDP and GPP Simplification – ’highly symmetric’ graphs . . . SotirovR 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

SDP and GPP Simplification – ’highly symmetric’ graphs . . . SotirovR 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 }

SDP and GPP Simplification – ’highly symmetric’ graphs . . . SotirovR 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 )

SDP and GPP Simplification – ’highly symmetric’ graphs . . . SotirovR 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 ) 1 min 2 κ n (1 − z 1 ) k κ z 1 + ( n − κ − 1) z 2 = 1 m 2 � s . t . i − 1 n i =1 (GPP m ) 1 + rz 1 − ( r + 1) z 2 ≥ 0 1 + sz 1 − ( s + 1) z 2 ≥ 0 z 1 , z 2 ≥ 0

SDP and GPP SRG SotirovR 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 1 � i m 2 max � i < j m i m j , n ( κ + 1) − � i n 2 Similarly, the SDP bound for the maximum k -partition is � � κ − s 1 min � i < j m i m j , 2 κ n . n

SDP and GPP SRG SotirovR 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 1 � i m 2 max � i < j m i m j , n ( κ + 1) − � i n 2 Similarly, the SDP bound for the maximum k -partition is � � κ − s 1 min � i < j m i m j , 2 κ n . n 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 SRG SotirovR � n � after aggregating, 3 ∆ constraints remain: 3 z 1 ≤ 1 z 2 ≤ 1 2 z 1 − z 2 ≤ 1 − z 1 + 2 z 2 ≤ 1

SDP and GPP SRG SotirovR � n � after aggregating, 3 ∆ constraints remain: 3 z 1 ≤ 1 z 2 ≤ 1 2 z 1 − z 2 ≤ 1 − z 1 + 2 z 2 ≤ 1 Prop . For SRG with n > 5 the ∆ ineq. are redundant in GPP m .

SDP and GPP SRG SotirovR � n � after aggregating, 3 ∆ constraints remain: 3 z 1 ≤ 1 z 2 ≤ 1 2 z 1 − z 2 ≤ 1 − z 1 + 2 z 2 ≤ 1 Prop . For SRG with n > 5 the ∆ ineq. are redundant in GPP m . However, the independent set constraints improve GPP m .

SDP and GPP Simplification – ’not a special’ graph . . . SotirovR

SDP and GPP Simplification – ’not a special’ graph . . . SotirovR closed form expression for the GPP for ’any’ graph

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

SDP and GPP Simplification – ’not a special’ graph . . . SotirovR 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

SDP and GPP Simplification – ’not a special’ graph . . . SotirovR 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 U T i , ∀ 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

SDP and GPP Simplification – ’not a special’ graph . . . SotirovR in GPP m : relax diag( Y ) = u n � tr( Y ) = n remove nonnegativity constraints

SDP and GPP Simplification – ’not a special’ graph . . . SotirovR in GPP m : relax diag( Y ) = u n � tr( Y ) = n remove nonnegativity constraints 1 min 2 tr LY s . t . tr( Y ) = n (GPP eig ) k m 2 tr( JY ) = � i i =1 kY − J n � 0

SDP and GPP Simplification – ’not a special’ graph . . . SotirovR in GPP m : relax diag( Y ) = u n � tr( Y ) = n remove nonnegativity constraints 1 min 2 tr LY s . t . tr( Y ) = n (GPP eig ) k m 2 tr( JY ) = � i i =1 kY − J n � 0 d Y = � z i F i , z i ∈ R ( i = 0 , . . . , d ) i =0

SDP and GPP Simplification – ’not a special’ graph . . . SotirovR in GPP m : relax diag( Y ) = u n � tr( Y ) = n remove nonnegativity constraints 1 min 2 tr LY s . t . tr( Y ) = n (GPP eig ) k m 2 tr( JY ) = � i i =1 kY − J n � 0 d Y = � z i F i , z i ∈ R ( i = 0 , . . . , d ) i =0 d d d � � � tr( LY ) = tr( λ j F j ( z i F i )) = λ i f i z i j =0 i =0 i =0 where 0 = λ 0 ≤ . . . ≤ λ d distinct eigenvalues of L etc . . . .

SDP and GPP Simplification – ’not a special’ graph . . . SotirovR 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 λ 1 � m i m j , n i < j and the bound GPP eig for the maximum k -partition of G equals λ d � m i m j . n i < j

SDP and GPP Simplification – ’not a special’ graph . . . SotirovR 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 λ 1 � m i m j , n i < j and the bound GPP eig for the maximum k -partition of G equals λ d � m i m j . 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.

SDP and GPP . SotirovR computational results . . .

SDP and GPP Quality of the presented bounds SotirovR 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.

SDP and GPP Quality of the presented bounds SotirovR 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

SDP and GPP . SotirovR vector lifting for the GPP . . .

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

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

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

SDP and GPP Vector lifting for GPP SotirovR let m = ( m 1 , . . . , m k ) T , � i m i = n X ∈ R n × k : Xu k = u n , X T u n = m , x ij ∈ { 0 , 1 } � � X ∈ P k := define y := vec ( X ), Y := yy T � relax Y − yy T � 0 1 min 2 tr (( J k − I k ) ⊗ A ) Y s . t . tr (( J k − I k ) ⊗ I n ) Y = 0 k � m 2 tr ( I k ⊗ J n ) Y + tr ( Y ) = − ( i + n ) (GPP v ) i =1 + 2 y T (( m + u k ) ⊗ u n ) � y T � 1 ∈ S + nk +1 , Y ≥ 0 y Y 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

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

SDP and GPP Vector lifting for GPP SotirovR 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 .

SDP and GPP Vector lifting for GPP SotirovR 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

SDP and GPP . SotirovR How to strengthen GPP v ?

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

SDP and GPP New bound for the bisection SotirovR ⇛ assign a pair of vertices of G to different parts of the partition

SDP and GPP New bound for the bisection SotirovR ⇛ assign a pair of vertices of G to different parts of the partition Which pair of vertices?

SDP and GPP New bound for the bisection SotirovR ⇛ 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 ) }

SDP and GPP New bound for the bisection SotirovR ⇛ 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

SDP and GPP New bound for the bisection SotirovR ⇛ 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

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

SDP and GPP New bound for the bisection SotirovR 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 h 1 , r h 2 ) be an arbitrary pair of vertices in O h ( h = 1 , 2 , . . . , t ).

SDP and GPP New bound for the bisection SotirovR 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 h 1 , r h 2 ) be an arbitrary pair of vertices in O h ( h = 1 , 2 , . . . , t ). Then Z ∈P 2 tr Z T AZ ( J 2 − I 2 ) = X ∈P 2 ( h ) tr X T AX ( J 2 − I 2 ) , min min min h =1 , 2 ,..., t where P 2 ( h ) = { X ∈ P 2 : X r h 1 , 1 = 1 , X r h 2 , 2 = 1 } ( h = 1 , 2 , . . . , t ).

SDP and GPP New bound for the bisection SotirovR 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 h 1 , r h 2 ) be an arbitrary pair of vertices in O h ( h = 1 , 2 , . . . , t ). Then Z ∈P 2 tr Z T AZ ( J 2 − I 2 ) = X ∈P 2 ( h ) tr X T AX ( J 2 − I 2 ) , min min min h =1 , 2 ,..., t where P 2 ( h ) = { X ∈ P 2 : X r h 1 , 1 = 1 , X r h 2 , 2 = 1 } ( h = 1 , 2 , . . . , t ). ⇒ for each h , compute: µ ∗ h := { GPP v with two additional constraints }

SDP and GPP New bound for the bisection SotirovR 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 h 1 , r h 2 ) be an arbitrary pair of vertices in O h ( h = 1 , 2 , . . . , t ). Then Z ∈P 2 tr Z T AZ ( J 2 − I 2 ) = X ∈P 2 ( h ) tr X T AX ( J 2 − I 2 ) , min min min h =1 , 2 ,..., t where P 2 ( h ) = { X ∈ P 2 : X r h 1 , 1 = 1 , X r h 2 , 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: h =1 ,..., t µ ∗ GPP fix := min h

SDP and GPP . SotirovR computational results . . .

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

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

�������� �������� SDP and GPP Example: the bandwidth problem SotirovR

�������� �������� SDP and GPP Example: the bandwidth problem SotirovR The Bandwidth Problem in graphs: label the vertices v i of G with distinct integers φ ( v i ) s.t. ( v i , v j ) ∈ E | φ ( v i ) − φ ( v j ) | max minimal

SDP and GPP Example: the bandwidth problem SotirovR The Bandwidth Problem in graphs: label the vertices v i of G with distinct integers φ ( v i ) s.t. ( v i , v j ) ∈ E | φ ( v i ) − φ ( v j ) | max minimal 2 3 8 1 �������� �������� 5 4 7 6

SDP and GPP Example: the bandwidth problem SotirovR The Bandwidth Problem in graphs: label the vertices v i of G with distinct integers φ ( v i ) s.t. ( v i , v j ) ∈ E | φ ( v i ) − φ ( v j ) | max minimal 2 3 3 7 5 8 1 1 �������� �������� 5 4 2 6 7 6 4 8

SDP and GPP Example: the bandwidth problem SotirovR The Bandwidth Problem in graphs: label the vertices v i of G with distinct integers φ ( v i ) s.t. ( v i , v j ) ∈ E | φ ( v i ) − φ ( v j ) | max minimal 2 3 3 7 5 8 1 1 �������� �������� 5 4 2 6 7 6 4 8 0 1 0 0 0 0 1 1 0 0 1 1 1 0 0 0     1 0 1 0 1 0 0 0 0 0 1 1 0 1 0 0     0 1 0 1 0 0 0 1 1 1 0 0 0 0 1 0         0 0 1 0 0 1 0 1 1 1 0 0 0 0 0 1         0 1 0 1 0 0 1 0 1 0 0 0 0 0 1 1         0 0 0 1 0 0 1 1 0 1 0 0 0 0 1 1         1 0 0 1 0 1 0 0 0 0 1 0 1 1 0 0     1 0 1 0 0 1 0 0 0 0 0 1 1 1 0 0

SDP and GPP The bandwidth problem SotirovR ⇛ the bandwidth problem is related to the following GPP problem