Applications of semidefinite programming, symmetry and algebra to graph partitioning problems

Transcription

1 Applications of semidefinite programming, symmetry and algebra to graph partitioning problems Edwin van Dam and Renata Sotirov Tilburg University, The Netherlands Summer 2018 Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

2 Semidefinite programming... generalization of linear programming (LP) unifies linear and quadratic programming problems arise naturally as relaxation of discrete optimization problems can be efficiently solved by interior-point-methods applications: global and combinatorial optimization eigenvalue optimization robust optimization circuit design coding theory finance signal processing chemical engineering sensor network localization, etc. Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

3 Where is SDP? NLP convex SDP LP Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

4 Primal SDP Primal problem: min tr(cx ) s.t. tr(a i X ) = b i, i = 1,..., m X 0 where C, A i S n, b i R (i = 1,..., m). S n... space of symmetric n n matrices X 0... positive semidefinite iff z T Xz 0, z R n iff all eigenvalues of X are 0 N.B. SDP reduces to LP when all matrices are diagonal. Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

5 Historical events related to SDP Lyapunov (1890) stability of dynamic systems Bellman and Fan (1963) first SDP formulated Lovász (1979) upper bound Shannon capacity of a graph Lovász and Schrijver (1991) SDP can provide tighter relaxations of 0-1 problems than LP Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

6 Historical events related to SDP Lyapunov (1890) stability of dynamic systems Bellman and Fan (1963) first SDP formulated Lovász (1979) upper bound Shannon capacity of a graph Lovász and Schrijver (1991) SDP can provide tighter relaxations of 0-1 problems than LP Goemans, Williamson (1995) SDP-based approximation for max-cut Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

7 On solving SDP... Polynomial time algorithms: Ellipsoid method Grötschel, Lovász and Schrijver (1988) first to solve SDP in polynomial time not practical Interior-point methods (IPM) Nesterov and Nemirovski (1994), Alizadeh (1995) practical, suitable for medium size available software: CSDP DSDP SDPA SDPT3 SeDuMi Mosek since 1995 the interest in SDP has grown tremendously Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

8 Max-cut Given: G = (V, E),... an undirected graph with V = n w ij = w ji 0... the weight of edge (i, j) E MC problem. Find partition of V into S and V \ S s.t. the total weight of the edges joining S and V \ S is maximized. Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

9 Max-cut Given: G = (V, E),... an undirected graph with V = n w ij = w ji 0... the weight of edge (i, j) E MC problem. Find partition of V into S and V \ S s.t. the total weight of the edges joining S and V \ S is maximized. { 1 for j S x j := 1 for j V \ S (MC) max 1 4 n i,j=1 w ij (1 x i x j ) s.t. x j { 1, 1}, j = 1,..., n. NP-hard problem Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

10 Goemans-Williamson Let Y = xx T (MC) max 1 4 n i,j=1 w ij(1 Y ij ) s.t. x j { 1, 1}, j = 1,..., n Y = xx T Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

11 Goemans-Williamson Let Y = xx T (MC) (MC) max 1 4 n i,j=1 w ij(1 Y ij ) s.t. x j { 1, 1}, j = 1,..., n Y = xx T max 1 4 n i,j=1 w ij(1 Y ij ) s.t. Y jj = 1, j = 1,..., n Y 0, rank(y ) = 1 Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

12 Goemans-Williamson Let Y = xx T (MC) (MC) max 1 4 n i,j=1 w ij(1 Y ij ) s.t. x j { 1, 1}, j = 1,..., n Y = xx T max 1 4 n i,j=1 w ij(1 Y ij ) s.t. Y jj = 1, j = 1,..., n Y 0, rank(y ) = 1 relax the rank one constraint (SDP MC ) max 1 4 n i,j=1 w ij(1 Y ij ) s.t. Y jj = 1, j = 1,..., n Y 0 Goemans, Williamson (1995): this relaxation has an error 13.82% Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

13 Max-cut remarks strengthened SDP MC by adding 4 ( n 3) triangle constraints: MET y ij + y ik + y jk 1 y ij y ik y jk 1 y ij + y ik y jk 1 y ij y ik + y jk 1, i < j < k the resulting SDP is difficult to solve with IPM when n > 150 The bundle method computes nearly optimal solution for n 2000: Fischer, Gruber, Rendl, and Sotirov. Computational Experience with a Bundle Approach for Semidefinite Cutting Plane Relaxations of Max-Cut and Equipartition, Math. Program B, 105(2-3): , Branch and bound, SDP based solver for Max-cut: Rinaldi, Rendl and Wiegele. Biq Mac Solver - Binary quadratic and Max cut Solver, Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

14 GPP the graph partition problem... Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

15 The Graph Partition Problem G = (V, E)... an undirected graph V... vertex set, V = n E... edge set The k-partition problem: 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. when m i = V k, i the graph equipartition problem when k = 2 the bisection problem GPP is NP-hard (Garey and Johnson, 1976) applications: VLSI design, parallel computing, floor planning, telecommunications, etc. Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

16 The k-partition problem A... the adjacency matrix of G, m := (m 1,..., m k ) T, u n all-ones vector V k let X = (x ij ) R { 1, if node i Sj x ij := 0, if node i / S j P k := { X R n k : Xu k = u n, X T u n = m, x ij {0, 1} } Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

17 The k-partition problem A... the adjacency matrix of G, m := (m 1,..., m k ) T, u n all-ones vector V k let X = (x ij ) R { 1, if node i Sj x ij := 0, if node i / S j P k := { X R n k : Xu k = u n, X T u n = m, x ij {0, 1} } For X P k : 1 2 tr(x T AX ) = j weight of edges within S j : w(e cut ) = 1 2 tr(x T Diag(Au n )X X T AX ) = 1 2 tr(x T LX ), where L := Diag(Au n ) A is the Laplacian matrix of G Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

18 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} Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

19 SDP for GPP linearize the objective (matrix lifting): trace(lxx T ) trace(ly) Y conv{ỹ : X P k s.t. Ỹ = XX T } ky J n 0. Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

20 SDP for GPP linearize the objective (matrix lifting): trace(lxx T ) trace(ly) Y conv{ỹ : X P k s.t. Ỹ = XX T } ky J n 0. (GPP RS ) min 1 2 tr(ly ) s.t. diag(y ) = u n tr(jy ) = k mi 2 i=1 ky J n 0, Y 0 Sotirov, An efficient SDP relaxation for the GPP, INFORMS J. Comput. (2014) Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

21 Improvements?? How to improve GPP RS? impose the linear inequalities: constraints y ij + y ik 1 + y jk, (i, j, k) independent set type of constraints y ij 1, I s.t. I = k + 1 i<j,i,j I there are 3 ( ) ( n 3, and n k+1) independent set constraints Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

22 Some facts... for graphs with 100 vertices: the best known vector lifting relaxation is hopeless GPP RS + + independent set constraints computes bounds in about 3 hours GPP RS computes bounds in about 14 minutes? Can we compute GPP RS more efficiently? Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

23 Some facts... for graphs with 100 vertices: the best known vector lifting relaxation is hopeless GPP RS + + independent set constraints computes bounds in about 3 hours GPP RS computes bounds in about 14 minutes? Can we compute GPP RS more efficiently? yes Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

24 Symmetry and algebra matrix *-algebra: subspace of R n n that is closed under matrix multiplication and transposition Assumption: The data matrices of an SDP problem and I belong to a matrix *-algebra of dimension r, where r n 2 Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

25 Symmetry and algebra matrix *-algebra: subspace of R n n that is closed under matrix multiplication and transposition Assumption: The data matrices of an SDP problem and I belong to a matrix *-algebra of dimension r, where r n 2 Then, if the SDP relaxation has an optimal solution then it has an optimal solution in the matrix *-algebra. Schrijver, Goemans, Rendl, Parrilo, De Klerk, Pasechnik, Sotirov,... Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

26 Symmetry and algebra matrix *-algebra: subspace of R n n that is closed under matrix multiplication and transposition Assumption: The data matrices of an SDP problem and I belong to a matrix *-algebra of dimension r, where r n 2 Then, if the SDP relaxation has an optimal solution then it has an optimal solution in the matrix *-algebra. Schrijver, Goemans, Rendl, Parrilo, De Klerk, Pasechnik, Sotirov,... Coherent algebra with basis of 01-matrices (centralizer ring, for example): (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. Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

27 Simplification highly symmetric graphs... Y = r z i A i, i=1 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 (GPP m ) 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 Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

28 Simplification highly symmetric graphs... Y = r z i A i, 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 independent set constraints extend the approach from: M.X. Goemans, F. Rendl. Semidefinite Programs and Association Schemes. Computing, 63(4): , Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

29 On aggregating constraints... for a given (a, b, c) consider the constraint y ab + y ac 1 + y bc if (A i ) ab = 1, (A h ) ac = 1, (A j ) bc = 1 type (i, j, h) constraint Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

30 On aggregating constraints... for a given (a, b, c) consider the constraint y ab + y ac 1 + y bc if (A i ) ab = 1, (A h ) ac = 1, (A j ) bc = 1 type (i, j, h) constraint summing all constraints of type (i, j, h) aggregated constraint: 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, Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

31 On aggregating constraints... for a given (a, b, c) consider the constraint y ab + y ac 1 + y bc if (A i ) ab = 1, (A h ) ac = 1, (A j ) bc = 1 type (i, j, h) constraint summing all constraints of type (i, j, h) aggregated constraint: 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, of aggregated constraints is bounded by r 3 similar approach applies to independent set constraints when k = 2 Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

32 Strongly regular graphs Example. Strongly regular graph 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 ) Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

33 Strongly regular graphs Example. Strongly regular graph 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 sz 1 (s + 1)z 2 0 z 1, z 2 0 k mi 2 1 i=1 Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

34 Strongly regular graphs... 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 (lower) 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 (upper) bound for the maximum k-partition is { } κ s min n i<j m 1 im j, 2 κn. Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

35 Strongly regular graphs... 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 (lower) 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 (upper) 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, Sotirov, Dobre: On SDP relaxations of maximum k-section, Math. Program. Ser. B, 136(2): , Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

36 Adding constraints after aggregating constraints: z 1 1 z 2 1 2z 1 z 2 1 z 1 + 2z 2 1 For SRG with n > 5 the constraints are redundant in GPP m. However, the independent set constraints improve GPP m. Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

37 The Laplacian algebra 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 of G F i = U i U T i (eigenspace decomposition, LU i = λ i U i ) F i F j = δ ij F i for i j d i=0 F i = I F i = Fi T, i tr(f i ) = f i... the multiplicity of i-th eigenvalue of L Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

38 Simplification any graph... relax diag(y ) = u n tr(y ) = n remove nonnegativity constraint (GPP eig ) 1 min 2 tr LY s.t. tr(y ) = n tr(jy ) = k ky J n 0 mi 2 i=1 Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

39 Simplification any graph... relax diag(y ) = u n tr(y ) = n remove nonnegativity constraint (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 y i F i, y i R (i = 0,..., d) i=0 tr(ly ) = tr( d λ j F j ( d y i F i )) = d λ i f i y i j=0 i=0 i=0 where 0 = λ 0... λ d distinct eigenvalues of L,... Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

40 Eigenvalue bounds Theorem Let G = (V, E) be a graph, m i, i = 1,..., 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, and the bound GPP eig for the maximum k-partition of G equals m i m j. λ 1 n λ d n i<j i<j Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

41 Eigenvalue bounds Theorem Let G = (V, E) be a graph, m i, i = 1,..., 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, and the bound GPP eig for the maximum k-partition of G equals m i m j. λ 1 n λ d n i<j i<j Other known closed form expression only for the minimum k-partition when k = 2, 3: J. Falkner, F. Rendl, and H. Wolkowicz. A computational study of graph partitioning. Math. Program., 66: , Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

42 Computational times for presented bounds exploit symmetry if available G n m time, no symmetry r aut time grid graph 100 (50,25,25) Table: Computational time (s.) to solve GPP RS computational time to solve GPP RS + constraints, with n = 100: without symmetry, about 2 hours aggregating constraints, if possible, a few seconds Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

43 Quality of the presented bounds G n k GPP eig GPP RS r comb time r aut time Chang SRG(64, 18) Doob SRG(64, 18) e (45, 12, 3)-design G n k GPP eig GPP RS GPP RS + Desargues Foster Biggs-Smith Table: Lower bounds for the min k-partition. each computation in the last two columns of the last table < 1 s. Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

44 The max-k-cut G = (V, E)... an undirected graph V... vertex set, V = n E... edge set The max-k-cut problem Find a partition of V into into at most k subsets such that the total weight of edges joining different sets is maximized. the max-k-cut... is NP-hard k = 2 the max-cut Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

45 An SDP relaxation A... the adjacency matrix of G L := Diag(Au n ) A is the Laplacian matrix of G (k MC) max 1 2 tr(ly ) s.t. diag(y ) = u n ky J n 0, Y 0 where J n (resp. u n ) is all-ones matrix (resp. vector) restriction on sizes of the parts GPP and independent set constraints can be added (k MC) is equivalent to the relaxation from: Frieze, Jerrum. Improved approximation algorithms for max-k-cut and max bisection. Algorithmica 18:67-81, Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

46 Eigenvalue bounds: the max-k-cut relax diag(y ) = u n tr(y ) = n remove nonnegativity constraint Theorem Let G = (V, E) be a graph on n vertices and k an integer k 2. The eigenvalue (upper) bound for the max-k-cut problem is: n(k 1) 2k λ max (L). Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

47 Eigenvalue bounds: the max-k-cut relax diag(y ) = u n tr(y ) = n remove nonnegativity constraint Theorem Let G = (V, E) be a graph on n vertices and k an integer k 2. The eigenvalue (upper) bound for the max-k-cut problem is: n(k 1) 2k λ max (L). for k = 2 this result coincides with: Mohar and Poljak. Eigenvalues and the max-cut problem. Czechoslovak Mathematical Journal, 40: , there are few other eigenvalue bounds for the max-k-cut when k > 2 (Nikiforov) Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

48 Eigenvalue bounds: the max-k-cut Delorme and Poljak (1993) How to improve the eigenvalue bound n(k 1) 2k λ max (L)? Perturbations of L by a diagonal matrix with zero trace do not change the optimal value of the max-cut problem, but have an impact on λ max (L). Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

49 Eigenvalue bounds: the max-k-cut Delorme and Poljak (1993) How to improve the eigenvalue bound n(k 1) 2k λ max (L)? Perturbations of L by a diagonal matrix with zero trace do not change the optimal value of the max-cut problem, but have an impact on λ max (L). similarly for the max-k-cut ( ) min d T u n=0 where d is known as the correcting vector n(k 1) λ max (L + Diag(d)) 2k Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

50 Eigenvalue bounds: the max-k-cut Delorme and Poljak (1993) How to improve the eigenvalue bound n(k 1) 2k λ max (L)? Perturbations of L by a diagonal matrix with zero trace do not change the optimal value of the max-cut problem, but have an impact on λ max (L). similarly for the max-k-cut ( ) min d T u n=0 where d is known as the correcting vector ( ) is equivalent to: n(k 1) λ max (L + Diag(d)) 2k max{ 1 2 tr(ly ) : diag(y ) = u n, ky J n 0} no closed form! perturbations of objectives and SDP were studied by Alizadeh 1995 Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

51 The chromatic number The chromatic number of a graph A coloring of a graph is an assignment of colors to the vertices of G s.t. no two adjacent vertices have the same color. The smallest number of colors needed to color G is called its chromatic number χ(g). Figure: Petersen graph, χ(g) = 3 Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

52 The chromatic number The chromatic number of a graph A coloring of a graph is an assignment of colors to the vertices of G s.t. no two adjacent vertices have the same color. The smallest number of colors needed to color G is called its chromatic number χ(g). A coloring with k colors is the same as a partition V into k independent sets. For a given graph G = (V, E) and integer k, if max-k-cut < E then χ(g) k + 1. The eigenvalue bound for the max-k-cut a bound on χ(g) Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

53 Eigenvalue bound for the chromatic number Theorem Let G = (V, E) be a graph with Laplacian matrix L. Then χ(g) E nλ max (L) 2 E Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

54 Eigenvalue bound for the chromatic number Theorem Let G = (V, E) be a graph with Laplacian matrix L. Then χ(g) E nλ max (L) 2 E Hoffman bound, 1970: χ(g) 1 θmax(a) θ min (A), where θ max(a) and θ min (A) are largest and smallest eigenvalue of adjacency matrix A. For regular graphs, these two bounds coincide. Otherwise, they are incomparable. For the complete graph on 100 vertices minus an edge, our bound is 99 (= χ(g)) while the Hoffman bound is 51. Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

55 Walk-regular graphs Walk-regular graphs A graph with adjacency matrix A is called walk-regular if A l has constant diagonal for every nonnegative integer l. The class of walk-regular graphs contains: vertex-transitive graphs distance-regular graphs (including strongly regular graphs) graphs in an association scheme Walk-regular graphs: are regular graphs all matrices in L of a walk-regular graph have constant diagonal Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

56 Max-k-cut for walk-regular graphs the optimum correcting vector d in ( ) min d T u n=0 n(k 1) λ max (L + Diag(d)) 2k equals the zero vector. Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

57 Max-k-cut for walk-regular graphs the optimum correcting vector d in ( ) min d T u n=0 n(k 1) λ max (L + Diag(d)) 2k equals the zero vector. Theorem Let G be a walk-regular graph on n vertices and let k be an integer k 2. Then the eigenvalue bound for the max-k-cut equals the bound ( ). For k = 2 the eigenvalue bound equals the optimal value of the SDP rel. (k MC). Goemans and Rendl (1999) proved the latter result for the max-cut problem for graphs in an association scheme. Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

58 Strongly regular graph Theorem Let G = (V, E) be a SRG with eigenvalues κ, r, s. Then the SDP bound (k-mc) for the max-k-cut of G is given by { } min n(k 1) 1 2k (κ s), 2 κn. Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

59 Strongly regular graph Theorem Let G = (V, E) be a SRG with eigenvalues κ, r, s. Then the SDP bound (k-mc) for the max-k-cut of G is given by { } min n(k 1) 1 2k (κ s), 2 κn. For SRG with n > 5 the (aggregated) constraints are redundant in (k MC). the independent set constraints improve (k MC) Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

60 Hamming Graphs Hamming graph H(d, q, j) (j = 0,..., d) Vertex set S d, where S is a set of size q Vertices are adjacent if they differ in j coordinates Conjecture Let j d d 1 q, with j even if q = 2. Then K j(0) K j (1) (Kravchuk) is the largest Laplacian eigenvalue of H(d, q, j). Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

61 Hamming Graphs Hamming graph H(d, q, j) (j = 0,..., d) Vertex set S d, where S is a set of size q Vertices are adjacent if they differ in j coordinates Conjecture Let j d d 1 q, with j even if q = 2. Then K j(0) K j (1) (Kravchuk) is the largest Laplacian eigenvalue of H(d, q, j). Theorem Let k q, j d d 1 q, with j even if q = 2, and consider H(d, q, j). If the conjecture is true, then: for the max-k-cut problem, the eigenvalue and (k MC) bound are equal. the optimal value of the max-q-cut equals the eigenvalue bound. Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

62 Hamming Graphs Hamming graph H(d, q, j) (j = 0,..., d) Vertex set S d, where S is a set of size q Vertices are adjacent if they differ in j coordinates Conjecture Let j d d 1 q, with j even if q = 2. Then K j(0) K j (1) (Kravchuk) is the largest Laplacian eigenvalue of H(d, q, j). Theorem Let k q, j d d 1 q, with j even if q = 2, and consider H(d, q, j). If the conjecture is true, then: for the max-k-cut problem, the eigenvalue and (k MC) bound are equal. the optimal value of the max-q-cut equals the eigenvalue bound. The conjecture was recently proven by Brouwer, Cioabă, Ihringer, and McGinnis (JCTB, 201?) Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

63 The bandwidth issue Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

64 The Bandwidth Problem for graphs { } σ(g) := min max φ(i) φ(j) ; φ : V {1,..., n} (i,j) E find a permutation P such that in PAP T all nonzero entries are as close as possible to the main diagonal Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

65 The Bandwidth Problem for graphs { } σ(g) := min max φ(i) φ(j) ; φ : V {1,..., n} (i,j) E Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

66 Applications... The bandwidth problem... originated in the 1950s from sparse matrix computations NP-hard (Papadimitriou (1976)) engineering applications efficient storage and processing minimizing distortion in the multi-channel transmission Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

67 A quadratic assignment formulation of the bandwidth natural problem formulation Fix k... define B = (b ij ): b ij := the bandwidth related to the QAP: { 1 for i j > k 0 otherwise µ = min P Π n tr(apbp T ), where Π n is the set of permutation matrices if µ > 0 σ(g) > k Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

68 The QAP based bound α QAP := min tr(b A)Y s.t. tr(i E jj )Y = 1, tr(e jj I )Y = 1 j tr(i (J I )) + (J I ) I )Y = 0 tr(jy ) = n 2 Y 0, Y 0 ( ) E jj = e j e T j I is the identity matrix J is all-ones matrix QAP SDP formulation by Povh and Rendl (2009); Zhao, Karisch, Rendl, and Wolkowicz (1998) Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

69 The QAP based bound α QAP := min tr(b A)Y s.t. tr(i E jj )Y = 1, tr(e jj I )Y = 1 j tr(i (J I )) + (J I ) I )Y = 0 tr(jy ) = n 2 Y 0, Y 0 ( ) E jj = e j e T j I is the identity matrix J is all-ones matrix QAP SDP formulation by Povh and Rendl (2009); Zhao, Karisch, Rendl, and Wolkowicz (1998) since aut(b) = {P Π n : PBP T = B} has order only 2 look for other approaches Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

70 The min-cut problem S 1, S 2, S 3 V S i = m i for i = 1, 2, 3, i m i = n The min-cut problem is: (MC) OPT MC := min s.t. i S 1,j S 2 a ij (S 1, S 2, S 3 ) partitions V where A = (a ij ) is the adjacency matrix.? relation to the bandwidth problem? Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

71 The min-cut and bandwidth problem The cube, m = (m 1, m 2, m 3 ) = (2, 3, 3) m 1 m 3 m if OPT MC > 0 σ(g) m Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

72 The min-cut and bandwidth problem The cube, m = (m 1, m 2, m 3 ) = (2, 3, 3) m 1 m 3 m if OPT MC > 0 σ(g) m generalized bound (Povh-Rendl (2007), EvD-Sotirov): If for some m = (m 1, m 2, m 3 ) it holds that OPT MC α > 0, then σ(g) m α Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

73 Eigenvalue based bound for the min-cut Helmberg, Rendl, Mohar, Poljak (1995), derive the mc bound: α L = 1 2 (µ 2λ 2 + µ 1 λ n ), where λ 2 (λ n)... the second smallest (largest resp.) Laplacian eigenvalue of G µ 1 and µ 2 are constants depending on m = (m 1, m 2, m 3) α L is the closed form solution of a minimization problem over {X R n 3 : X T X = Diag(m), Xu 3 = u n, X T u n = m} Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

74 The QAP based bound for the min-cut the min-cut can be formulated as the QAP min tr(ax BX T ), X Π n where A is the adjacency matrix of G and B := 0 m 1 m 1 0 m3 m 1 0 m1 m 3 0 m3 m 3 J m1 m 2 0 m3 m 2 J m2 m 1 0 m2 m 3 0 m2 m 2 Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

75 The QAP based bound for the min-cut the min-cut can be formulated as the QAP min tr(ax BX T ), X Π n where A is the adjacency matrix of G and B := 0 m 1 m 1 0 m3 m 1 0 m1 m 3 0 m3 m 3 J m1 m 2 0 m3 m 2 J m2 m 1 0 m2 m 3 0 m2 m 2 B generates a coherent algebra of rank 12. Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

76 On solving ( ) relaxation where min 1 2 tr A(X 3 + X 5 ) s.t. X 1 + X 6 + X 11 = I n 2 12 i=1 X i = J n 2, tr(jx i ) = p i, X i 0, i = 1,..., 12, 12 i=1 p 1 i B i X i 0 X 3 = X T 5, X 4 = X T 9, X 8 = X T 11, X 1, X 2, X 6, X 7, X 11, X 12 S n 2, p i (i = 1,..., 12) are given constants related m = (m 1, m 2, m 3 ) this reduction was introduced by De Klerk and Sotirov (2010), see also: E. de Klerk, F.M. de Oliveira Filho, and D.V. Pasechnik. Relaxations of combinatorial problems via association schemes, in Handbook of Semidefinite, Cone and Polynomial Optimization, Miguel Anjos and Jean Lasserre (eds.), pp , Springer, Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

77 The QAP based bound for the min-cut Theorem. Let G be an undirected graph with n vertices and adjacency matrix A, and m = (m 1, m 2, m 3 ), i m 1 = n. Then, α QAP α L.? How can we further improve the lower bound for the min-cut? Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

78 Bounds for the min-cut Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

79 New bound for the min-cut assume: G A is edge transitive the graph with adjacency matrix B is edge transitive one can fix arbitrary edge in B and compute a lower bound for the original QAP from the SDP relaxation of the reduced QAP Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

80 New bound for the min-cut assume: G A is edge transitive the graph with adjacency matrix B is edge transitive one can fix arbitrary edge in B and compute a lower bound for the original QAP from the SDP relaxation of the reduced QAP Theorem. Let G A be an undirected graph with adjacency matrix A. Suppose for simplicity that aut(g A ) is transitive on both edges and non-edges. Then for any fixed edge (s 1, s 2 ) in G B, and any fixed edge (r 1, r 2 ) and non-edge (q 1, q 2 ) in the graph G A one has min tr(axbx T ) = min{ min tr(azbz T ), min tr(ayby T )} X Π n Z Π n Y Π n where Z r1,s 1 = 1, Z r2,s 2 = 1 and Y q1,s 1 = 1, Y q2,s 2 = 1. Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

81 New bound for the min-cut assume: G A is edge transitive the graph with adjacency matrix B is edge transitive one can fix arbitrary edge in B and compute a lower bound for the original QAP from the SDP relaxation of the reduced QAP Theorem. Let G A be an undirected graph with adjacency matrix A. Suppose for simplicity that aut(g A ) is transitive on both edges and non-edges. Then for any fixed edge (s 1, s 2 ) in G B, and any fixed edge (r 1, r 2 ) and non-edge (q 1, q 2 ) in the graph G A one has min tr(axbx T ) = min{ min tr(azbz T ), min tr(ayby T )} X Π n Z Π n Y Π n where Z r1,s 1 = 1, Z r2,s 2 = 1 and Y q1,s 1 = 1, Y q2,s 2 = 1. this can be extended to graphs that are edge-transitive and have several classes of non-edges Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

82 Bandwidth of Johnson graphs... Let Ω be a set of size v {vertices of the Johnson graph J(v, d)} = ( Ω d) and two subsets are adjacent if their intersections has size d 1 v nodes bw L bw QAP time(s) bw new time(s) u.b Table: Bounds on the bandwidth of J(v, 3) lower bounds: m α upper bounds obtained by improving Cuthill-McKee heuristic Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

83 Bandwidth of Hamming graphs... Hamming graph H(d, q) q nodes bw L bw QAP time(s) bw new time(s) u.b Table: Bounds on the bandwidth of H(3, q) lower bounds: m α Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53

84 The end Edwin van Dam (Tilburg) SDP, symmetry, algebra, GPP Summer / 53