1 relaxations of combinatorial problems via …1 relaxations of combinatorial problems via...

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Relaxations of combinatorial problems viaassociation schemes

Etienne de Klerk‡, Fernando M. de Oliveira Filho‡, and Dmitrii V.Pasechnik†

‡Tilburg University, The Netherlands; †Nanyang Technological University,Singapore

1.1 Introduction

Semidefinite programming relaxations of combinatorial problems date back tothe work of Lovasz [17] from 1979, who proposed a semidefinite programmingrelaxation for the maximum stable set problem which now is known as theLovasz theta number. More recently, Goemans and Williamson [9] showed howto use semidefinite programming to provide an approximation algorithm forthe maximum-cut problem; this algorithm achieves the best known approxi-mation ratio for the problem, which is moreover conjectured to be the bestpossible ratio under the unique games conjecture, a complexity-theoreticalassumption (cf. Khot, Kindler, Mossel, and O’Donnell [12]).

The usual approach to obtaining semidefinite programming relaxationsof combinatorial problems has been via binary variable reformulations. Forexample, the convex hull of the set {xxT : x ∈ {−1, 1}n } is approximated bythe convex elliptope

{X ∈ Rn×n : Xii = 1 for i = 1, . . . , n and X � 0 },

where X � 0 means that X is positive semidefinite. Stronger relaxations maybe obtained by the use of lift-and-project techniques, which provide hierarchiesof tighter and tighter semidefinite programming problems; see e.g. the articleby Laurent and Rendl [16] for a description of such methods.

Recently, de Klerk, Pasechnik, and Sotirov [13] presented a different ap-proach to derive semidefinite programming relaxations for the travelling sales-man problem; their approach is based on the theory of association schemes.

An association scheme is a set {A0, . . . , Ad} ⊆ Rn×n of 0–1 matrices thatsatisfy the following properties:

1. A0 = I and A0 + · · ·+Ad = J , where J is the all-ones matrix;2. AT

i ∈ {A0, . . . , Ad} for i = 1, . . . , d;3. AiAj = AjAi for i, j = 0, . . . , d;

2 Authors Suppressed Due to Excessive Length

4. AiAj ∈ span{A0, . . . , Ad} for i, j = 0, . . . , d.

We say that the association scheme {A0, . . . , Ad} has d classes. Associationschemes where all the matrices Ai are symmetric are called symmetric associ-ation schemes. In what follows, we will only work with symmetric associationschemes and will usually omit the word ”symmetric”. All the theory can beeasily extended to non-symmetric association schemes, though.

In this chapter we show how to obtain semidefinite programming relax-ations of the convex hull of an association scheme. To make ideas more precise,let {A0, . . . , Ad} be an association scheme. We show how to derive semidefiniteprogramming relaxations of the set

conv({A0, . . . , Ad}) = conv{ (PTA0P, . . . , PTAdP ) : P ∈ Rn×n is a

permutation matrix }.(1.1)

Notice that {PTA0P, . . . , PTAdP}, where P ∈ Rn×n is a permutation matrix,

is an association scheme as well.Many combinatorial problems can be expressed as the problem of finding

a maximum-weight matrix in an association scheme, and so by showing howto obtain relaxations of the convex hull of an association scheme, we provide aunified approach to obtaining semidefinite programming relaxations for manydifferent combinatorial problems.

As an example, consider the maximum bisection problem: We are givena symmetric matrix W ∈ R2m×2m with nonnegative entries, which we see asedge weights on the complete graph K2m on 2m vertices, and we wish to finda maximum-weight copy of the complete bipartite graph Km,m in K2m.

It is a simple matter to check that the 2m× 2m matrices

A0 = I, A1 =

(0 JmJm 0

), and A2 =

(Jm 00 Jm

), (1.2)

where Jm is the all-ones m ×m matrix, form an association scheme. Noticethat A1 is the adjacency matrix of Km,m, and that A2 is the matrix whosenonzero entries mark pairs of vertices of Km,m that are at distance 2 fromeach other.

One may now rewrite the maximum bisection problem as follows:

maximize 12 〈W,X1〉(X0, X1, X2) ∈ conv({A0, A1, A2}),

where for matrices X, Y ∈ Rn×n we write 〈X,Y 〉 for the trace inner productbetween X and Y .

In an analogous way, one may encode other combinatorial problems asoptimization problems over the convex hull of an association scheme, and soby providing a way to derive semidefinite programming relaxations for theconvex hull of an association scheme, we provide a way to derive semidefiniteprogramming relaxations for many combinatorial problems. In particular, we

1 Relaxations of combinatorial problems via association schemes 3

show how to obtain known relaxations for the travelling salesman and themaximum bisection problems, and new relaxations for the cycle covering andmaximum p-section problems.

Finally, our approach may be further generalized by considering coher-ent configurations instead of association schemes. Coherent configurations aredefined as association schemes, but there is no requirement for the matricesto commute. Problems such as the maximum (k, l)-cut problem, which can-not be encoded as optimization problems over the convex hull of associationschemes, can be seen as optimization problems over the convex hull of coherentconfigurations.

1.2 Preliminaries and notation

All the graphs considered in this chapter are simple: They have no loops andno parallel edges.

We denote by AT the transpose of a matrix A. For complex matrices, wedenote by A∗ the conjugate transpose of A. By e ∈ Rn we denote the all-onesvector.

When we say that a real matrix is positive semidefinite, we mean that itis symmetric and has nonnegative eigenvalues. If a complex matrix is positivesemidefinite, then it is Hermitian and has nonnegative eigenvalues. We oftenuse the notation A � 0 to denote that A is positive semidefinite.

For matrices A and B ∈ Rn×n, we denote by 〈A,B〉 the trace inner productbetween A and B, that is

〈A,B〉 = trace(BTA) =

n∑i,j=1

AijBij .

When A, B ∈ Cn×n, the trace inner product is given by

〈A,B〉 = trace(B∗A) =

n∑i,j=1

AijBij .

We also need some basic properties of tensor products of vector spaces.For background on tensor products, see the book by Greub [10].

If U and V are vector spaces over a same field, we denote by U ⊗ V thetensor product of U and V . If u ∈ U and v ∈ V , then u ⊗ v is an elementof U ⊗V . The space U ⊗V is spanned by elements of the form u⊗v for u ∈ Uand v ∈ V .

When U = Rm1×n1 and V = Rm2×n2 , we identify U ⊗ V with thespace Rm1m2×n1n2 , and for A ∈ Rm1×n1 and B ∈ Rm2×n2 we identify A⊗ Bwith the Kronecker product between matrices A and B. This is the m1 × n1block matrix with blocks of size m2 × n2, block (i, j) being equal to AijB.


So A ⊗ B is an m1m2 × n1n2 matrix. The same definitions hold when R isreplaced by C.

One property of the tensor product of matrices that we will use relates tothe trace inner product. Namely, for A1, A2 ∈ Rm×m and B1, B2 ∈ Rn×n wehave

〈A1 ⊗B1, A2 ⊗B2〉 = 〈A1, A2〉〈B1, B2〉.

The same holds when R is replaced by C, of course.Finally, we denote by δij the Kronecker delta function, which is equal to 1

if i = j, and equal to 0 otherwise.

1.3 Association schemes

We defined association schemes in the introduction. Now we give a briefoverview of the relevant properties of association schemes that will be of useto us; for background on association schemes we refer the reader to Chapter 12of the book by Godsil [7] or Section 3.3 in the book by Cameron [3].

1.3.1 Sources of association schemes

In the introduction we used an association scheme that was related to thecomplete bipartite graph Km,m. This idea can be generalized: Associationschemes can be obtained from some special classes of graphs. In this section,we discuss how to obtain association schemes from distance-regular graphs;these schemes will be important in Section 1.5 in the encoding of combinatorialproblems as problems over the convex hull of association schemes.

Let G = (V,E) be a graph of diameter d. We say that G is distance-regularif for any vertices x, y ∈ V and any numbers k, l = 0, . . . , d, the numberof vertices at distance k from x and l from y depends only on k, l, and thedistance between x and y, not depending therefore on the actual choice of xand y. A distance-regular graph of diameter 2 is called strongly regular.

Now, say G is a distance-regular graph of diameter d and label its ver-tices 1, . . . , n. Then the n× n matrices A0, . . . , Ad such that

Ai(x, y) =

{1 if distG(x, y) = i,0 otherwhise

form an association scheme with d classes (cf. Cameron [3], Theorem 3.6), towhich we refer as the association scheme of the graph G. The matrices Aiabove are the distance matrices of the graph G, and A1 is just the adjancecymatrix of G. The association scheme we used in the introduction to encodethe maximum bisection problem was the association scheme of Km,m.

So from distance-regular graphs one may obtain association schemes. Ex-amples of distance-regular graphs are the complete bipartite graphs Km,m

and the cycles.


1.3.2 Eigenvalues of association schemes

Let {A0, . . . , Ad} ⊆ Rn×n be an association scheme. The span of the ma-trices A0, . . . , Ad is a matrix ∗-algebra, that is, it is a subspace of Cn×nclosed under matrix multiplication and taking complex conjugate transposes.This algebra is the so-called Bose-Mesner algebra of the association scheme,and A0, . . . , Ad is an orthogonal basis of this algebra. As we observed in the in-troduction, if P ∈ Rn×n is a permutation matrix, then {PTA0P, . . . , P

TAdP}is also an association scheme, and its Bose-Mesner algebra is isomorphic asan algebra to that of {A0, . . . , Ad}.

The Bose-Mesner algebra of an association scheme is commutative, andtherefore it can be diagonalized, that is, there is a unitary matrix U ∈ Cn×nsuch that the matrices

U∗A0U , . . . , U∗AdU

are diagonal. This result implies that the algebra has a basis of idempotents,as we state below.

Theorem 1. The Bose-Mesner algebra of an association scheme {A0, . . . , Ad}has an orthonormal basis E0, . . . , Ed of Hermitian idempotents (that is, Eiis Hermitian and E2

i = Ei for i = 0, . . . , d). Moreover, E0 = 1nJ

and E0 + · · ·+ Ed = I.

Note that the matrices Ei are positive semidefinite, since they are Hermi-tian and have zero-one eigenvalues.

We may write the basis A0, . . . , Ad of the Bose-Mesner algebra in termsof the basis E0, . . . , Ed and vice versa. So there are constants pij for i, j = 0,. . . , d such that

Aj =

d∑i=0

pijEi

for j = 0, . . . , d. Notice that pij is actually the i-th eigenvalue of the matrix Aj .The constants pij are the eigenvalues of the association scheme {A0, . . . , Ad}.

We may also write the basis E0, . . . , Ed in terms of A0, . . . , Ad. So thereare constants qij for i, j = 0, . . . , d, the dual eigenvalues of the associationscheme, such that

Ej =1

n

d∑i=0

qijAi (1.3)

for j = 0, . . . , d. Our normalization above ensures that qi0 = 1 for all i = 0,. . . , d.

It is customary to define primal and dual matrices of eigenvalues of theassociation scheme as follows:

P = (pij)di,j=0 and Q = (qij)

di,j=0.

It is easy to check that PQ = nI.


Since we work with symmetric association schemes, we have that the num-bers pij and qij are all real. Moreover, we have the identity

pjimi

=qij

n traceEj. (1.4)

Finally, notice that if P ∈ Rn×n is a permutation matrix, then the eigen-values and dual eigenvalues of both {A0, . . . , Ad} and {PTA0P, . . . , P

TAdP}are the same.

There is a closed formula for the eigenvalues and dual eigenvalues of theassociation scheme of a strongly regular graph. If A is the adjacency matrixof a strongly regular graph, then e is an eigenvector of A, since a stronglyregular graph is regular. Eigenvalues associated with eigenvectors orthogonalto e are said to be the restricted eigenvalues of the graph. Every stronglyregular graph has exactly two distinct restricted eigenvalues. We then havethe following result.

Theorem 2. Let G be a strongly regular graph with n vertices and of degree khaving restricted eigenvalues r and s with r > s. Suppose that the eigenvalue roccurs with multiplicity f and that s occurs with multiplicity g. Then

P =

1 k |V | − k − 11 r −r − 11 s −s− 1

and Q =

1 f g1 fr/k gs/k

1 − f(r+1)|V |−k−1 −

g(s+1)|V |−k−1

.

1.4 Semidefinite programming relaxations of associationschemes

Let {A0, . . . , Ad} ⊆ Rn×n be an association scheme. We now present asemidefinite programming relaxation of its convex hull conv({A0, . . . , Ad}),which we defined in (1.1).

Recall that we have expression (1.3) for the dual eigenvalues of an as-sociation scheme, and also that each Ei is positive semidefinite. Since theeigenvalues and dual eigenvalues of an association scheme are the same as theeigenvalues and dual eigenvalues of any permuted version of the associationscheme, we see that any (X0, . . . , Xd) ∈ conv({A0, . . . , Ad}) satisfies∑d

i=0Xi = J,∑di=0 qijXi � 0 for j = 1, . . . , d,

X0 = I, Xi nonnegative and symmetric for i = 1, . . . , d.

(1.5)

The theorem below shows that many important constraints are alreadyimplied by (1.5).


Theorem 3. Suppose X0, . . . , Xd satisfy (1.5). Write

Yj =1

n

d∑i=0

qijXi.

Then the following holds:

1.∑dj=0 Yj = I;

2. Yje = 0 for j = 0, . . . , d;

3.∑di=0 pijYi = Xj for j = 0, . . . , d;

4. Xje = Aje = p0je for j = 0, . . . , d.

Proof. Direct from (1.5) and the properties of the matrices of eigenvalues anddual eigenvalues of the association scheme. ut

Also a whole class of linear matrix inequalities is implied by (1.5), as weshow next.

Theorem 4. Let {A0, . . . , Ad} ⊆ Rn×n be an association scheme and supposethat the linear matrix inequality

d∑i=0

αiXi � 0, (1.6)

where α0, . . . , αd are given scalars, is satisfied by Xi = Ai. Then (1.6) maybe written as a conic combination of the linear matrix inequalities

d∑i=0

qijXi � 0 for j = 0, . . . , d.

Proof. We have

Y =

d∑i=0

αiAi � 0.

But then since Y is in the Bose-Mesner algebra of the association scheme,there exist nonnegative numbers β0, . . . , βd such that

Y =

d∑j=0

βjEj ,

where E0, . . . , Ed is the idempotent basis of the algebra. (Notice that, in fact,the numbers βj are the eigenvalues of Y .) Substituting

Ej =1

n

d∑i=0

qijAi


yieldsd∑i=0

αiAi =

d∑j=0

βjn

( d∑i=0

qijAi

),

and since the Ai are pairwise orthogonal we see that

αi =

d∑j=0

βjnqij ,

as we wanted. ut

1.5 Semidefinite programming relaxations ofcombinatorial problems

We now show how to derive semidefinite programming relaxations of manydifferent combinatorial problems from the relaxation for the convex hull of anassociation scheme which we presented in the previous section.

The basic idea is to restate the combinatorial problem as the problem offinding a maximum/minimum-weight distance-regular subgraph of a weightedcomplete graph, and then use the fact that the distance matrices of a distance-regular graph give rise to an association scheme. For example, the travellingsalesman problem is the problem of finding a minimum-weight Hamiltoniancycle in a graph, and the maximum bisection problem is the problem of findinga maximum-weight copy of Km,m in the complete graph K2m.

More precisely, suppose we are given a symmetric matrix W ∈ Rn×n thatwe see as giving weights to the edges of the complete graph Kn, and we arealso given a distance-regular graph H on n vertices. Our problem is to find amaximum-weight copy of H in Kn.

Say H has diameter d, and let A0, . . . , Ad be its distance matrices, sothatA1 is the adjacency matrix ofH. SinceH is distance-regular, {A0, . . . , Ad}is an association scheme. Our problem can be equivalently stated as the fol-lowing quadratic assignment problem (QAP):

maximize 12 〈W,P

TA1P 〉P ∈ Rn×n is a permutation matrix.

The problem above can be equivalentely stated in terms of the convex hullof the association scheme {A0, . . . , Ad}:

maximize 12 〈W,X1〉(X0, . . . , Xd) ∈ conv({A0, . . . , Ad}).

Finally, one may consider relaxation (1.5) for the above problem, obtainingthe following semidefinite programming relaxation for it:


maximize 12 〈W,X1〉∑di=0Xi = J,∑di=0 qijXi � 0 for j = 1, . . . , d,

X0 = I, Xi nonnegative and symmetric for i = 1, . . . , d.

This is the approach we employ in the following sections.

1.5.1 The travelling salesman problem and the k-cycle coveringproblem

We are given a symmetric nonnegative matrix W ∈ Rn×n, wich we view asgiving weights to the edges of the complete graph Kn. The travelling salesmanproblem asks us to find a minimum-weight copy of Cn, the cycle on n vertices,in the weighted graph Kn.

Now, Cn is a distance-regular graph, and its association scheme has bn/2cclasses and is know as the Lee scheme, which is also the association scheme ofsymmetric circulant matrices. The dual eigenvalues of the association schemeof Cn are given by

qij = 2 cos(2ijπ/n) for i, j = 1, . . . , bn/2c.

Moreover, qi0 = 1 for i = 0, . . . , bn/2c and q0j = 2 for j = 1, . . . , bn/2c.So, if we consider relaxation (1.5) for the convex hull of the Lee scheme,

we obtain the following semidefinite programming relaxation for the travellingsalesman problem:

minimize 12 〈W,X1〉I +

∑di=1Xi = J,

I +∑di=1 cos(2ijπ/n)Xi � 0 for j = 1, . . . , bn/2c,

Xi nonnegative and symmetric for i = 1, . . . , bn/2c.

(1.7)

This is precisely the semidefinite programming relaxation for the travellingsalesman problem introduced by de Klerk, Pasechnik, and Sotirov [13]. Onemay use Theorem 4 to show that this relaxation is tighter than the one givenby Cvetkovic, Cangalovic, and Kovacevic-Vujcic [2]. This fact was alreadyknown, but the proof can be greatly simplified by using the more generalTheorem 4.

A simple modification in the objective function of (1.7) allows us to providea relaxation for the minimum k-cycle covering problem. This is the problem ofpartitioning a weighted complete graph into k vertex-disjoint cycles of equallength so that the total weight of the cycles is minimized.

This problem has been studied by Goemans and Williamson [8] (see alsoManthey [18]), who showed that a 4-approximation algorithm exists when theweights satisfy the triangle inequality.

Now suppose n is a multiple of k. One may check that the matrix Ak, thek-th distance matrix of the cycle Cn, is actually the incidence matrix of k


vertex-disjoint cycles of length n/k each. So, to obtain a relaxation of the k-cycle covering problem, one only has to replace the objective function of (1.7)by 1

2 〈W,Xk〉.

1.5.2 The maximum bisection problem

We already considered the maximum bisection problem in the introduction.The problem consists of, given a nonnegative symmetric matrix W ∈ R2m×2m,which we see as giving weights to the edges of the complete graph K2m, findinga maximum-weight copy of the complete bipartite graph Km,m in K2m.

The graph Km,m is a strongly regular graph; the association scheme in-duced by it was described in (1.2). Since Km,m is strongly regular, its dualeigenvalues have a closed formula, as given in Theorem 2. Indeed, the re-stricted eigenvalues of Km,m are the eigenvalues r = 0 and s = −m of A1,and so the matrices of eigenvalues and dual eigenvalues are given by

P =

1 m m− 11 0 −11 −m m− 1

and Q =

1 2(m− 1) 11 0 −11 −2 1

.

Now, by adapting (1.5), we obtain the following relaxation of the maximumbisection problem:

maximize 12 〈W,X1〉I +X1 +X2 = J,

(m− 1)I −X2 � 0,

I −X1 +X2 � 0,

X1, X2 nonnegative and symmetric.

(1.8)

Frieze and Jerrum [6] (see also Ye [24]) considered another semidefiniteprogramming relaxation of the maximum bisection problem. Their relaxationis the following:

maximize 14 〈W,J −X〉Xii = 1 for i = 1, . . . , 2m,Xe = 0,X � 0.

(1.9)

To see that this is a relaxation of the problem, notice that X = vvT is afeasible solution, where v ∈ {−1, 1}2m gives a bisection of the vertex set.

Actually, the relaxation we give is equivalent to the one given by Friezeand Jerrum, as we show in the next theorem.

Theorem 5. The optimal values of problems (1.8) and (1.9) coincide.


Proof. Given a feasible solution X1, X2 of (1.8), set

X = I −X1 +X2 � 0.

Using (4) of Theorem 3 we have that

Xe = e−X1e+X2e = e−me+ (m− 1)e = 0,

where we have used that A1e = me and A2e = (m − 1)e for the associationscheme of Km,m. It is also easy to verify that the diagonal entries of X areall equal to 1 and that

1

4〈W,J −X〉 =

1

2〈W,X1〉.

Conversely, suppose X is a feasible solution of (1.9). We claim that

X1 =1

2(J −X) and X2 =

1

2(J +X)− I

is a feasible solution of (1.8) with the same objective value.Indeed, it is easy to see that X1, X2 has the same objective value as X.

To see that it is a feasible solution of (1.8), notice that since X is positivesemidefinite and has diagonal entries equal to 1, all entries of X have absolutevalue at most 1, and therefore both X1 and X2 are nonnegative. Moreover,it is easy to see that X1, and X2 are symmetric, and that I + X1 + X2 = Jand I −X1 +X2 � 0. So we argue that (m− 1)I −X2 � 0.

Indeed, (m − 1)I − X2 = mI − 12 (J + X). Now, the matrix M = 1

2 (J +X) is positive semidefinite, and e is one of its eigenvectors, with associatedeigenvalue m. Since M has trace 2m and is positive semidefinite, this impliesthat any other eigenvalue of M is at most m. So it follows that mI −M � 0,as we wanted. ut

1.5.3 The maximum p-section problem

Given a nonnegative symmetric matrix W ∈ Rpm×pm, which we see as giv-ing weights to the edges of the complete graph Kpm, our goal is to find amaximum-weight copy of the complete p-partite graph in Kpm.

The complete p-partite graph is strongly regular, and it generates an as-sociation scheme whose matrix Q of dual eigenvalues is

Q =

1 p(m− 1) p− 11 0 −11 −p p− 1

.

Then relaxation (1.5) simplifies to


maximize 12 〈W,X1〉I +X1 +X2 = J,

(m− 1)I −X2 � 0,

(p− 1)I −X1 + (p− 1)X2 � 0,

X1, X2 nonnegative and symmetric.

Note that this coincides with relaxation (1.8) for the maximum bisection prob-lem when p = 2. A different semidefinite programming relaxation for themaximum p-section problem was proposed by Andersson [1].

1.6 Semidefinite programming relaxations of coherentconfigurations

A coherent configuration is a set {A1, . . . , Ad} ⊆ Rn×n of 0–1 matrices withthe following properties:

1. There is a set N ⊆ {1, . . . , d} such that∑i∈N Ai = I and A1 + · · ·+Ad =

J ;2. AT

i ∈ {A1, . . . , Ad} for i = 1, . . . , d;3. AiAj ∈ span{A1, . . . , Ad} for i, j = 1, . . . , d.

Thus, association schemes are commutative coherent configurations.Let {A1, . . . , Ad} be a coherent configuration. For i ∈ {1, . . . , d} we let i∗ ∈

{1, . . . , d} be such that Ai∗ = ATi . We also write

mi = 〈J,Ai〉 = 〈Ai, Ai〉

for i = 1, . . . , d.Matrices A1, . . . , Ad generate a matrix ∗-algebra containing the identity,

which however need not be commutative, and therefore it might not be possi-ble to diagonalize this algebra. It is always possible to block-diagonalize the al-gebra of a coherent configuration, however, as was shown by Wedderburn [23].To precisely state Wedderburn’s result we first introduce some notation.

Let A, B ⊆ Cn×n. We write

A⊕ B =

{(A 00 B

): A ∈ A and B ∈ B

}.

For A ⊆ Cn×n and for any positive integer r we write

r �A = { Ir ⊗A : A ∈ A},

where Ir is the r × r identity matrix, so that Ir ⊗ A is an rn × rn matrixwith r repeated diagonal blocks equal to A and zeros everywhere else. Finally,let U ∈ Cn×n. Then for A ⊆ Cn×n we write

U∗AU = {U∗AU : A ∈ A}.

Wedderburn has shown the following theorem.


Theorem 6. Let A ⊆ Cn×n be a matrix ∗-algebra containing the identity.Then there is a unitary matrix U ∈ Cn×n and positive integers r1, . . . , rkand s1, . . . , sk such that

U∗AU =

k⊕i=1

ri � Csi×si .

Notice that, if {A1, . . . , Ad} ⊆ Rn×n is a coherent configuration and P ∈Rn×n is a permutation matrix, then also {PTA1P, . . . P

TAdP} is a coherentconfiguration. Our goal is to provide a semidefinite programming relaxationfor the convex hull of a coherent configuration, which is the set

conv({A1, . . . , Ad}) = conv{ (PTA1P, . . . , PTAdP ) : P ∈ Rn×n is a

permutation matrix }.

To provide a relaxation for this set, we use the following theorem, whichdescribes a linear matrix inequality satisfied by a coherent configuration.

Theorem 7. Let {A1, . . . , Ad} ⊆ Rn×n be a coherent configuration. Then

d∑i=1

m−1i Ai ⊗Ai � 0. (1.10)

We prove the theorem in a moment; first we use it to describe our semidef-inite programming relaxation of the convex hull of a coherent configuration.So let {A1, . . . , Ad} ⊆ Rn×n be a coherent configuration. From Theorem 7, itis clear that for (X1, . . . , Xd) ∈ conv({A1, . . . , Ad}) we have

d∑i=1

m−1i Ai ⊗Xi � 0. (1.11)

Let N ⊆ {1, . . . , d} be such that∑i∈N Ai = I. Then any (X1, . . . , Xd) ∈

conv({A1, . . . , Ad}) satisfies∑i∈N Xi = I,∑di=1Xi = J,∑di=1m

−1i Ai ⊗Xi � 0,

〈J,Xi〉 = mi, Xi∗ = XTi , and Xi ≥ 0 for i = 1, . . . , d.

(1.12)

We observe that, in the linear matrix inequality (1.11), one may replace thematrices Ai from the coherent configuration by their block-diagonalizations,obtaining an equivalent constraint. Also, repeated blocks may be eliminated:Only one copy of each set of repeated blocks is necessary. So, from Wedder-burn’s theorem we see that in constraint (1.11) it is possible to replace thematrices Ai, which are n× n matrices, by matrices whose dimensions depend


only on the dimension of the algebra generated by A1, . . . , Ad, which is equalto d.

In an analogous way to Theorem 4, the linear matrix inequality (1.11)implies a whole class of linear matrix inequalities that are valid for the convexhull of a coherent configuration, as we will see after the proof of Theorem 7.

To prove Theorem 7 we need the following theorem that holds in the moregeneral context of C∗-algebras. We give a proof here for the case of matrix∗-algebras to make our presentation self-contained.

Theorem 8. Let A ⊆ Cn×n be a matrix ∗-algebra containing the identity andlet X ∈ Cn×n be a positive semidefinite matrix. Then the orthogonal projectionof X onto A is also positive semidefinite.

Proof. From Wedderburn’s theorem (Theorem 6) we may assume that

A =

k⊕i=1

ri � Csi×si . (1.13)

For i = 1, . . . , k and j = 1, . . . , ri, let Xij be the si × si diagonal blockof X corresponding to the j-th copy of Csi×si in the decomposition of A givenin (1.13).

Since X is positive semidefinite, every matrix Xij as defined above is alsopositive semidefinite. Moreover, the projection of X onto A is explicitly givenas

k⊕i=1

ri � r−1iri∑j=1

Xij ,

which is then also seen to be positive semidefinite. ut

We now can give a proof of Theorem 7.

Proof (Proof of Theorem 7). Consider the matrix

X =

n∑k,l=1

Ekl ⊗ Ekl,

where Ekl is the n× n matrix with 1 in position (k, l) and 0 everywhere else.We claim that X is positive semidefinite. To see this, let e1, . . . , en be the

canonical orthonormal basis of Rn. Then ei ⊗ ej for i, j = 1, . . . , n is a basisof Rn ⊗ Rn.

Since Eklei = δliek, if i 6= j, then X(ei ⊗ ej) = 0. So all vectors ei ⊗ ejwith i 6= j are in the kernel of X.

On the other hand, for i = 1, . . . , n we have that

X(ei ⊗ ei) =

n∑k=1

ek ⊗ ek.


Then it is clear that also the vectors e1⊗e1−ei⊗ei, for i = 2, . . . , n, lie in thekernel of X. Now, from the above identity we see that e1⊗e1 + · · ·+en⊗en isan eigenvector of X with associated eigenvalue n, and hence it follows that Xis positive semidefinite.

Now notice that, since {A1, . . . , Ad} is a coherent configuration, the ma-trices

(mimj)−1/2Ai ⊗Aj for i, j = 1, . . . , d (1.14)

span a matrix ∗-algebra which moreover contains the identity. Actually, thematrices above form an orthonormal basis of this algebra.

By Theorem 8, the projection of X onto the algebra spanned by the matri-ces in (1.14) is a positive semidefinite matrix. We show now that this projec-tion is exactly the matrix in (1.10). Indeed, the projection is explicitly givenas

d∑i,j=1

(mimj)−1Ai ⊗Aj

⟨ d∑k,l=1

Ekl ⊗ Ekl, Ai ⊗Aj⟩

=

d∑i,j=1

(mimj)−1Ai ⊗Aj

d∑k,l=1

〈Ekl, Ai〉〈Ekl, Aj〉.

Notice that, for i, j ∈ {1, . . . , d} with i 6= j we have 〈Ekl, Ai〉〈Ekl, Aj〉 =0, because the matrices A1, . . . , Ad have disjoint supports. So the aboveexpression simplifies to

d∑i=1

m−2i Ai ⊗Ain∑

k,l=1

〈Ekl, Ai〉2 =

d∑i=1

m−1i Ai ⊗Ai,

which is exactly the matrix in (1.10), as we wanted. ut

Theorem 8 also suggests a whole class of linear matrix inequalities that arevalid for the convex hull of a coherent configuration. Indeed, let {A1, . . . , Ad} ⊆Rn×n be a coherent configuration. Given a positive semidefinite matrix Y ∈Rn×n, Theorem 8 implies that the projection of Y onto the algebra spannedby A1, . . . , Ad is positive semidefinite, that is

d∑i=1

m−1i 〈Y,Ai〉Ai � 0.

So we see that any (X1, . . . , Xd) ∈ conv({A1, . . . , Ad}) satisfies

d∑i=1

m−1i 〈Y,Ai〉Xi � 0.

All these infinitely many linear matrix inequalities are already implied by (1.11),however, as we show next.


Theorem 9. Let {A1, . . . , Ad} ⊆ Rn×n be a coherent configuration. Supposematrices X1, . . . , Xd ∈ Rn×n which are such that Xi∗ = XT

i for i = 1, . . . , dsatisfy (1.11). Then we also have

d∑i=1

m−1i 〈Y,Ai〉Xi � 0, (1.15)

where Y ∈ Rn×n is any positive semidefinite matrix.

Proof. Let M be the matrix in (1.15). It is clear that M is symmetric. To seethat it is actually positive semidefinite, let Z ∈ Rn×n be a positive semidefinitematrix. We show that 〈Z,M〉 ≥ 0.

Indeed, we have

〈Z,M〉 =

d∑i=1

m−1i 〈Y,Ai〉〈Z,Xi〉.

Now, notice that the matrix Y ⊗ Z is positive semidefinite, since both Yand Z are positive semidefinite. Then, since X1, . . . , Xd satisfy (1.11) we havethat

0 ≤⟨Y ⊗ Z,

d∑i=1

m−1i Ai ⊗Xi

⟩

=

d∑i=1

m−1i 〈Y,Ai〉〈Z,Xi〉

= 〈Z,M〉,

as we wanted. ut

We also have the following theorem, which is analogous to Theorem 4.

Theorem 10. Let {A1, . . . , Ad} ⊆ Rn×n be a coherent configuration and sup-pose that the linear matrix inequality

d∑i=1

αiXi � 0, (1.16)

where α1, . . . , αd are scalars such that αi∗ = αi for i = 1, . . . , d, is satisfiedby Xi = Ai.

If matrices X1, . . . , Xd ∈ Rn×n such that Xi∗ = XTi for i = 1, . . . , d

satisfy (1.11), then X1, . . . , Xd also satisfy (1.16).

Proof. We claim that there is a positive semidefinite matrix Y ∈ Rn×n suchthat

〈Y,Ai〉 = αimi for i = 1, . . . , d.


Indeed, from Farkas’ lemma we know that either there exists such a ma-trix Y , or there are numbers β1, . . . , βd such that

d∑i=1

βiAi � 0 and

d∑i=1

αiβimi < 0.

Now, since (1.16) is satisfied by Xi = Ai, we know that

0 ≤⟨ d∑i=1

αiAi,

d∑j=1

βjAj

⟩

=

d∑i,j=1

αiβj〈Ai, Aj〉

=

d∑i=1

αiβimi

< 0,

a contradiction. Here, we used the fact that 〈Ai, Aj〉 = δijmi for i, j = 1,. . . , d. So there must be a matrix Y as claimed.

Now, if Y is given as in our claim, then we have that

d∑i=1

m−1i 〈Y,Ai〉Xi =

d∑i=1

αiXi,

and since X1, . . . , Xd satisfy (1.11), from Theorem 9 we see that the matrixabove is positive semidefinite, as we wanted. ut

Finally, we observe that for an association scheme {A0, . . . , Ad} ⊆ Rn×n,relaxation (1.12) is equivalent to (1.5).

To see this, it suffices to show that if {A0, . . . , Ad} is an association scheme,then the constraint

d∑i=0

m−1i Ai ⊗Xi � 0

is equivalent to the constraints

d∑i=0

qijXi � 0 for j = 0, . . . , d. (1.17)

Indeed, since {A0, . . . , Ad} is an association scheme, the matrices A0,. . . , Ad can be simutaneously diagonalized via a unitary transformation. Thenwe may write

Ai =

d∑j=0

pjiEj ,


where the matrices E0, . . . , Ed are 0–1 diagonal matrices that sum to theidentity. So we see that the linear matrix inequality

d∑i=0

m−1i

d∑j=0

pjiEj ⊗Xi =

d∑i=0

m−1i Ai ⊗Xi � 0

is equivalent to the linear matrix inequalities

d∑i=0

m−1i pjiXi � 0 for j = 0, . . . , d,

and by using (1.4) we get that the above constraints are equivalent to (1.17),as we wanted.

1.6.1 Relation to a relaxation of the quadratic assignment problem

Given symmetric matrices A and W ∈ Rn×n, a simplified version of thequadratic assignment problem is as follows:

maximize 〈W,PTAP 〉P ∈ Rn×n is a permutation matrix.

Povh and Rendl [20] proposed the following semidefinite programmingrelaxation for this problem:

maximize 〈A⊗W,Y 〉〈I ⊗ Ejj , Y 〉 = 〈Ejj ⊗ I, Y 〉 = 1 for j = 1, . . . , n,〈I ⊗ (J − I) + (J − I)⊗ I, Y 〉 = 0,〈J ⊗ J, Y 〉 = n2,Y ≥ 0 and Y � 0,

(1.18)

where Ejj is the matrix with 1 in position (j, j) and 0 everywhere else. Thisrelaxation is in fact equivalent to an earlier one due to Zhao et al. [25].

When the matrixA belongs to the span of a coherent configuration {A1, . . . , Ad} ⊆Rn×n, then one may use (1.12) to provide a semidefinite programming relax-ation of the quadratic assignment problem as follows

maximize⟨W,∑di=1m

−1i 〈A,Ai〉Xi

⟩∑i∈N Xi = I,∑di=1Xi = J,∑di=1m

−1i Ai ⊗Xi � 0,


(1.19)

Actually, when A belongs to the span of a coherent configuration, relax-ations (1.18) and (1.19) are equivalent. This result is essentially due to deKlerk and Sotirov [14], but we give a proof for the sake of completeness.


Theorem 11 (cf. de Klerk and Sotirov [14]). If A belongs to the spanof a coherent configuration {A1, . . . , Ad} ⊆ Rn×n, then the optimal valuesof (1.18) and (1.19) coincide.

Proof. We first show that the optimal value of (1.18) is at most that of (1.19).To this end, let Y ∈ Rn×n ⊗ Rn×n be a feasible solution of (1.18).

Write A = span{A1, . . . , Ad} and consider the matrices

m−1/2i Ai ⊗ Ekl for i = 1, . . . , d and k, l = 1, . . . n,

where Eij ∈ Rn×n is the matrix with 1 at position (i, j) and 0 everywhere else.The matrices above form an orthonormal basis of the matrix ∗-algebra A ⊗Cn×n. From Theorem 8, we know that the orthogonal projection of Y ontothis algebra is a positive semidefinite matrix. In view of the basis given abovefor the algebra, this projection may be explicitly written as

Y =

d∑i=1

m−1i Ai ⊗Xi,

where Xi ∈ Rn×n, and this is therefore a positive semidefinite matrix.We now show that X1, . . . , Xd is a feasible solution of (1.19). Indeed,

it is immediate that Xi∗ = XTi for i = 1, . . . , d and, since the Ai are 0–1

matrices with disjoint supports and Y is nonnegative, we also see that the Xi

are nonnegative. Moreover, by construction we have

d∑i=1

m−1i Ai ⊗Xi � 0.

To see that∑i∈N Xi = I, notice that for all j = 1, . . . , n, we have I⊗Ejj ∈

A ⊗ Cn×n. Then, since Y is feasible for (1.18) and since {A1, . . . , Ad} is acoherent configuration, we have for j = 1, . . . , n that

1 = 〈I ⊗ Ejj , Y 〉 =

d∑i=1

m−1i 〈I, Ai〉〈Ejj , Xi〉 =∑i∈N〈Ejj , Xi〉. (1.20)

Now, since Y is feasible for (1.18) we also have that

0 = 〈I ⊗ (J − I) + (J − I)⊗ I, Y 〉= 〈I ⊗ (J − I) + (J − I)⊗ I, Y 〉

=

d∑i=1

m−1i 〈I, Ai〉〈J − I,Xi〉+ 〈J − I, Ai〉〈I,Xi〉.(1.21)

Since each Xi is nonnegative, this implies that whenever i ∈ N , all off-diagonal entries of Xi are equal to zero. But then together with (1.20) wehave that

∑i∈N Xi = I.


We now claim that for i ∈ N one has 〈J,Ai〉 = mi. Indeed, since 〈Ejj ⊗I, Y 〉 = 1 for j = 1, . . . , n, for i ∈ N we have that 〈Ai⊗ I, Y 〉 = mi. But thenwe have that

mi = 〈Ai ⊗ I, Y 〉 =

d∑j=1

m−1j 〈Ai, Aj〉〈I,Xj〉 = 〈I,Xi〉.

But then, since∑i∈N Xi = I and since each Xi is nonnegative, we must have

that 〈J,Xi〉 = 〈I,Xi〉 = mi for i ∈ N , as we wanted.Now we show that in fact 〈J,Xi〉 = mi for i = 1, . . . , d. To see this, notice

that the matrix

d∑i=1

m−1i 〈J,Xi〉Ai = (I ⊗ e)T( d∑i=1

m−1i Ai ⊗Xi

)(I ⊗ e)

is positive semidefinite. But since 〈J,Xi〉 = mi for i ∈ N , the diagonal en-tries of this matrix are all equal to 1. So since this is a positive semidefinitematrix, it must be that all of its entries have absolute value at most 1. Butthen, since {A1, . . . , Ad} is a coherent configuration, and since each Xi isnonnegative, it must be that 〈J,Xi〉 ≤ mi for i = 1, . . . , n.

Now, we know that

n2 = 〈J ⊗ J, Y 〉 = 〈J ⊗ J, Y 〉 =

d∑i=1

〈J,Xi〉,

and it follows that 〈J,Xi〉 = mi for i = 1, . . . , n.To prove that X1, . . . , Xd is feasible for (1.19), we still have to show

that X1 + · · · + Xd = J . To this end, notice that, since the Xi are nonnega-tive, (1.21) implies that every Xi with i /∈ N has only zeros on the diagonal.So the matrix X1 + · · ·+Xd has diagonal entries equal to 1 and then, since itis positive semidefinite (as Y is positive semidefinite), and since

d∑i=1

〈J,Xi〉 = n2,

we see immediately that X1 + · · ·+Xd = J , as we wanted.So X1, . . . , Xd is feasible for (1.19). Finally, the objective value of X1,

. . . , Xd coincides with that of Y , and we are done.To see now that the optimal value of (1.19) is at most that of (1.18), one

has to show that, given a solution X1, . . . , Xd of (1.19), the matrix

Y =

d∑i=1

m−1i Ai ⊗Xi

is a feasible solution of (1.18) with the same objective value, and showing thisis rather straight-forward. ut


1.6.2 Coherent configurations and combinatorial optimizationproblems

In Section 1.5 we saw how the problem of finding a maximum/minimum-weight subgraph H of a complete graph can be modelled with the help ofassociation schemes when the graph H is distance-regular. We now show howto model this problem as an optimization problem over the convex hull of acoherent configuration for general graphs H.

Let H = (V,E) be a graph, with V = {1, . . . , n}. An automorphism of His a permutation π : V → V that preserves the adjacency relation. The set ofall automorphisms of H, denoted by Aut(H), is a group under the operationof function composition.

For (x, y), (x′, y′) ∈ V × V , we write

(x, y) ∼ (x′, y′)

if there is π ∈ Aut(H) such that π(x) = x′ and π(y) = y′. Relation ∼ is anequivalence relation that partition V ×V into equivalence classes called orbits.Say V × V/∼ = {C1, . . . , Cd} and consider the matrices A1, . . . , Ad ∈ Rn×nsuch that

Ai(x, y) =

{1 if (x, y) ∈ Ci,0 otherwhise.

We claim that {A1, . . . , Ad} is a coherent configuration.Indeed, it is easy to see that {A1, . . . , Ad} satifies items (1)–(2) in the

definition of a coherent configuration. We show that item (3) is also satisfied,that is, that the span of {A1, . . . , Ad} is a matrix ∗-algebra.

To see this, associate to each permutation π ∈ Aut(H) a permutationmatrix Pπ ∈ Rn×n in the obvious way. Then the set of matrices

{A ∈ Rn×n : APπ = PπA for π ∈ Aut(H) }

is exactly the span of {A1, . . . , Ad}, and it can be then easily seen that thisspan is also a matrix ∗-algebra. So {A1, . . . , Ad} is a coherent configuration.

Now suppose we are given a symmetric matrix W ∈ Rn×n which we seeas giving weights to the edges of the complete graph Kn, and suppose we areasked to find a maximum-weight copy of H in Kn. Let A be the adjacencymatrix of H. The we may write this problem equivalently as

maximize 12 〈W,P

TAP 〉P ∈ Rn×n is a permutation matrix,

It can be easily seen that, for some subset M ⊆ {1, . . . , d}, one has

A =∑i∈M

Ai.

So the problem above can be rewritten as


maximize 12

⟨W,∑i∈M Xi

⟩(X1, . . . , Xd) ∈ conv({A1, . . . , Ad}).

Finally, when we consider relaxation (1.12) for this problem, we obtain

maximize 12

⟨W,∑i∈M Xi

⟩∑i∈N Xi = I,∑di=1Xi = J,∑di=1m

−1i Ai ⊗Xi � 0,


In our discussion it was not in any moment necessary to work with the fullautomorphism group of H; any subgroup of this group will do. In any case,the larger the group one uses, the smaller the matrices Ai can be made to beafter the block-diagonalization.

If the automorphism group of H is transitive, then it is possible to obtainstronger SDP relaxations by using a stabilizer subgroup in stead of the thefull automorphism group; see de Klerk and Sotirov [15]. A similar idea wasused by Schrijver [22] earlier to obtain improved SDP bounds for binary codesizes.

1.6.3 The maximum (k, l)-cut problem

The maximum (k, l)-cut problem is a generalization of the maximum bisectionproblem which consists of the following: Given a symmetric nonnegative ma-trix W ∈ Rn×n, which we see as giving weights to the edges of the completegraph Kn, where n = k + l, find a maximum-weight copy of Kk,l in Kn.

We may use our semidefinite programming relaxation of the convex hullof a coherent configuration to give a relaxation of the maximum (k, l)-cutproblem by considering the following coherent configuration associated withthe complete bipartite graph Kk,l:

A1 =

(Ik 0k×l

0l×k 0l×l

), A2 =

(Jk − Ik 0k×l

0l×k 0l×l

), A3 =

(0k×k Jk×l0l×k 0l×l

),

A4 =

(0k×k 0k×lJl×k 0l×l

), A5 =

(0k×k 0k×l0l×k Il×l

), A6 =

(0k×k 0k×l0l×k Jl − Il

).

(1.22)

Now, relaxation (1.12) simplifies to:

maximize 12 〈W,X3 +X4〉X1 +X5 = I,∑6i=1Xi = J,∑6i=1m

−1i Ai ⊗Xi � 0,

〈J,Xi〉 = mi, Xi∗ = XTi , and Xi ≥ 0 for i = 1, . . . , 6.

(1.23)


The algebra spanned by A1, . . . , A6 is isomorphic to the algebra C⊕C⊕C2×2, so one may replace the matrices A1, . . . , A6 above by their respectiveimages under this isomorphism, which we denote by φ. Their images are givenby

φ(A1) =

(101 00 0

), φ(A2) =

(−10k−1 00 0

), φ(A3) =

√kl

(000 10 0

),

φ(A4) =√kl

(000 01 0

), φ(A5) =

(010 00 1

), φ(A6) =

(0−1

0 00 l−1

).

(1.24)Feige and Langberg [4] and Han, Ye, and Zhang [11] proposed another

semidefinite programming relaxation for the maximum (k, l)-cut problem,namely

maximize 14 〈W,J −X〉Xii = 1 for i = 1, . . . , n,〈J,X〉 = (k − l)2,X � 0.

(1.25)

Our relaxation (1.23) is actually as strong as (1.25). Indeed, if (X1, . . . , X6)is a feasible solution of (1.23), one may construct a feasible solution of (1.25)by setting

X = X1 +X2 −X3 −X4 +X5 +X6.

Indeed, it can be easily seen that X has diagonal entries equal to 1 andsatisfies 〈J,X〉 = (k − l)2. To see that X is positive semidefinite, notice thatthe matrices A1, . . . , A6 satisfy

A1 +A2 −A3 −A4 +A5 +A6 � 0,

as one may easily check by using the isomorphism φ. Then, from Theorem 10one has that X is positive semidefinite.

Now, also 14 〈W,J − X〉 = 1

2 〈W,X3 + X4〉, so X has the same objectivevalue as X1, . . . , X6, as we wanted. Finally, our relaxation is actually stronger,as can be checked numerically in some examples. In particular, we have thefollowing theorem.

Theorem 12. The optimal value of (1.23) is at most that of (1.25), and canbe strictly smaller for specific instances.

1.6.4 The maximum stable set problem

Let G = (V,E) be a graph with adjacency matrix A and stability numberα(G) and let an integer k > 1 be given. One has α(G) ≥ k, if and only if

0 = min 〈A1 +A2, PTAP 〉

P ∈ Rn×n is a permutation matrix,


where A1 and A2 are from the coherent configuration in (1.22).Applying our procedure we obtain the SDP problem:

minimize 〈A,X1 +X2〉X1 +X5 = I,∑6i=1Xi = J,∑6i=1m

−1i φ(Ai)⊗Xi � 0,

〈J,Xi〉 = mi, Xi∗ = XTi , and Xi ≥ 0 for i = 1, . . . , 6,

(1.26)

where the φ(Ai) are as in (1.24). We denote by σk(G) de optimal valueof (1.26) when A is the adjacency matrix of the graph G.

If the optimal value of this problem is strictly positive, then α(G) < k. Soan upper bound for α(G) is given by

max{ k : σk(G) = 0 }.

The bound provided above is at least as strong as the ϑ′ bound of McEliece,Rodemich, and Rumsey [19] and Schrijver [21]. Recall that, given a graph G =(V,E), we let ϑ′(G) be the optimal value of the semidefinite programmingproblem

maximize 〈J,X〉traceX = 1,Xuv = 0 if uv ∈ E,X : V × V → R is nonnegative and positive semidefinite.

(1.27)

It is easy to see that

max{ k : σk(G) = 0 } ≤ bϑ′(G)c.

To see this, suppose σk(G) = 0. We show that ϑ′(G) ≥ k. So let X1, . . . , X6 bean optimal solution of (1.26) for the parameter k and set X = k−1(X1 +X2).Then X is a feasible solution of (1.27).

Indeed, X is nonnegative, and it is also positive semidefinite since A1 +A2 is a positive semidefinite matrix. Since the Xi sum up to J and X1 +X5 = I, we have that traceX = k−1 traceX1 = k−1〈J,X1〉 = 1. Finallysince 〈A,X〉 = k−1〈A,X1 +X2〉 = 0 and X is nonnegative, we see that Xuv =0 whenever uv ∈ E, and therefore X is feasible for (1.27).

Now, notice that 〈J,X〉 = k−1〈J,X1+X2〉 = k, and we see that ϑ′(G) ≥ k,as we wanted.

1.6.5 The vertex separator problem

Let G = (V,E) again be a graph with adjacency matrix A and |V | = n, andlet n1, n2 > 0 be given integers satisfying n1 + n2 < n.


The vertex separator problem is to find disjoint subsets V1, V2 ⊂ V suchthat |Vi| = ni (i = 1, 2) and no edge in E connects a vertex in V1 with avertex in V2, if such sets exist. (The set of vertices V \{V1 ∩ V2} is called avertex separator in this case; see Feige et al. [5] for a review of approximationresults for various separator problems.)

Such a vertex separator exists if and only if

0 = min 〈W,PTAP 〉P ∈ Rn×n is a permutation matrix,

where

W =

0(n−n1−n2)×(n−n1−n2) 0(n−n1−n2)×n10(n−n1−n2)×n2

0n1×(n−n1−n2) 0n1×n1Jn1×n2

0n2×(n−n1−n2) Jn2×n10n2×n2

,

and where the subscripts again indicate the matrix sizes. It is easy to verifythat the matrix W belongs to a coherent configuration with d = 12 relations,say {A1, . . . , A12}.

Thus one may again obtain an SDP problem where a strictly positiveoptimal value gives a certificate that the required vertex separator does notexist. The details of this example are left as an exercise to the reader.

1.7 Conclusion

In this chapter we have given a unified framework for deriving SDP relax-ations of combinatorial problems. This approach is still in its infancy, and therelationships to various existing SDP relaxations is not yet fully understood.Moreover, since SDP relaxations play an important role in approximation al-gorithms, it would be very desirable to extend this framework with a generalrounding technique that is amenable to analysis. It is the hope of the au-thors that these topics will spark the interest of other researchers in the SDPcommunity.

Acknowledgement

Etienne de Klerk would like to thank Chris Godsil for many valuable discus-sions on the contents of this chapter.

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