Matrices attaining the minimum semidefinite rank of a chordal graph

Linear Algebra and its Applications 438 (2013) 3804–3816

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Matrices attaining the minimum semidefinite rank of a chordal graph Naomi Shaked-Monderer Emek Yezreel College, Emek Yezreel 19300, Israel

ARTICLE INFO

ABSTRACT

Article history: Received 20 January 2011 Accepted 11 August 2011 Available online 9 September 2011

Every positive semidefinite matrix whose graph G is chordal, may be represented as a sum of rank 1 positive semidefinite matrices whose graphs are subgraphs of G. We show that if the matrix is of minimum rank, then this representation is unique. This result is used to give a full characterization of the completely positive matrices of minimum rank with a given chordal graph. Every such matrix is shown to have a unique, easily computable, minimal rank 1 representation, and cprank equal to its rank. We also characterize all chordal graphs with the property that every minimum rank doubly nonnegative matrix realization of the graph is completely positive. © 2011 Elsevier Inc. All rights reserved.

Submitted by W. Barrett Dedicated with admiration and gratitude to Avi Berman, Moshe Goldberg and Raphi Loewy AMS classification: 15B48 05C50 Keywords: Chordal graph (Edge-) clique covering number Completely positive matrix cp-Rank Cholesky decomposition

1. Introduction For a real symmetric n × n matrix A, the graph of A is denoted by G(A). Its vertex set is {1, . . . , n}, and ij is an edge if and only if aij  = 0. For a (simple, undirected) graph G = (V , E ), with V = {1, . . . , n}, and a set S of symmetric n × n matrices, let mrS (G) be the minimum rank of matrices in S with graph G, i.e., mrS (G)

= min{rank A | A ∈ S , G(A) = G}.

E-mail address: [email protected] 0024-3795/$ - see front matter © 2011 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.laa.2011.08.025

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Denote by PSD the set of real positive semidefinite matrices, by DNN the set of doubly nonnegative matrices, and by CP the set of completely positive matrices (definitions of the latter two follow shortly). Thus mrPSD (G) denotes here the minimum semidefinite rank of G, more commonly denoted by mr+ (G) (the notation mrPSD (G) is preferred here for the sake of uniformity of notations). The main results of the paper deal with completely positive matrices with a chordal graph, which are of minimum rank, but we prove first a result on positive semidefinite matrices, which seems to be interesting for its own sake. An n × n matrix A is completely positive if A = BBT for some (entrywise) nonnegative matrix B ∈ Rn×k (notation: B ≥ 0). Equivalently, A is completely positive if

A

=

k  i=1

bi bTi ,

bi

∈ Rn , bi ≥ 0,

(1)

where the bi ’s are B’s columns. A representation of A in the form (1) is called a rank 1 representation of A. The minimal number of columns in such B (or summands in a rank 1 representation) is the cp-rank of A. Every completely positive matrix is doubly nonnegative, that is, it is both positive semidefinite and nonnegative. The converse is true for matrices of order at most 4, but is not generally true for larger matrices. For every completely positive matrix A, cp-rank A ≥ rank A, but cp-rank A may be much larger than rank A. There are two basic problems concerning complete positivity: determining which doubly nonnegative matrices are completely positive, and computing, or estimating, the cp-rank of a given completely positive matrix. Both these problems are open. In particular, a tight upper bound for the cp-ranks of n × n completely positive matrices is still unknown. According   to the DJL conjecture,

made in [9], the cp-rank of an n × n completely positive matrix is at most

n2 4

. This bound is known to

hold in some special cases, and no counter example is known at this time (although such an example was announced in [3], it was never published). Details on completely positive matrices can be found in [6]. The state of the research of these matrices may be summed up in short: Given a doubly nonnegative matrix, there is no systematic way to find out whether it is completely positive or not; when a matrix is recognized as completely positive, it might not be clear what its cp-rank is; and when the cp-rank is known, it might not be easy to find a factorization A = BBT , where B is an n × k nonnegative matrix and k = cp-rank A. Nonetheless, results on all these aspects of the subject do exist, and quite a few of these results deal with matrices with certain types of graphs. For example, graphs for which every doubly nonnegative matrix realization is completely positive (completely positive graphs) were characterized in a series of papers [5,4,12,1]; In [9] completely positive matrices with triangle free graphs were characterized, and their cp-ranks were determined; In [13] the DJL conjecture was shown to hold for 5 × 5 matrices whose graphs are not complete; In [17] graphs with the property that every completely positive realization has cp-rank equal to the rank were characterized. We continue here this line of research by studying completely positive matrices with a chordal graph, which are of minimum rank. A graph G = (V , E ) is chordal if every cycle of length 4 or more in G has a chord. Chordal graphs are discussed in the next section. For now we mention only one characterization of chordal graphs, that was proved in [2], independently in [15], and again recently in [11]: Let PSD(G) be the convex cone of positive semidefinite matrices whose graphs are subgraphs of G. A graph G is chordal if and only if the extreme rays of PSD(G) are generated by the rank 1 matrices in G. In other words, G is chordal if and only if every positive semidefinite matrix whose graph is a subgraph of G may be represented as a sum of rank 1 positive semidefinite matrices whose graphs are subgraphs of G. Usually, there are several such representations. We will show here that if A is a positive semidefinite matrix with G(A) = G, G chordal, and rank A = mrPSD (G), then the representation of A in this form is unique. It should be mentioned that the minimum semidefinite rank of a chordal graph G is known: By [16], when G is a chordal graph, mrPSD (G)

= mrDNN (G) = mrCP(G) = cc (G),

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where cc (G) is the minimal number of complete subgraphs of G needed to cover G’s edges. (mrPSD (G) = cc (G) was also proved in [7], for both the real and complex case.) The paper is organized as follows: in Section 2 we cover terminology, notations and basic facts. In Section 3 we prove the result on positive semidefinite matrices with chordal graphs. In Section 4 we use this result to study doubly nonnegative matrices A with graph G, where G is a chordal graph, and rank A = mrDNN (G) = r. It turns out that in this case, the problems of determining whether A is completely positive, computing cp-rank A if it is, and finding a factorization A = BBT where B ≥ 0 has cp-rank A columns, have relatively simple answers: A is completely positive if and only if for a certain permutation matrix P a Cholesky-type procedure on P T AP yields a nonnegative n × r factor L such that P T AP = LLT . In that case, A is of course completely positive with cp-rank equal to the rank, and B = PL ≥ 0 satisfies A = BBT . It is also shown that this is the unique completely positive factorization of A. We also characterize in Section 4 all chordal graphs with the property that every doubly nonnegative matrix of minimum rank is completely positive. 2. Preliminaries If A is an n × n matrix and α ⊆ {1, . . . , n} is a set of k indices we write A[α] to denote the principal submatrix of A on rows and columns α . Similarly, for a vector x ∈ Rn , x[α] denotes the vector of length k consisting of the entries of x indexed by α . The support of a vector x ∈ Rn is supp x = {1 ≤ i ≤ n | xi  = 0}. For a set of indices C ⊆ {1, . . . , n}, we denote by 1C the vector supported by C whose nonzero elements are all equal to 1. The subgraph of a graph G = (V , E ) induced by a set of vertices K ⊆ V is denoted G[K ], and the subgraph induced on the complement of K is denoted by G \ K. A clique in G is a nonempty set C ⊆ V such that every two vertices in C are adjacent, i.e., G[C ] is a complete graph. For graph theoretic terminology and notations see [8]. A collection C = {C1 , . . . , Ck } of cliques in G is a clique covering 1 of G, if each edge of G is an edge of one of the k complete subgraphs G[C1 ], . . . , G[Ck ]. The (edge-)clique covering number cc (G) is defined to be the minimal number of cliques in a clique covering of G. For a survey of results on this parameter see [14]. A graph G is chordal if and only if there exists a perfect elimination ordering (PEO) of G’s vertices. An order on G’s vertices is a PEO if for every vertex v the set Xv

= {u ∈ adj (v) | u > v}

is a clique. A vertex v of G is simplicial, if its adjacency set is a clique. Every chordal graph which is not the complete graph is known to have at least two nonadjacent simplicial vertices. A PEO of G’s vertices may be obtained by first choosing any simplicial vertex and labeling it 1. When i vertices have already been labeled 1, . . . , i, any simplicial vertex of G \ {1, . . . , i} may be chosen to be labeled i + 1. Thus there exist many different PEO’s for any given chordal graph G on n > 1 vertices. Details on chordal graphs may be found in [10]. Clearly, for every graph G, mrPSD (G)

≤ mrDNN (G) ≤ mrCP (G) ≤ cc (G).

As mentioned in the introduction, when G is chordal all these parameters are equal: mrPSD (G)

= mrDNN (G) = mrCP (G) = cc (G).

Note that if a graph G consists of a single vertex, all these parameters are zero, and if a graph G (not necessarily chordal) is not connected, each of these graph parameters’ value for G is the sum of its values for the components of G. 1 The complete subgraph induced by a clique is sometimes referred to also as a “clique". It is technically the complete subgraphs that cover G.

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Let A be any positive semidefinite matrix of rank r. Then A can be factored as A = LLT , where L = [x1 . . . xr ] is an n × r matrix, such that if vi = min supp xi , then v1 < · · · < vr , and vi ≥ i. This can be achieved by the following version of the Cholesky factorization: Let a be the first nonzero column of A. Suppose it is the ith column of A. Then aii  = 0 (since in a positive semidefinite matrix the rank of a principal submatrix A[α] is equal to the rank of the submatrix consisting of columns α ). Let x1

1

= √ a. aii

The matrix A1 = A − x1 x1T is a positive semidefinite matrix of rank r − 1, whose ith column is zero, as well as any other column which in A was a scalar multiple of a (A1 is the direct sum of the Schur complement A/A[i] with a 1 × 1 zero matrix). When x1 , . . . , xk have already been calculated, and  Ak = A − ki=1 xi xiT , perform the same step on Ak to obtain xk+1 and Ak+1 . Due to the rank reduction  at each step, in step r = rank A the process ends, with Ar = A − ri=1 xi xiT = 0. For example, if ⎡

A

=

4 2 6 2 0

⎤

⎥ ⎢ ⎥ ⎢ ⎢2 1 3 1 0⎥ ⎥ ⎢ ⎥ ⎢ ⎢ 6 3 10 2 1 ⎥ ⎥ ⎢ ⎥ ⎢ ⎢2 1 2 3 1⎥ ⎦ ⎣ 0 0 1 1 5

this process yields ⎡

L

= [x1 x2 x3 ] =

2

0 0

⎤

⎥ ⎥ 0 0⎥ ⎥ ⎥ 1 0⎥. ⎥ ⎥ −1 1 ⎥ ⎦ 0 1 2

⎢ ⎢ ⎢1 ⎢ ⎢ ⎢3 ⎢ ⎢ ⎢1 ⎣

To stress that we are not looking for a square lower triangular L with some zero columns, nor permuting A’s rows and columns so that its leading r × r principal submatrix is nonsingular, as is sometimes done in the process of computing a “regular" Cholesky factorization, we will refer to the specific version of the Cholesky factorization described above as the Cholesky* factorization. Given a chordal graph G such that 1 < · · · < n is a PEO of its vertices, if A is positive semidefinite and E (G(A)) ⊆ E (G), then in the Cholesky* factor L = [x1 . . . xr ] of A, the support of each of the columns is a clique in G. Indeed, if the first nonzero column of A is column i, then the first i − 1 rows of A are zero, and thus the support of the ith column of A, and therefore also that of x1 , is contained in Xi ∪ {i}, which is a clique in G. Suppose we have already shown that supp xj is a clique in G, 1 ≤ j ≤ k. Then for every such j we  have E (G(xj xjT )) ⊆ E (G). This, combined with the fact that E (G(A)) ⊆ E (G) and Ak = A − kj=1 xj xjT , implies that E (G(Ak ))

⊆ E (G(A)) ∪

k

∪

j=1

E (G(xj xjT ))

⊆ E (G).

Repeat the argument above to show that the support of the first nonzero column of Ak , and therefore supp xk+1 , is a clique in G. (See similar arguments in [15,11].)

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If G is a chordal graph, and v1 , . . . , vn is any PEO of its vertices, let P





= ev1 . . . evn ,

where e1 , . . . , en are the standard basis vectors in Rn . We refer to P as the permutation matrix corresponding to the PEO v1 , . . . , vn . If G(A) = G, the graph G(P T AP ) is G, with each vertex vi re-labeled i. 3. Positive semidefinite matrices Let A be a nonzero positive semidefinite matrix with a chordal graph G. There are usually several different representations of A as a sum of rank 1 matrices whose graphs are subgraphs of G, as in the following simple example, of a 2 × 2 matrix whose graph is the complete graph on 2 vertices: ⎡ ⎣

2 1

⎤

⎡

⎦

⎣

1 1

=

⎤ 1 0 0 0

⎡

⎦+⎣

1 1 1 1

⎤

⎡

⎦

⎣

=

⎤ 2 1 1

1 2

⎡

⎦+⎣

⎤ 0 0 0

1 2

⎦.

In this section it is shown that when rank A = mrPSD (G), A has a unique representation as a sum of rank 1 matrices whose graphs are subgraphs of G. It should be noted that every rank 1 summand in a given representation may be represented as a convex combination of several copies of itself, thus creating more representations of A. We consider only representations with no such redundancies, i.e.,

A

=

m  i=1

bi bTi ,

∀i supp bi is a clique in G, and ∀i = j bi is not a scalar multiple of bj

(2)

We will refer to a representation of a positive semidefinite matrix in the form (2) as a rank 1 Grepresentation. Two rank 1 G-representations which differ only in the order of the summands are regarded as the same representation. m T T B = [b1 . . . bm ]. And if A = BBT , then the Recall that if A = i=1 bi bi , then A = BB , where m column space of A is equal to that of B. Thus if A = i=1 bi bTi , then b1 , . . . , bm span the column space of A. The notion of a rank 1 G-representation of a positive semidefinite matrix is similar to the notion of a rank of a completely positive matrix. Indeed, if A is completely positive, and  1 representation T A= m i=1 bi bi is a rank 1 representation of A (i.e., the vectors bi are all nonnegative), then there are no cancelations in the sum. So E (G(A)) ⊇ E (G(bi bTi )) for every i. That is, every rank 1 representation of a completely positive matrix A with no redundancies is a rank 1 G-representation, where G = G(A). We will need the following lemma, which appeared in [16]. For the sake of completeness we include a proof (a somewhat shorter version of the proof in [16]). Lemma 1. Let G be a chordal graph on n vertices, x1 , . . . , xk be nonzero vectors in Rn , and C1 , . . . , Ck be maximal cliques in G such that supp xi ⊆ Ci , i = 1, . . . , k, and supp xi ∪ supp xj is not a clique for every i  = j. Then for every 1 ≤ q ≤ k, if x ∈ span {x1 , . . . , xk } and supp x ⊆ Cq , then x is a scalar multiple of xq . Proof. If k = 1 there is nothing to prove. If k > 1, by the assumption on the supports of the vectors xi , C1 , . . . , Ck are k different maximal cliques. In particular, G is not a complete graph, and given 1 ≤ q ≤ k we may choose a PEO of G’s vertices in which Cq ’s vertices are the last vertices (when constructing the PEO choose at each step a simplicial vertex which is not in Cq , until Cq ’s vertices are the only ones left). Assume 1 < 2 < · · · < n is already such a PEO. Let vi = min supp xi . We may assume that v1 ≤ v2 ≤ · · · ≤ vk . But vi  = vj for i  = j – otherwise, supp xi ∪ supp xj ⊆ Xvi ∪ {vi } would have been

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a clique. Thus v1

< v2 < · · · < vk ,

and for every i

< j, vi ∈ / supp xj .

Also, in this PEO every v ≥ vq is in Cq , and thus in this ordering of the xi ’s we have that q Now if x ∈ span {x1 , . . . , xk }, x

=

k 

=

By (3) xk [α]

= k.

ai xi .

i=1

In particular, for α x[α]

(3)

k  i=1

= {v1 , . . . , vk−1 } we have

ai xi [α].

= 0, so k −1

x[α]

=

x[α]

= La,

i=1

ai xi [α],

(4)

or (5)

where a = [a1 , . . . , ak−1 ]T and L = [x1 [α] . . . xk−1 [α]]. By (3) the matrix L is lower triangular, and vi ∈ supp xi implies that L has nonzero diagonal elements. Hence L is nonsingular. If supp x ⊆ Ck , then / Ck , i < k, we have that x[v1 ] = · · · = x[vk−1 ] = 0, i.e. x[α] = 0. Thus (5) implies that since vi ∈ a = 0, that is x = ak xk .  We can now prove Theorem 1. Let G be a chordal graph with r = cc (G) = mrPSD (G), and let A be a positive semidefinite matrix with graph G and rank r. Then A has a unique rank 1 G-representation, and there are r summands in this representation. Moreover, this representation may be obtained as follows: A = (PL)(PL)T , where P is any permutation matrix corresponding to a PEO of G’s vertices, and P T AP = LLT is the Cholesky* factorization of P T AP. Proof. Assume w.l.o.g. that 1 < · · · < n is a PEO of G’s vertices, and let A = LLT , L = [x1 . . . xr ], be the Cholesky* factorization of A. For every i = 1, . . . , r let vi = min supp xi , and let Ci be a maximal clique in G such that supp xi ⊆ Xvi ∪{vi } ⊆ Ci . The vertex v1 is a simplicial vertex of G. (This is certainly the case if v1 = 1. If v1 > 1, then the first v1 − 1 rows and columns of A are zero. Thus the first v1 − 1 vertices of G are isolated vertices, and v1 is simplicial in this case too.) Therefore C1 = Xv1 ∪ {v1 }. Since  A = ri=1 xi xiT , we have E (G(A))

r

 

⊆ ∪ E G xi xiT i=1



.

But G(A) = G, and each G(xi xiT ) is the complete graph on vertices supp xi , so {supp x1 , . . . , supp xr } is a clique covering of G by r cliques. For i  = j we have that supp xi ∪ supp xj is not a clique, otherwise two of the cliques in this covering could be replaced by one, contradicting the fact that r = cc (G).

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p Let A = i=1 bi bTi be any rank 1 G-representation of A. Necessarily p is both span {b1 , . . . , bp } and span {x1 , . . . , xr }, and thus bi

≥ r. Then A’s column space

∈ span {x1 , . . . , xr }, i = 1, . . . , p.

Let wi = min supp bi . We may assume that w1 ≤ . . . ≤ wp . We prove by induction on k, 1 ≤ k ≤ r, that bi = ±xi for every 1

≤ i ≤ k. If k = 1, let

 = {i ∈ {1, . . . , p} | v1 ∈ supp bi }. Since av1 v1 > 0, and this is the first nonzero diagonal entry of A, we get by the labeling of the bi ’s that 1 ∈ . The maximal clique C1 = Xv1 ∪ {v1 } is the only maximal clique containing v1 , hence supp bi ⊆ C1 for every i ∈ . For each i ∈ , we may apply Lemma 1 with k = r, x1 , . . . , xr and C1 , . . . , Cr as constructed above, x = bi and q = 1. By the lemma, every bi , i ∈ , is a scalar multiple of x1 . This together with the assumption of no redundancy in the representation implies that  is a singleton, that is, b1 is the only vector in the representation whose support includes v1 . Thus av1 v1 = (b1 bT1 )v1 v1 = (x1 x1T )v1 v1 , and b1 = ±x1 . Assume the claim has been proved for 1 ≤ k < r. Then Ak

=A−

k  i=1

xi xiT

=

r  i=k+1

xi xiT

=

p  i=k+1

bi bTi .

Recall that rank Ak = rank A−k, and by the construction of the Cholesky* factorization the first vk+1 −1 rows in Ak are zero, and row vk+1 is nonzero. Setting  = {i ∈ {k + 1, . . . , p} | vk+1 ∈ supp bi }, we get that k + 1 ∈  and supp bi ⊆ Xvk+1 ∪ {vk+1 } ⊆ Ck+1 for every i ∈ . Repeating the previous argument for the current  we get that bk+1 = ±xk+1 . r r p T T T Finally we get A = i=1 xi xi = i=1 bi bi , and thus i=r +1 bi bi = 0, which implies that this remainder sum is empty.  Remark 1. Let G be a chordal graph with n vertices, m maximal cliques, and cc (G) assume that G has no isolated vertices. Then:

= r. For simplicity,

• For every r < k ≤ n there exist matrices of rank k with more than one rank 1 G-representation. To see that, first consider the case that r = 1 and k = 2. If A is a rank 2 positive semidefinite

•

matrix whose graph G is complete, then A has more than one rank 1 G-representation. One such representation is obtained by the Cholesky* factorization of A, A = LLT . To obtain another, choose an i for which the ith column of A is not a scalar multiple of the first column, let P be a permutation matrix corresponding to an ordering of the vertices in which the ith vertex of G is labeled 1, and let L LT be the Cholesky* factorization of P T AP. The factorizations A = LLT and A = (P L)(P L)T P T AP =  yield different rank 1 G-representations of A. For example, both L and P L have a unique column which has no zero entry. In L this column is a scalar multiple of the first column of A, and in P L this column is a scalar multiple of the ith column of A. But since these two columns of A are linearly independent, these unique zero-free columns in L and P L are different (just as in the 2 × 2 example at the beginning of the section). For all other cases of r < k ≤ n, here is one way to construct a matrix with at least two rank 1 G-representations: Choose maximal cliques C1 , . . . , Cr covering G. Let bi = 1Ci , i = 1, . . . , r. For each r < i ≤ k let bi be any standard basis vector not in span {b1 , . . . , bi−1 }. Then b1 , . . . , bk are  linearly independent and A = ki=1 bi bTi has graph G. If, say, supp br +1 ⊆ Cr , then by the rank 2 T case discussed above, br br + br +1 bTr+1 has more than one representation, and therefore so does A. For every r ≤ k ≤ m there exist matrices of rank k with a unique rank 1 G-representation. To construct such a matrix, let C1 , . . . , Ck be k different maximal cliques covering G. Assume w.l.o.g. that 1 < · · · < n is a PEO of G’s vertices, and that the k maximal cliques are labeled so that vi = min Ci satisfy v1 < · · · < vk (vi = vi+1 is impossible, since it would have implied that both Ci

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and Ci+1 are equal to the clique Xvi ∪{vi }, contradicting the fact that these are two different maximal  cliques). Let A = ki=1 1Ci 1TCi . Since the k cliques cover G, G(A) = G. We compute the Cholesky* factorization A = LLT : By the labeling of the cliques, the first nonzero column of A is column v1 , / Ci , i ≥ 2, column v1 of A is equal to column v1 of 1C1 1TC1 , and adj (v1 ) ∪ {v1 } = C1 . Since v1 ∈  hence the first column in L is x1 = 1C1 , and A1 = A − x1 x1T = ki=2 1Ci 1TCi . Continue by induction:  k T after proving that xj = 1Cj for 1 ≤ j ≤ i, we have that Ai = A − ij=1 xj xjT = j=i+1 1Cj 1Cj . Again we deduce by the labeling of the cliques that the first nonzero column in Ai is column vi+1 , and it is equal to the first nonzero column of 1Ci+1 1TCi+1 , that is, to 1Ci+1 . Hence xi+1 = 1Ci+1 . This

shows that in the Cholesky* factorization A = LLT , L = [1C1 , . . . , 1Ck ]. In particular, rank A = k. To show that the Cholesky* factorization of A yields a unique rank 1 G-representation of A, we repeat the arguments in the proof of Theorem 1, with one difference: In the proof of Theorem 1, L had r columns x1 , . . . , xr , and there were r maximal cliques C1 , . . . , Cr s.t. supp xi ⊆ Xvi ∪ {vi } ⊆ Ci for every i. We had r = cc (G), and this implied that supp xi ∪ supp xj is not a clique for i  = j (and therefore Lemma 1 could be used). Here L has k, r ≤ k ≤ m, columns x1 , . . . , xk . C1 , . . . , Ck are k different maximal cliques s.t. supp xi = Xvi ∪ {vi } = Ci for every i. For i  = j we have that supp xi ∪ supp xj = Ci ∪ Cj is not a clique since Ci and Cj are different maximal cliques in G. Again Lemma 1 can be used, and the proof continues exactly as the proof of Theorem 1. 4. Completely positive matrices In this section we consider positive semidefinite and doubly nonnegative matrices A with a chordal graph, which are of minimum rank. Theorem 2. Let G be a chordal graph with r = cc (G) = mrPSD (G), and let A be a positive semidefinite matrix with graph G and rank r. Then the following are equivalent: (a) A is completely positive. (b) There exists a PEO of G’s vertices, such that for the corresponding permutation matrix P, the matrix L in the Cholesky* factorization P T AP = LLT is nonnegative. (c) For every PEO of G’s vertices and corresponding permutation matrix P, the matrix L in the Cholesky* factorization P T AP = LLT is nonnegative. Proof. Clearly (c) ⇒ (b) ⇒ (a). We prove  thatT (a) ⇒ (c). If A is a completely positive matrix, then any minimal rank 1 representation A = m i=1 bi bi , where bi ≥ 0 for every i, is a rank 1 G-representation of A. By Theorem 1, there is a unique such representation of A, and if P is any permutation matrix corresponding to a PEO of G’s vertices, the factorization A = (PL)(PL)T yields this unique representation. This implies that m = r and (since each column in PL has a positive element) PL is equal, up to the order of its columns, to [b1 . . . br ]. Thus PL, and L, are nonnegative.  In other words, Theorem 2 states that a positive semidefinite matrix A with a chordal graph, which is of minimum rank, is completely positive if and only if the rank 1 matrices in the unique minimal rank 1 G-representation of A are nonnegative. In particular, Corollary 1. Let G be a chordal graph. Every completely positive matrix A with G(A) = G and rank A = mrCP (G) satisfies cp-rank A = rank A, and has a unique minimal rank 1 representation. This representation is obtained by the factorization A = (PL)(PL)T , where P is any permutation matrix corresponding to a PEO of G’s vertices, and P T AP = LLT is the Cholesky* factorization of P T AP. Next we characterize all chordal graphs with the property that every doubly nonnegative matrix of rank cc (G) = mrDNN (G) is completely positive. Note that if cc (G) = 0, then G has no edges. In that case, the zero matrix is the unique positive semidefinite matrix with graph G and minimum rank, and it is completely positive. We therefore consider from now on only graphs with cc (G) > 0.

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For the characterization, we use the following notations: Let C = {C1 , . . . , Cr } be a minimal covering of G’s edges by maximal cliques of G. For every 1 ≤ i ≤ r, define Hi = Hi (C ) to be the graph on vertices Ci , whose edges are the edges of G covered by Ci , which are not covered by any other Cj . That is, E (Hi )

= {uv ∈ E (G[Ci ]) | uv ∈ / E (G[Cj ]) for 1 ≤ j ≤ r , j = i}

By the minimality of the covering, each Hi has at least one edge. Theorem 3. Let G be a chordal graph with r

= cc (G) > 0. Then the following conditions are equivalent:

(a) Every doubly nonnegative matrix with graph G and rank r is completely positive. (b) For every minimal clique covering of G by maximal cliques, C = {C1 , . . . , Cr }, each of the graphs Hi (C ), 1 ≤ i ≤ r, is connected. Proof. We prove first that (a) ⇒ (b). Let G be a chordal graph, which satisfies (a). Let C = {C1 , . . . , Cr } be a minimal covering of G by maximal cliques and suppose one of the graphs Hi = Hi (C ), w.l.o.g. Hr , is not connected. Let x be a vector such that supp x = Cr , x has both positive and negative entries, and the sign of the entries of x is constant on each connected component of Hr . For a small enough positive scalar a, the positive semidefinite matrix

A

=

r −1 i=1

1Ci 1TCi

+ axxT

(6)

is nonnegative, and has G(A) = G. Also r = mrPSD (G) ≤ rank A ≤ r, so rank A = r. By Theorem 1, (6) is the unique representation of A as a sum of r rank 1 matrices supported by cliques in G. Since there is a rank 1 summand in this representation that is not nonnegative, A is not completely positive. To prove that (b) ⇒ (a), suppose that 1 < · · · < n is a PEO of G’s vertices, and (b) holds. Let A be a doubly nonnegative matrix with G(A) = G, rank A = r, and Cholesky* factorization A = LLT , L = [x1 . . . xr ]. Then supp x1 , . . . , supp xr are r cliques covering the edges of G = G(A). For each 1 ≤ i ≤ r let Ci be a maximal clique such that supp xi ⊆ Ci . These r maximal cliques cover G. Since r = cc (G), these are necessarily r different maximal cliques, and C = {C1 , . . . , Cr } is a minimal covering of G by maximal cliques. By the assumption, each Hi = Hi (C ) is connected. For every 1 ≤ i ≤ r, and every pq ∈ E (Hi ), we have apq = (xi )p (xi )q . Then apq > 0 implies that (xi )p and (xi )q have equal sign. And since Hi is connected, all the nonzero entries of the vector xi have the same sign – they are positive, since the first nonzero entry of every column of L is positive. Thus L ≥ 0, and by Theorem 2, A is completely positive. 

Theorem 3 may be somewhat refined, due to the following result on minimal coverings of a chordal graph by maximal cliques. Theorem 4. Let G be a chordal graph and r = cc (G) > 0. Let C = {C1 , . . . , Cr } be a minimal clique covering of G by maximal cliques. If the graphs Hi = Hi (C ), 1 ≤ i ≤ r, are all connected, then C is the unique minimal covering of G by r maximal cliques. If v is a simplicial vertex of G, then C = {v} ∪ adj (v) is a maximal clique of G, and it is the unique maximal clique of G containing v. Thus every minimal covering of G by maximal cliques includes C. We may try to prove Theorem 4 by induction on r: discard v and all or part of the edges of G[C ] and consider the remaining graph. However, this does not always work well, as illustrated by the following example: Let G be the chordal graph shown here:

N. Shaked-Monderer / Linear Algebra and its Applications 438 (2013) 3804–3816

t5
@ A
A@
A @
At2 @t8 t t 7 1 @ A
@ A
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t t t 4

3

t

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10t

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6

It has 7 maximal cliques: C1 = {1, 2, 9, 10}, C2 = {3, 7, 9}, C3 = {4, 7, 10}, C4 = {5, 8, 10}, C5 = {6, 8, 9}, C6 = {7, 9, 10} and C7 = {8, 9, 10}. Any covering of G must include C1 , C2 , C3 , C4 , C5 , since each of these cliques is the unique maximal clique containing one (or more) of the simplicial vertices of G, and covering the edges incident to it (or them). But these 5 cliques cover the whole G, so cc (G) = 5, and {C1 , C2 , C3 , C4 , C5 } is a minimal clique covering of G by maximal cliques. In this case, each Hi is equal to G[Ci ], so it is a complete graph (in particular, it is connected). The vertex 1 is a simplicial vertex. Suppose we try to obtain a chordal graph with smaller clique covering number by removing the vertex 1 with some or all of the edges of C1 . If we discard both the vertices 1 and 2 and all the edges of C1 , then we obtain a graph with clique covering number 4, which is not chordal. But if we discard only the vertex 1 and/or the vertex 2 with all the edges incident to it/them, we obtain a chordal subgraph of G, but one whose clique covering number is still 5. In any case, we have trouble in the induction step. In this example, erasing the edges in any of the other cliques in the covering would yield a chordal graph with clique covering number 4, but in the general case, identifying a clique in the covering whose removal will yield a chordal subgraph with a smaller clique covering number is not so obvious (nor is it obvious that such a clique always exists). To by-pass this difficulty, we consider a more general setting, aided by some additional terminology and notations. Let G = (V , E ) be a subgraph of a graph G = (V , E ). We will say that a collection C = {C1 , . . . , Ck } of cliques in G is a covering of G by cliques of G, if each edge of G is an edge of one of the complete graphs G[C1 ], . . . , G[Ck ]. The minimal number of cliques of G in a clique covering of G will be denoted by cc G (G ). Then cc G (G ) is also the minimal number of maximal cliques of G needed to cover the edges of G , and cc G (G ) = 0 if and only if G has no edges. Suppose C = {C1 , . . . , Ck } is a covering of G by maximal cliques of G. For each 1 ≤ i ≤ k, let Hi (C , G ) be the graph with vertex set Ci , whose edges consist of all the edges of G with both ends in Ci , which are not covered by any other clique Cj , 1 ≤ j ≤ k, j  = i. Using these notations we prove the following lemma: Lemma 2. Let G = (V , E ), be a chordal graph with at least two vertices and with no isolated vertices. Let G = (V , E ) be a spanning subgraph of G with no isolated vertices, and let k = cc G (G ). Let C = {C1 , . . . , Ck } be a minimal covering of G by maximal cliques of G. Let Hi = Hi (C , G ), 1 ≤ i ≤ k. If each Hi , i = 1, . . . , k, is connected, then C is the unique minimal covering of G by maximal cliques of G. Proof of Lemma 2. We prove the lemma by induction on k. Since G has edges, k ≥ 1. The case k = 1 is quite trivial: If C1 is one maximal clique covering the spanning subgraph G , which has no isolated vertices, then C1 is necessarily equal to V . Thus G is a complete graph, so there is only one maximal clique and the covering is unique. (Since there are at least two vertices in V , G has at least one edge, so an empty covering is not an option.) Assume that k > 1 and the claim holds for k − 1. Let G, G , C = {C1 , . . . , Ck } and Hi = Hi (C , G ) be as in the statement of the lemma. Let v be a simplicial vertex of G. Since G is a spanning subgraph with no isolated vertices, it has an edge incident to v. This edge is covered by one of the maximal cliques in C, w.l.o.g. by C1 . In particular, v ∈ C1 . But since v is simplicial, adj G (v) ∪ {v} is the unique maximal clique of G containing

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N. Shaked-Monderer / Linear Algebra and its Applications 438 (2013) 3804–3816

v. Thus C1 = adj G (v) ∪ {v}. By that argument any collection of maximal cliques of G covering G includes C1 . E ) is chordal. Since G has no isolated G = G[K ]. The graph  G = (K ,  Now let K = ∪ki=2 Ci and  G has at least two vertices. vertices, any maximal clique of G (in particular, C2 ) is of size 2 or more. Thus  E is covered by one of the Let  E be the set of edges of G , which are not covered by C1 . Each edge in     Ci ’s, 2 ≤ i ≤ k, so both its ends are in K. Let G = (K , E ). By definition, G is a spanning subgraph of  G, and we clearly have  E ⊇ E

k

⊇ ∪ E (Hi ). i=2

Since each Hi is a connected graph on vertices Ci , both  G and  G have no isolated vertices.  G covering  G . Thus The collection C = {C2 , . . . , Ck } is a collection of k − 1 maximal cliques of  ) ≤ k − 1. On the other hand, by adding C to any clique covering of  by cliques of   ( G G, we get G cc  1 G ) + 1. Together these inequalities  G ( a clique covering of G by cliques of G. Hence k = cc G (G ) ≤ cc  G  imply that cc  G (G ) = k − 1.  = Hi (  is Ci , and its edge set consists of C,  G ). Then the vertex set of H For every 2 ≤ i ≤ k let H i i the edges of G with both ends in Ci , which are not covered by C1 , and also not covered by any other  ) = E (H ), so H  is connected. Cj , 2 ≤ j ≤ k, j  = i. Thus for every 2 ≤ i ≤ k, E (H i i i By the induction hypothesis,  C = {C2 , . . . , Ck } is the unique minimal covering of  G by maximal  cliques of G. We are now ready to prove the uniqueness of the minimal covering of G by maximal cliques of G. Suppose C = {C 1 , . . . , C k } is a covering of G by maximal cliques of G. As shown above, one of the cliques in C has to be C1 . W.l.o.g. suppose C 1 = C1 . For every 2 ≤ i ≤ k, let Ui = C i ∩ K. Each Ui is a clique in  G. G by maximal cliques of G. Each edge of  G has both The collection {C 2 , . . . , C k } is a covering of    ends in K, hence {U2 , . . . , Uk } is a covering of G by cliques of G. For every 2 ≤ i ≤ k let Wi be a G by k − 1 maximal cliques maximal clique of  G containing Ui . Then {W2 , . . . , Wk } is a covering of    C. Suppose w.l.o.g. of G. By the uniqueness of the minimal covering C, we have that {W2 , . . . , Wk } =  that Wi = Ci , 2 ≤ i ≤ k. Then Ui ⊆ Ci for each 2 ≤ i ≤ k. We now show that the reverse inclusion holds also, and therefore each of these k − 1 inclusions is actually an equality. Let 2 ≤ i ≤ k, and let v be any vertex in Ci . Since Hi is a connected graph on vertices Ci , there exists an edge of Hi incident to v. This edge is an edge of  G not covered by any Cj , j ≥ 2, j  = i, and therefore (since Uj ⊆ Cj ) also not covered by any Uj , j ≥ 2, j  = i. Hence this edge is covered by Ui , and in particular v ∈ Ui . This proves that Ci ⊆ Ui . Added to the previous inclusion, this implies that Ci = Ui = C i ∩ K. Finally, since Ci = C i ∩ K ⊆ C i for every 2 ≤ i ≤ k, and since both Ci and C i are maximal cliques in G, they are actually equal. Together with the equality C 1 = C1 this completes the proof.  Theorem 4 now follows from Lemma 2: Proof of Theorem 4. Let V0 ⊆ V be the (possibly empty) set of isolated vertices of G. A clique covering of G is a clique covering of G \ V0 . Since cc (G) > 0, G \ V0 has at least two vertices. Apply Lemma 2 with G \ V0 as both G and G of the lemma, and k = r (note that in this case Hi (C , G )) = Hi (C )).  In view of Theorems 3 and 4, we can state the following: Corollary 2. Let G be a chordal graph with r = cc (G). Then every doubly nonnegative matrix with graph G and rank r is completely positive if and only if there exists a unique covering of G by r maximal cliques, C = {C1 , . . . , Cr }, and each of the graphs Hi (C ), 1 ≤ i ≤ r, is connected. Remark 2. If C = {C1 , . . . , Cr } is any minimal clique covering of a graph G by maximal cliques, then for any Ci that contains a simplicial vertex, the graph Hi (C ) is connected.

N. Shaked-Monderer / Linear Algebra and its Applications 438 (2013) 3804–3816

3815

Example 1. The graph G1 7t

t6

@

@

@ @t5

t 1 @ @

@

@t

t

t

2 3 4 has cc (G1 ) = 4. It has a unique minimal clique covering by maximal cliques, consisting of the cliques C1 = {1, 2, 3}, C2 = {3, 4, 5}, C3 = {5, 6, 7}, C4 = {1, 3, 5, 7}. The graphs Hi , i = 1, . . . , 4 are: 7t

1

t

t5

t

t

2

3

t 3

7t

t6

t

t

4

5

1

t5

t

t 3

H1 H2 H3 H4 They are all connected, so every doubly nonnegative matrix with graph G1 and rank 4 is completely positive. Example 2. The graph G2 8t 7t t6

@

@

1

@ @t5

t @ @ t

@

@t

t

2 3 4 has cc (G2 ) = 5. A minimal clique covering by maximal cliques consists of the cliques C1 = {1, 2, 3}, C2 = {3, 4, 5}, C3 = {5, 6, 7}, C4 = {1, 7, 8}, and C5 is either the clique {1, 3, 7} or {3, 5, 7}. Since there are two different minimal coverings, it is already clear that there exists a doubly nonnegative matrix with graph G2 and rank 5, which is not completely positive. Indeed, consider the first covering. In this case, the graph H5 is not connected: 7t

1

t

t5

t

t

2

3 H1

t 3 H2

7t

t6

t

t

4

5 H3

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t7

1

t

t

t 1

3 H4

H5

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N. Shaked-Monderer / Linear Algebra and its Applications 438 (2013) 3804–3816

We remark that a graph may have a unique covering by maximal cliques, but with a non-connected Hi (and thus with a doubly nonnegative realization of minimum rank, which is not completely positive). For example, consider G3 :



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A
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A At

Acknowledgement The author is grateful to an anonymous referee, whose careful reading and useful comments led to improvements to the paper. References [1] T. Ando, Completely Positive Matrices, Lecture Notes, The University of Wisconsin, Madison, 1991. [2] J. Agler, J.W. Helton, S. McCullough, L. Rodman, Positive semidefinite matrices with a given sparsity pattern, Linear Algebra Appl. 107 (1988) 101–149. [3] F. Barioli, Complete positivity of small and large acute sets of vectors, in: Talk at the 10th ILAS Conference in Auburn, 2002. [4] A. Berman, R. Grone, Completely positive bipartite matrices, Math. Proc. Cambridge Philos. Soc. 103 (1988) 269–276. [5] A. Berman, D. Hershkowitz, Combinatorial results on completely positive matrices, Linear Algebra Appl. 95 (1987) 111–125. [6] A. Berman, N. Shaked-Monderer, Completely Positive Matrices, World Scientific, 2003. [7] M. Booth, P. Hackney, B. Harris, C.R. Johnson, M.T. Lay, L.H. Mitchell, S.K. Narayan, A. Pascoe, K. Steinmetz, B.D. Sutton, W. Wang, On the minimum rank among positive semidefinite matrices with a given graph, SIAM J. Matrix Anal. Appl. 30 (2008) 731–740. [8] R. Diestel, Graph Theory, fourth ed., Springer-Verlag, Heidelberg, 2010. [9] J.H. Drew, C.R. Johnson, R. Loewy, Completely positive matrices associated with M-matrices, Linear and Multilinear Algebra 37 (1994) 303–310. [10] M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, second ed., Annals of Discrete Mathematics, vol. 57, Elsevier, 2004. [11] N. Kakimura, A direct proof for the matrix decomposition of chordal-structured positive semidefinite matrices, Linear Algebra Appl. 433 (2010) 819–823. [12] N. Kogan, A. Berman, Characterization of completely positive graphs, Discrete Math. 114 (1993) 298–304. [13] R. Loewy, B-S. Tam, CP rank of completely positive matrices of order five, Linear Algebra Appl. 363 (2003) 161–176. [14] S.D. Monson, N.J. Pullman, R. Rees, A survey of clique and biclique coverings and factorizations of (0, 1)-matrices, Bull. Inst. Combin. Appl. 14 (1995) 17–86. [15] V.I. Paulsen, S.C. Power, R.R. Smith, Schur products and matrix completions, J. Funct. Anal. 85 (1989) 151–178. [16] N. Shaked-Monderer, Extreme chordal doubly nonnegative matrices with given row sums, Linear Algebra Appl. 183 (1993) 23–39. [17] N. Shaked-Monderer, Minimal cp rank, Electron. J. Linear Algebra 8 (2001) 140–157.