Markov bases and generalized Lawrence liftings
aa r X i v : . [ m a t h . A C ] O c t Markov bases and generalized Lawrenceliftings
Hara Charalambous, Apostolos Thoma and Marius Vladoiu
Abstract.
Minimal Markov bases of configurations of integer vectors cor-respond to minimal binomial generating sets of the assocciated latticeideal. We give necessary and sufficient conditions for the elements of aminimal Markov basis to be (a) inside the universal Gr¨obner basis and(b) inside the Graver basis. We study properties of Markov bases ofgeneralized Lawrence liftings for arbitrary matrices A ∈ M m × n ( Z ) and B ∈ M p × n ( Z ) and show that in cases of interest the complexity of anytwo Markov bases is the same. Mathematics Subject Classification (2010).
Primary 14M25; Secondary14L32,13P10,62H17.
Keywords.
Toric ideals, Markov basis, Graver basis, Lawrence liftings.
1. Introduction
Let A be an element of M m × n ( Z ), for some positive integers m, n . Theobject of interest is the lattice L ( A ) := Ker Z ( A ). A Markov basis M of A isa finite subset of L ( A ) such that whenever w , u ∈ N n and w − u ∈ L ( A ) (i.e. A w = A u ), there exists a subset { v i : i = 1 , . . . , s } of M that connects w to u . This means that for 1 ≤ p ≤ s , w + P pi =1 v i ∈ N n and w + P si =1 v i = u .A Markov basis M of A gives rise to a generating set of the lattice ideal I L ( A ) := h x u − x v : A u = A v i . Each u ∈ Z n can be uniquely written as u = u + − u − where u + , u − ∈ N n arevectors with non-overlapping support. In the seminal work of Diaconis andSturmfels in [4], it was shown that M is a Markov basis of A if and only ifthe set { x u + − x u − : u ∈ M } is a generating set of I L ( A ) . A Markov basis M of A is minimal if no subset of M is a Markov basis of A . We say that L ( A )is positive if L ( A ) ∩ N n = { } and non positive if L ( A ) ∩ N n = { } . When L ( A ) is positive then the graded Nakayama Lemma applies and all minimal . Charalambous, Thoma and VladoiuMarkov bases have the same cardinality. When L ( A ) is non positive, it ispossible to have minimal Markov bases of A of different cardinalities, see [3].It is important to note that the study of non positive lattices has importantimplications in the study of positive ones, see for example the proof of [10,Theorem 3], [7, Lemma 5] and [8, Theorem 3.5]. The universal Markov basis of A will be denoted by M ( A ) and is defined as the union of all minimal Markovbases of A of minimal cardinality, where we identify a vector u with − u ,see [3, 10]. The sublattice of L ( A ) generated by all elements of L ( A ) ∩ N n iscalled the pure sublattice of L ( A ) and is important when considering minimalMarkov bases of A , see [3]. The pure sublattice of L ( A ) is zero exactly when L ( A ) is positive.Let u , v , w be non zero vectors in Z n . If u = v + w we write u = v + c w todenote that this sum gives a conformal decomposition of u i.e. u + = v + + w + and u − = v − + w − . The set consisting of all non zero elements of L ( A ) whichhave no conformal decomposition is denoted by G ( A ) and is called the Graverbasis of A . When u ∈ G ( A ) the binomial x u + − x u − is called primitive . G ( A )is always a finite set, see [6, 11]. In this paper we examine in detail when anelement of a minimal Markov basis belongs to G ( A ). We show in Theorem 2.3that M ( A ) ⊂ G ( A ) holds in just two cases: when L ( A ) is positive and when L ( A ) is pure of rank 1. We point out that even though the inclusion forpositive lattices is well known, we could not locate its proof in the literature,so we provide it here for completeness of the exposition.By U ( A ) we denote the universal Gr¨obner basis of A , i.e. the set whichconsists of all vectors u ∈ L ( A ) such that x u + − x u − is part of a reducedGr¨obner basis of I L ( A ) for some term order on N n . The inclusion U ( A ) ⊂ G ( A )always hold (see [11, Lemma 4.6]). In this paper we examine the relationbetween M ( A ) and U ( A ). In general M ( A ) is not a subset of U ( A ) evenwhen L ( A ) is positive as Example 2.8 shows. In Theorem 2.7 we give anecessary and sufficient condition for M ( A ) to be contained in U ( A ) when L ( A ) is positive.In Section 3, for r ≥ B ∈ M p × n ( Z ), we study the generalizedLawrence lifting Λ( A, B, r ):Λ(
A, B, r ) = r − times z }| { A A
0. . .0 0
AB B · · · B . When B = I n one gets the usual r –th Lawrence lifting A ( r ) , see [10]. Suchliftings were used to prove for example the finiteness of the Graver basisof A and are connected to hierarchical models in Algebraic Statistics, see[10, 8]. We denote the columns of A by a , . . . , a n and the columns of B byarkov bases and generalized Lawrence liftings 3 b , . . . , b n . The ( rm + p ) × rn matrix Λ( A, B, r ) has columns the vectors { a i ⊗ e j ⊕ b i : 1 ≤ i ≤ n, ≤ j ≤ r } , where e , . . . , e n represents the canonical basis of Z n . Note that L (Λ( A, B, r ))is a sublattice of Z rn . Let C ∈ L (Λ( A, B, r )). We can assign to C an r × n matrix C such that C i,j = C ( i − n + j . Each row of C corresponds to an elementof L ( A ) and the sum of the rows of C corresponds to an element in L ( B ). Thenumber of nonzero rows of C is the type of C . The complexity of any subsetof Λ( A, B, r ) is the largest type of any vector in that set.For r ≥ A, B, r ), G (Λ( A, B, r )). Welet the
Graver complexity of (
A, B ) be the supremum over r of the complexi-ties of G (Λ( A, B, r )) and denote it by g ( A, B ). By [8, Theorem 3.5], g ( A, B )is finite and equals the maximum 1-norm of the elements in the Graver ba-sis of the matrix B · G r ( A ), where G r ( A ) is the matrix whose columns arethe elements of G ( A ). In the literature there are two definitions of Markovcomplexity of ( A, B ). The first introduced in [10] defines the Markov com-plexity of (
A, B ) as the smallest integer m such that there exists a Markovbasis of Λ( A, B, r ) of type less than or equal to m for any r ≥
2. It is al-ways finite and bounded by g ( A, B ). The second definition given in [8] definesthe Markov complexity of (
A, B ) as the largest type of any element in theuniversal Markov basis of Λ(
A, B, r ) as r varies. The main result of Section3 is Theorem 3.3. It states that when L (Λ( A, B, r )) is positive, all minimalMarkov bases of Λ(
A, B, r ) have the same complexity. This is computation-ally essential, since to compute the complexity of any minimal Markov basisof Λ(
A, B, r ) one can start with any monomial order in N n , compute thereduced Gr¨obner basis of Λ( A, B, r ), eliminate extraneous elements to ob-tain a minimal Markov basisof Λ(
A, B, r ) and then read the largest type ofthe remaining elements. We note that in general L (Λ( A, B, r )) is positive forsome r ≥ L (Λ( A, B, r )) is positive for all r ≥ Z ( A ) ∩ Ker Z ( B )is positive (see Lemma 3.2). When Ker Z ( A ) ∩ Ker Z ( B ) is non positive thenTheorem 3.5 shows that the Markov complexities of Λ( A, B, r ) cannot bebounded as r varies. In Example 3.7 we give matrices A, B such that for all r ≥ L (Λ( A, B, r )) has a minimal Markov basis of complexity k for every2 ≤ k ≤ r . Finally, in Remark 3.8 we discuss implications of our work to thetwo notions of Markov complexity.
2. Universal Markov and Gr¨obner bases
For simplicity of notation we write L for L ( A ), M for M ( A ), G for G ( A ), U for U ( A ) and L pure for the pure sublattice of L generated by the elements in L ∩ N n . Theorem 2.1. [3, Theorem 4.18]
If i) rank( L pure ) > or ii) rank( L pure ) = 1 and L 6 = L pure then M is infinite. Charalambous, Thoma and VladoiuNext we consider the fibers F u of I L for any u ∈ L . We let F u := { t ∈ N n : u + − t ∈ L} . We note that if L is positive then F u is a finite set. Weconstruct a graph G u with vertices the elements of F u . Two vertices w , w are joined by an edge if there is an index i such that i -th component of w and w are nonzero. Thus w , w are joined by an edge if and only if( w − w ) + is componentwise smaller than w , meaning that at least onecomponent of their difference is strictly positive. The following necessarycondition for u ∈ L to be in M was observed in [2, Theorem 2.7] and [5,Theorem 1.3.2] when L is positive. Theorem 2.2. If L is positive then u is in the universal Markov basis of A ifand only if u + and u − belong to different connected components of G u . F u is called a Markov fiber when there exists an element v in the uni-versal Markov basis of A such that v + ∈ F u . The Markov polyhedra of F u arethe convex hulls of the elements of the connected components of G u . When L is positive, the Markov polyhedra of F u are actually polytopes. For u ∈ L we let P [ u ] be the convex hull of all elements of G u . When u ∈ M the ver-tices of P [ u ] are vertices of the Markov polyhedra of F u and we will see thatthe vertices of the Markov polyhedra are decisive in determining whether u belongs to the universal Gr¨obner basis of A . The criterion of Theorem 2.2 isused in the proof of the next theorem. Theorem 2.3.
The universal Markov basis of A is a subset of the Graver basisof A if and only if one of the following two conditions hold: i) L is positiveor ii) L = L pure and rank L = 1 .Proof. Assume first that
M ⊂ G . Theorem 2.1 says that if rank( L pure ) > L pure ) = 1 and L 6 = L pure then the universal Markov basis M of A is infinite. Since the Graver basis of A is finite the desired conclusion follows.For the converse, assume first that L is non positive. By Theorem 2.1 we arein the case where rank( L pure ) = 1 and L = L pure . We let = w ∈ N n besuch that L = h w i . It is immediate that w ∈ G and thus M = G . Next weexamine the case where L is positive. We will show that if u ∈ L , u / ∈ G then u / ∈ M . Since u / ∈ G there exist nonzero vectors v , w ∈ L such that u = v + c w . Thus u + = v + + w + and u − = v − + w − . It follows that u + , u − and u + − v = w + + v − are all in F u . Next we show that v + isnonzero. Indeed suppose not. Since v − = v − v + = v ∈ L ∩ N n = { } ,it follows that v = , a contradiction. Similarly w + , v − , w − are nonzero.Thus in the fiber F u , the elements u + , w + + v − are connected by an edgebecause ( u + − ( w + + v − )) + = v + , which is smaller than u + . Similarly u − , w + + v − are connected by an edge because (( w + + v − ) − u − ) + = w + , whichis smaller than w + + v − . It follows that u + , u − belong to the same connectedcomponent of G u and thus u / ∈ M . (cid:3) Remark 2.4.
Let E be the union of all minimal Markov bases of A , notnecessarily of minimal cardinality, where we identify a vector u with − u .Note that M ⊂ E . Next we show that E is a subset of G if and only if L isarkov bases and generalized Lawrence liftings 5a positive lattice. Indeed when L is a positive lattice then E = M ⊂ G aspointed out in the introduction (see also [3]). Suppose now that L is a nonpositive lattice and consider the only case left unanswered by Theorem 2.3:namely consider the case where L = L pure of rank 1. This means that L = h w i for some = w ∈ N n . It is easy to see that if k, l ≥ { k w , l w } is a minimal Markov basis of A . Thus k w ∈ E for every k and therefore E is infinite and cannot be a subset of the Graverbasis of A , which is equal to { w } .Next we examine the relation between the universal Markov basis of A and the universal Gr¨obner basis of A . We recall the following characterizationgiven in [11]. Theorem 2.5. [11, Theorem 7.8] If L is positive and u ∈ L then u is in theuniversal Gr¨obner basis of A if and only if the greatest common divisor ofthe coordinates of u is one and the line segment [ u + , u − ] is an edge of thepolytope P [ u ] . For u ∈ R n , we let supp( u ) := { i : u i = 0 } . For X ⊂ R n , we letsupp( X ) := [ u ∈ X supp( u ) . We note that if u ∈ M and L is positive then it is not hard to prove thatthe supports of different connected components of G u are disjoint. Hence theMarkov polytopes of F u are disjoint. Lemma 2.6.
Let L be positive. An element u of the universal Markov basis of A belongs to the universal Gr¨obner basis of A if and only if u + and u − arevertices of two different Markov polytopes.Proof. Suppose that u ∈ U . Since u ∈ M , it follows by Theorem 2.2 that u + and u − are elements of two different Markov polytopes of F u . Since [ u + , u − ]is an edge of P [ u ] it follows that u + , u − are vertices of P [ u ] and thus of theirMarkov polytopes as well.For the converse assume u + , u − are vertices of the disjoint Markovpolytopes P , P respectively. Since u + , u − are vertices of P , P we can findvectors c , c such that supp( c i ) ⊂ supp( P i ) for i = 1 , c · u + = 0, c · v > v ∈ P \ { u + } and c · u − = 0, c · v > v ∈ P \ { u − } . We define c as follows c i = ( c ) i , if i ∈ supp( c ) , ( c ) i , if i ∈ supp( c ) , , otherwise . From the definition of c it follows that c · v = 0 for all v ∈ [ u + , u − ] and c · v > v ∈ conv( P ∪ P ) \ { [ u + , u − ] } . On the other hand c · v > v / ∈ conv( P ∪ P ), since P [ u ] ⊂ R n + . Therefore [ u + , u − ] is an edge of P [ u ] and by Theorem 2.5 we have u ∈ U , as desired. (cid:3) Charalambous, Thoma and VladoiuNote that if L is non positive and rank L pure >
1, then M is infiniteby Theorem 2.1 and thus M 6⊂ U . On the other hand if L = L pure andrank L = 1 then M = U = { w } where w is the generator of L . The proof ofthe next theorem follows immediately by these remarks and Lemma 2.6. Theorem 2.7.
Let A be an arbitrary integer matrix. The universal Markovbasis of A is a subset of the universal Gr¨obner basis of A if and only if oneof the following two conditions holds L is positive and every element of a Markov fiber is a vertex of a Markovpolytope, L = L pure and rank L = 1 . We finish this section with an example which shows specific elements of M not in U . Example 2.8.
Let A = . Then L is a positive lattice. One can prove that { u , v , w } is a minimal Markovbasis of A where u = (1 , , − , − , , , , v = (0 , , , , , , − , − w = (2 , , , , − , − , − , − F u = { u + , u − } , F v = { v + , v − } and F w = { u + , w + , u + + 2 u − , u − , v + , w − , v − } . The Markov polytopes of F u and F v are zero dimensional, theyconsist of points. The Markov polytopes of F w are one dimensional: theyare the line segments conv(3 u + , u − ) and conv(2 v + , v − ). We note that w + = 2 u + + u − is in conv(3 u + , u − ), w − = v + + v − is in conv(2 v + , v − )but they are not vertices. Thus w is not in the universal Gr¨obner basis of A .Note that A has 12 different minimal Markov bases. Of those bases, exactly 4are subsets of U . Moreover |M| = 14 and |M ∩ U| = 6. Moreover, computingwith 4ti2[1], we get that G = M and |U| = 6.
3. Generalized Lawrence liftings
Let
L ⊂ Z n be a lattice. We say that = u is L –primitive if Q u ∩ L = Z u .Suppose that L is non positive. In [3] it was shown that there exists an L – primitive element u ∈ L∩ N n such that supp( u ) = supp L pure , [3, Proposition2.7, Proposition 2.10]. If L = L ( A ) then this element can be extended to aminimal basis of L pure and then to a minimal Markov basis of A of minimalcardinality by [3, Theorem 2.12, Theorem 4.1, Theorem 4.11]. This is thepoint of the next lemma. Lemma 3.1. If L is non positive, there exists an L –primitive element v ∈ N n such that v is in the universal Markov basis of A and supp( v ) = supp L pure . arkov bases and generalized Lawrence liftings 7Let A ∈ M m × n ( Z ), B ∈ M p × n ( Z ) and an integer r ≥
2. We let L r := L (Λ( A, B, r )) , L A,B := Ker Z ( A ) ∩ Ker Z ( B ) . We note that L r ⊂ Z rn while L A,B ⊂ Z n . Proposition 3.2. L A,B is positive if and only if L r is positive for any r ≥ .Proof. Let C ∈ L A,B ∩ N n . We think of the elements of L r as r × n matrices, asexplained in the introduction. We have that [ C · · · C ] T ∈ L r ∩ N rn . Conversely,if [ C · · · C r ] T ∈ L r ∩ N rn then C + · · · + C r ∈ L A,B ∩ N n . (cid:3) Suppose that L r is positive. Let W ∈ L r and let W the corresponding r × n matrix with w i as its i -th row. We define σ ( W ) = { i : w i = 0 , ≤ i ≤ r } . Thus the type of W is the cardinality of σ ( W ). The Λ( A, B, r )– degree of W is the vector Λ( A, B, r ) W + . Thus the Λ( A, B, r )–degree of W is in thespan N ( a i ⊗ e j ⊕ b i : 1 ≤ i ≤ n, j ∈ σ ( W )). It is well known that theΛ( A, B, r )–degrees of any minimal Markov basis of Λ(
A, B, r ) are invariantsof Λ(
A, B, r ), see [11].
Theorem 3.3.
When L r is positive the complexity of a minimal Markov basisof Λ( A, B, r ) is an invariant of Λ( A, B, r ) .Proof. Let M , M be two minimal Markov bases of I L r . It is enough to showthat the complexity of M is less than or equal to the complexity of M . Let W = [ w · · · w r ] T ∈ M be such that the type of W is equal to the complexityof M . We let V = [ v · · · v r ] T ∈ M be such that the Λ( A, B, r )-degree of V is the same as the Λ( A, B, r )-degree of W . Thus the Λ( A, B, r )–degree of V isin N ( a i ⊗ e j ⊕ b i : 1 ≤ i ≤ n, j ∈ σ ( W )). This implies that v + i = 0 for every i σ ( W ). Since every nonzero element in Ker Z ( A ) has a nonzero positive part(and a nonzero negative part) it follows that v i = 0 for every i σ ( W ). Thus σ ( V ) ⊂ σ ( W ). Reversing the argument we get that σ ( W ) = σ ( V ). Thereforethe complexity of M is less than or equal to the complexity of M . (cid:3) As in [8, Theorem 3.5] one can prove the following statement for arbi-trary integer matrices A ∈ M m × n ( Z ), B ∈ M p × n ( Z ). We denote by G r ( A )the matrix whose columns are the vectors of the Graver basis of A . Theorem 3.4.
The Graver complexity g ( A, B ) is the maximum –norm of anyelement in the Graver basis G ( B ·G r ( A )) . In particular, we have g ( A, B ) < ∞ . Suppose that L r is non positive. Next we show that Λ( A, B, r ) has aminimal Markov basis (of minimal cardinality) whose complexity is r . Theorem 3.5.
Suppose that L r is non positive. There exists a minimal Markovbasis of Λ( A, B, r ) of minimal cardinality, that contains an element of type r .Proof. We first show that L r ∩ N rn has an element of type r . By Lemma 3.2, L A,B is non positive. We let w ∈ L A,B ∩ N n be such that supp( w ) = Charalambous, Thoma and Vladoiusupp(( L A,B ) pure ). It follows that w ... w ∈ L r ∩ N rn has type r . Since ( L r ) pure = hL r ∩ N rn i , we are done by Lemma 3.1. (cid:3) Remark 3.6.
Suppose that r ≥ L r is non positive. Let v be the L r -primitive element of Lemma 3.1. By adding positive multiples of v to theother elements of the Markov basis of Lemma 3.1 the new set is still a minimalMarkov basis of Λ( A, B, r ) with the property that all of its elements are oftype r (see [3]).In the next example we give matrices A, B so that for any r ≥ L (Λ( A, B, r )) is non positive and has the following interesting property: itpossesses minimal Markov bases of complexity ranging from 2 to r . Example 3.7.
We let A ∈ M ( Z ), A ∈ M ( Z ), B ∈ M ( Z ) and A, B ∈M × ( Z ) be the following matrices: A = (cid:18) (cid:19) , A = (cid:18) − (cid:19) , A = ( A | A ) , B = (cid:18) −
10 0 (cid:19) ,B = ( I | B ) . We consider the matrix Λ(
A, B, r ). After column permutations it followsthat Λ(
A, B, r ) = A A
0. . .0 0
AB B · · · B → A A A A
0. . . . . .0 0 A A I I · · · I B B · · · B . We note that the lattice L (Λ( A , I , r ) | Λ( A , B , r )) is isomorphic to thedirect sum of the lattices L (Λ( A , I , r )) and L (Λ( A , B , r )).The matrix Λ( A , I , r ) is the defining matrix of the toric ideal of thecomplete bipartite graph K ,r and has a unique minimal Markov basis cor-responding to cycles of length 4: all its elements have type 2, see [9] and [12].We denote by C i the columns of Λ( A , B , r ), for i = 1 , . . . , r . We note that C , C , . . . , C r − are linearly independent while C l − = − C l for 1 ≤ l ≤ r .It follows that the lattice L (Λ( A , B , r )) has rank r and is pure. Thus it hasinfinitely many Markov bases ( see [3]). We consider the following minimalMarkov basis of Λ( A , B , r ) consisting of elements of type 1: { , , . . . , } . arkov bases and generalized Lawrence liftings 9For fixed 1 ≤ a ≤ r and 1 ≤ b ≤ E a,b be the matrix of M r × ( Z ) which has 1 on the ( a, b )-th entry and 0 everywhere else. Moreoverfor 1 ≤ i < j ≤ r and 1 ≤ s ≤ r , we let P i,j ∈ M r × ( Z ) and T s ∈ M r × ( Z )be the matrices P i,j = E i, − E i, − E j, + E j, , T s = E s, + E s, . It follows that the set M = { T , . . . , T r } ∪ { P i,j : 1 ≤ i < j ≤ r } is aminimal Markov basis of Λ( A, B, r ) of cardinality r + (cid:0) r (cid:1) . The elements of M have type 1 and 2, therefore the complexity of this Markov basis is 2.Note that the set { T , T + T , . . . , T + · · · + T k , T k +1 , · · · , T r } ∪ { P i,j : 1 ≤ i < j ≤ r } is a minimal Markov basis of Λ( A, B, r ) and the type of its elements rangefrom 1 to k , where 2 ≤ k ≤ r . Therefore for any integer k between 2 and r there are minimal Markov bases of Λ( A, B, r ) of complexity k .Moreover if T = P rs =1 T s , then the set { T, T + T , . . . , T + T r } ∪ { T + P i,j : 1 ≤ i < j ≤ r } is a minimal Markov basis of Λ( A, B, r ) such that all its elements are of type r (see [3]).We remark that if S is any integer linear combination of the elements T s , 1 ≤ s ≤ r and 1 ≤ i < j ≤ r then the element S + P i,j belongs to theinfinite universal Markov basis of Λ( A, B, r ). Remark 3.8.
As pointed out in the introduction, in the literature there aretwo definitions of Markov complexity. The one introduced in [10], namely thesmallest integer m such that there exists a Markov basis of Λ( A, B, r ) of typeless than or equal to m for any r ≥
2, is always finite: there exists a mini-mal Markov basis inside the Graver basis and thus (this) Markov complexityis always smaller than the Graver complexity. When L A,B is a positive lat-tice then Theorem 3.3 becomes essential in the computation of this Markovcomplexity: it guarantees that all minimal Markov bases have the same com-plexity. Even when L A,B is a non positive lattice this Markov complexity isfinite. For example, the Markov complexity of (
A, B ) of Example 3.7 is equalto 2. The second definition was given in [8] where the Markov complexityof (
A, B ) is the largest type of any element in the universal Markov basis ofΛ(
A, B, r ) as r varies. It is clear form Theorem 3.5 that this Markov complex-ity is infinite if L A,B is a non positive lattice. We point out that when L A,B is a positive lattice which is the main case of interest in Algebraic Statistics,by Theorem 3.3, all minimal Markov bases of Λ(
A, B, r ) have the same com-plexity for r ≥ Acknowledgment . This paper was partially written during the visit ofthe first and third author at the University of Ioannina. The third authorwas supported by a Romanian grant awarded by UEFISCDI, project number83 / / References , 3443–3451 (2007).[3] H. Charalambous, A. Thoma, M. Vladoiu, Markov bases of lattice ideals,arXiv:1303.2303v2.[4] P. Diaconis, B. Sturmfels, Algebraic algorithms for sampling from conditionaldistributions, Ann. Statist. , 363–397 (1998).[5] M. Drton, B. Sturmfels, S. Sullivant, Lectures on algebraic statistics , Oberwol-fach Seminars, 39. Birkh¨auser Verlag, Basel, viii+171 pp (2009).[6] J. E. Graver, On the foundations of linear and integer linear programming I,Math. Program. , 207–226 (1975).[7] R. Hemmecke, K. Nairn, On the Gr¨obner complexity of matrices, J. Pure Appl.Alg. , 1558–1563 (2009).[8] S. Ho¸sten, S. Sullivant, A finiteness theorem for Markov bases of hierarchicalmodels, J. Combin. Theory Ser. A , 311–321 (2007).[9] H. Ohsugi, T. Hibi, Indispensable binomials of finite graphs, J. Algebra Appl. , 421–434 (2005).[10] F. Santos, B. Sturmfels, Higher Lawrence configurations, J. Combin. TheorySer. A , 151–164 (2003).[11] B. Sturmfels, Gr¨obner Bases and Convex Polytopes , University Lecture Series,No 8, AMS, R.I. (1995).[12] R. Villarreal, Rees algebras of edge ideals, Comm. Algebra , 3513–3524(1995).Hara CharalambousDepartment of MathematicsAristotle University of ThessalonikiThessaloniki 54124Greecee-mail: [email protected] Apostolos ThomaDepartment of MathematicsUniversity of IoanninaIoannina 45110Greecee-mail: [email protected]