A lower bound for the Graver complexity of the incidence matrix of a complete bipartite graph

The Graver complexity of an integer matrix is currently actively investigated for its importance to integer programming, algebraic statistics and other applications ([2], [4], [5], [1]). In particular, from the universality of the three-way transportation program to general integer programs (De Loera and Onn [3]), the Graver complexity of the incidence matrix of the complete bipartite graph $K_{3,r}$ is particularly important. Berstein and Onn [2] proved that the Graver complexity $g(r)$ for the incidence matrix of $K_{3,r}$, $r\geq 3$, is bounded below as $g(r)=\Omega(2^{r})$, where $g(r)\geq 17\cdot 2^{r-3}-7$. It is a natural question to generalize this result to the complete bipartite graph $K_{t,r}$ of arbitrary size $t,r$. We prove that the Graver complexity for $K_{t,r}$ is $\Omega((t-1)^{r})$, where $t\geq 4$ is fixed and $r$ diverges to infinity. For proving our result, we employ double induction on $r$ and $t$ starting from the result of [2].

Let $A_{t,r}$ denote the incidence matrix of the complete bipartite graph $K_{t,r}$ and let $g(A_{t,r})$ denote its Graver complexity. Here we state our main theorem. Relevant notations and definitions will be given in the next section.

We give a proof of this theorem in Section 3 after giving necessary definitions and reviewing relevant known results in Section 2. We conclude the paper with some discussion in Section 4.

In this section we summarize our notation and review relevant known results on the Graver complexity following Berstein and Onn [2].

The integer kernel of an $s\times t$ integer matrix $A$ is denoted by $\ker_{\mathbb{Z}}(A)=\{x\in{\mathbb{Z}}^{t}\mid Ax=0\}$. Define a partial order $\sqsubseteq$ on ${\mathbb{Z}}^{t}$, which extends the coordinate-wise order $\leq$ on ${\mathbb{Z}}_{+}^{t}$, as follows: For two vectors $u,v\in{\mathbb{Z}}^{t}$, $u\sqsubseteq v$ if $|u_{i}|\leq|v_{i}|$ and $u_{i}v_{i}\geq 0$ for $i=1,\dots,t$. The Graver basis ${\mathcal{G}}(A)$ of $A$ is the finite set of $\sqsubseteq$-minimal elements in the set $\ker_{\mathbb{Z}}(A)\setminus\{0\}$.

For any fixed positive integer $h$, write an $ht$-dimensional integer vector $x\in{\mathbb{Z}}^{ht}$ as $x=(x^{1},\dots,x^{h})$ with each block $x^{i}$ belonging to ${\mathbb{Z}}^{t}$. The type of $x=(x^{1},\dots,x^{h})$ is the number ${\rm type}(x):=\#(\{i\mid x^{i}\neq 0\})$ of nonzero blocks of $x$. The $h$-th Lawrence lifting of an $s\times t$ matrix $A$ is the following $(t+hs)\times ht$ matrix, with $I_{t}$ denoting the $t\times t$ identity matrix:

The Graver complexity of $A$ is defined as

Let ${\mathcal{G}}({\mathcal{G}}(A))$ denote the Graver basis of a matrix whose columns are the elements of ${\mathcal{G}}(A)$ ordered arbitrarily. The following result shows that the Graver complexity of $A$ is determined by ${\mathcal{G}}({\mathcal{G}}(A))$.

A circuit of an integer matrix $A$ is a nonzero integer vector $x\in\ker_{\mathbb{Z}}(A)$, that has inclusion-minimal support with respect to $\ker_{\mathbb{Z}}(A)$, and whose nonzero entries are relatively prime. Let ${\mathcal{C}}(A)$ denote the set of circuits of a matrix $A$. Then ${\mathcal{C}}(A)\subseteq{\mathcal{G}}(A)$ (cf. [7]). An integer relation $h=(h_{1},\dots,h_{k})$ on integer vectors $v^{1},\dots,v^{k}\in{\mathbb{Z}}^{t}$

is primitive if $h_{1},\dots,h_{k}$ are relatively prime positive integers and no $k-1$ of the $\{v^{i}\}_{i=1}^{k}$ satisfy any nontrivial linear relation. By ${\mathcal{C}}(A)\subseteq{\mathcal{G}}(A)$ and Proposition 2.1 we have the following result.

In this paper we consider the Graver complexity of the incidence matrix $A_{t,r}$ for the complete bipartite graph $K_{t,r}$. Let ${\bf 1}_{t}=(1,1,\dots,1)$ denote the $1\times t$ matrix consisting of $1$’s. Then the $r$-th Lawrence lifting $A_{t,r}={\bf 1}_{t}^{(r)}$ of ${\bf 1}_{t}$ is the incidence matrix of $K_{t,r}$. In algebraic statistics, $A_{t,r}$ is the design matrix specifying the row sums and the column sums of a two-way contingency table. Another Lawrence lifting $({\bf 1}_{t}^{(r)})^{(h)}$ of ${\bf 1}_{t}^{(r)}$ is the design matrix for no-three-factor interaction model for $t\times r\times h$ three-way contingency tables ([1], [5]). It is also the coefficient matrix for the three-way transportation program. The Graver complexity $g(A_{t,r})=g({\bf 1}_{t}^{(r)})$ gives the bound of complexity of the Graver basis for the toric ideal associated with the no-three-factor interaction model for $t\times r\times h$ three-way contingency tables as $h\rightarrow\infty$.

We employ below the following notation, where $t,r$ are positive integers. Let

Then $V\oplus U$ and $V\times U$ denote the set of vertices and the set of edges of the complete bipartite graph $K_{t,r}$, respectively. They index the rows and the columns of the incidence matrix $A_{t,r}$ of $K_{t,r}$. Here we explain interpretations of a circuit of $A_{t,r}$ referring to [2]. We interpret each vector $x\in{\mathbb{Z}}^{V\times U}$ as:

1. 1.

   an integer valued function on the set of edges $V\times U$;

2. 2.

   a $t\times r$ matrix with its rows and columns indexed by V and U.

With these interpretations, $x$ is in ${\mathcal{C}}(A_{t,r})$ if and only if:

1. 1.

   as a function on $V\times U$ with the following properties: its support is a circuit of $K_{t,r}$, along which its values $\pm 1$ alternate. It can be expressed by the sequence $(v_{i_{1}},u_{i_{1}},v_{i_{2}},u_{i_{2}},\dots,v_{i_{l}},u_{i_{l}})$ of vertices of the circuit of $K_{t,r}$ on which it is supported, with the convention that its value is $+1$ on the first edge $(v_{i_{1}},u_{i_{1}})$.

2. 2.

   as a nonzero matrix with the following properties: its elements are $0,\pm 1$, its row sums and columns sums are zeros, and it has an inclusion-minimal support with respect to these properties.

The following example is the base case for our inductive argument for the lower bound of the Graver complexity.

In this section we give a proof of our main theorem. Our proof is based on recursive construction of primitive relations for circuits of $A_{t,r}$. We need recursions for $t$ and $r$, separately. In Lemma 3.1 we give a recursion for $t$ and in Lemma 3.2 we give a recursion for $r$.

We are now ready to prove Theorem 1.1. In the proof we use the following notation. Let ${\mathscr{A}}(\{x^{i}\}_{i=1}^{k})=\{\bar{x}^{i}\}_{i=1}^{k+t}$ and ${\mathscr{B}}(h)=\bar{h}=(\bar{h}_{1},\dots,\bar{h}_{k+t-1},1)$ denote circuits of $A_{t+1,t+1}$ and the primitive relation which are obtained by the operation of Lemma 3.1 to circuits $\{x^{i}\}_{i=1}^{k}$ of $A_{t,t+1}$ and the primitive relation $h$. Note that $\|{\mathscr{B}}(h)\|_{1}=\|h\|_{1}+t$. Furthermore let ${\mathscr{A}}^{\prime}(\{x^{i}\}_{i=1}^{k})=\{\bar{x}^{i}\}_{i=1}^{k+t-1}$ and ${\mathscr{B}}^{\prime}(h)=\bar{h}=(\bar{h}_{1},\dots,\bar{h}_{k+t-2},1)$ denote circuits of $A_{t,r+1}$ and the primitive relation which are obtained by the operation of Lemma 3.2 to circuits $\{x^{i}\}_{i=1}^{k}$ of $A_{t,r}$ and the primitive relation $h$. Note that $\|{\mathscr{B}}^{\prime}(h)\|_{1}=(t-1)(\|h\|_{1}-1)+t$.

Our proof uses induction on $t,r$. We will construct a primitive relation $h^{(t\times r)}$ on circuits ${\mathscr{X}}_{(t\times r)}$ of $A_{t,r}$ by induction. Therefore we obtain $g(A_{t,r})\geq\|h^{(t\times r)}\|_{1}$. Our induction is illustrated in Figure 1. There, a down arrow corresponds to the operation of Lemma 3.1, and a right arrow corresponds to the operation of Lemma 3.2.