Clutters II

The last time I wrote on this subject, I made the case that clutters are of fairly natural interest to matroid theorists, since they are combinatorial objects with a duality involution, along with operations of deletion and contraction that are swapped by duality. They also arise fairly naturally in integer programming. In one formulation, integer programming is the problem of finding an integral vector $x$ that minimises the quantity $w^{T}x$ ($w$ is a fixed vector of weights) subject to the constraints $x\geq \mathbf{0}$ and $Mx\geq b$, where $M$ is a matrix of constraints. (Every entry of $\mathbf{0}$ is zero, and all comparisons of vectors are done entrywise – $x\geq \mathbf{0}$ means that every entry of $x$ is non-negative.)

Many optimisation problems in combinatorics can be expressed as integer programs. In particular, several problems are equivalent to integer programs where $b$ is equal to the vector of all ones, and $M$ contains only zeroes and ones. It’s an easy exercise to show that finding a minimum-size vertex cover in a graph is one of these problems. So is finding a minimum-size $s$-$t$ cut in a network.

In these cases, we may as well assume that the support of no row in $M$ properly contains the support of another row: if $r$ and $r’$ are rows of $M$, and $r$ is equal to $1$ in every position that $r’$ is equal to $1$, then we might as well remove the row $r$. This does not change the solution space of the program at all. In other words, we may as well assume that the rows of $M$ are the characteristic vectors of the edges in a clutter.

So clutters arise from combinatorial optimisation problems, via integer programming, and this leads to the following Venn diagram.

ClutterVenn

My aim today is modest: to define these classes of clutters. The illustration, along with essentially all the content found here, is in Combinatorial Optimisation: Packing and Covering, by Gérards Cornuéjols.

First of all, let $H=(S,\mathcal{A})$ be a clutter, so that $S$ is a finite set, and the members of $\mathcal{A}$ are subsets of $S$, none of which is properly contained in another. Let $M(H)$ be the incidence matrix of $H$. That is, the columns of $M(H)$ are labelled by the elements of $S$, and the rows are labelled by members of $\mathcal{A}$. An entry of $M(H)$ is one if and only if the corresponding element of $S$ is contained in the corresponding member of $\mathcal{A}$, otherwise it is zero.

Now assume that $w$ and $x$ are $|S|\times 1$ vectors with rows labelled by elements of $S$ (in the same order as the columns of $M(H)$), and the entries of $w$ non-negative. The vector $w$ can be seen as assigning a weight to each element of $S$. We will consider the following integer program.

(1) Minimise $w^{T}x$ subject to the constraints $x\geq \mathbf{0}$ and $M(H)x\geq \mathbf{1}$.

(Here $\mathbf{1}$ is the vector of all ones.)

If $x$ is an integer solution to this problem, then there is no harm in assuming that the entries of $x$ are all $0$ or $1$. (If any entry is greater than $1$, then replace it with $1$. Then $M(H)x\geq \mathbf{1}$ will still hold, and $w^{T}x$ will not have increased.) So an integer solution can be seen as a characteristic vector of a subset of $S$. This subset must intersect every member of $\mathcal{A}$ (by virtue of the constraint $M(H)x\geq \mathbf{1}$). Thus we are looking for a minimum-weight hitting set, and (1) can be thought of as the relaxation of the hitting set problem that allows non-integral solutions.

Next we consider the dual problem. Here $y$ is a $|\mathcal{A}|\times 1$ vector with entries labelled by the members of $\mathcal{A}$.

(2) Maximise $y^{T}\mathbf{1}$ subject to the constraints $y\geq \mathbf{0}$ and $y^{T}M(H)\leq w$.

In this problem we are thinking of $w$ as a capacity on each element of $S$. The vector $y$ assigns a non-negative flow value to each edge in $\mathcal{A}$, and the constraint $y^{T}M(H)\leq w$ says that the sum of flows over the edges that contain an element $s\in S$ cannot exceed the capacity that $w$ assigns to $s$. A solution to (2) maximises the sum of the flow values.

Now we can define the four classes in the Venn diagram.

$H$ has the Max Flow Min Cut property if, for every non-negative integer vector $w$, the programs (1) and (2) both have optimal solutions, $x$ and $y$, that are integral.

$H$ has the packing property if, for every vector $w$ whose entries are $0$, $1$, or $+\infty$, the programs (1) and (2) both have optimal solutions that are integral. In program (1), a weight of $+\infty$ means that we try to never include that element of $S$ in our hitting set. A weight of $0$ means that we can include that element for free.

$H$ packs if programs (1) and (2) have optimal solutions that are integral vectors when $w$ is set to be $\mathbf{1}$.

$H$ is ideal if, for any choice of vector $w$ with non-negative entries, not necessarily integral, program (1) has an optimal solution that is an integral vector.

Next time I will say more about the relations between these properties. For now, I will just note that, rather unfortunately, there really are clutters that pack, but which do not have the packing property. On the other hand, the shaded area of the Venn diagram is believed to be empty. So it is conjectured that a clutter has the Max Flow Min Cut property if and only if it has the packing property. Cornuéjols offers a bounty of $5000 for a resolution of this conjecture (or one of 17 others) before 2020.

Whittle’s Stabilizer Theorem

Representable matroids are an attractive subclass of matroids, because in their study you have access to an extra tool: a matrix representing this matroid. This is a concise way to describe a matroid: $O(n^2)$ numbers as opposed to $O(2^{2^n})$ bits declaring which subsets are (in)dependent. Let $M$ be a matroid, and $A$ a representation matrix of $M$. The following operations do not change the matroid:

  • Add a multiple of a row of $A$ to another row of $A$;
  • Scale a row of $A$ by a nonzero constant;
  • Scale a column of $A$ by a nonzero constant;
  • Add or remove all-zero rows;
  • Apply a field automorphism to each entry of $A$.

If a matrix $A_1$ can be turned into a matrix $A_2$ through such operations, then we say $A_1$ and $A_2$ are equivalent. If we don’t use any field automorphisms, then we say they are projectively equivalent. Generally, a matroid can have multiple inequivalent representations over a field. The exceptions are the finite fields $\textrm{GF}(2)$ and $\textrm{GF}(3)$ (shown in [BL]).

When we try to prove some theorem about a matroid or class of matroids, inequivalent representations can be a major complicating factor. For instance, the excluded-minor characterization of ternary matroids can be proven in under five pages [Oxley, pp. 380-385], whereas the excluded-minor characterization of quaternary matroids takes over fifty [GGK]. It is not surprising, then, that significant efforts have been made to get a handle on inequivalent representations. In this post I will focus on one such effort, namely a very attractive theorem by Geoff Whittle, who recently celebrated his 65th birthday with a wonderful workshop. First, a definition.

Definition. Let $\mathbb{F}$ be a field, and $\mathcal{M}$ a minor-closed class of $\mathbb{F}$-representable matroids. Let $N \in \mathcal{M}$. We say $N$ is a stabilizer for $\mathcal{M}$ if, for every 3-connected matroid $M \in \mathcal{M}$ that has $N$ as a minor, each representation of $N$ (over $\mathbb{F}$) extends to at most one representation of $M$ (up to the equivalence defined above).

In other words, once we select a representation for $N$, we have uniquely determined a representation of $M$. A small example: let $\mathbb{F} = \textrm{GF}(5)$, let $\mathcal{M}$ be the set of all minors of the non-Fano matroid $F_7^-$, and let $N$ be the rank-3 wheel. Now $N$ has the following representation:

$$
\begin{bmatrix}
1 & 0 & 0 & 1 & 0 & 1\\
0 & 1 & 0 & 1 & 1 & 0\\
0 & 0 & 1 & 0 & 1 & a
\end{bmatrix}
$$

where $a \in \{1, 2, 3\}$. The only 3-connected matroids in $\mathcal{M}$ that have $N$ as a minor are $N$ itself and $F_7^-$. We need to check that each representation of $N$ extends to at most one representation of $F_7^-$. Up to equivalence, the latter representation must look like

$$
\begin{bmatrix}
1 & 0 & 0 & 1 & 0 & 1 & 1\\
0 & 1 & 0 & 1 & 1 & 0 & b\\
0 & 0 & 1 & 0 & 1 & a & c
\end{bmatrix}
$$

and it is readily checked that we must have $b = c = a = 1$. Hence two of the representations of $N$ do not extend to a representation of $M$, whereas one extends uniquely to a representation of $M$. So $N$ is a stabilizer for $\mathcal{M}$.

If $\mathcal{M}$ is an infinite class, we cannot do an exhaustive check as in the example to verify a stabilizer. But Geoff Whittle managed to prove that a finite check still suffices:

Whittle’s Stabilizer Theorem [Whi]. Let $\mathcal{M}$ be a minor-closed class of $\mathbb{F}$-representable matroids, and $N \in \mathcal{M}$ a 3-connected matroid. Exactly one of the following holds:

  • $N$ is a stabilizer for $\mathcal{M}$ over $\mathbb{F}$;
  • There is a 3-connected matroid $M \in \mathcal{M}$ such that either:
    • $N = M\backslash e$ and some representation of $N$ extends to more than one representation of $M$;
    • $N = M / e$ and some representation of $N$ extends to more than one representation of $M$;
    • $N = M / e \backslash f$, $M / e$ and $M \backslash f$ are 3-connected, and some representation of $N$ extends to more than one representation of $M$.

I will conclude this post with two applications. I will leave the finite case checks to the reader.

Lemma. The matroid $U_{2,4}$ is a stabilizer for the class of quaternary matroids.

Corollary [Kah]. A 3-connected, quaternary, non-binary matroid has a unique representation over $\textrm{GF}(4)$.

Proof. A non-binary matroid $M$ has a $U_{2,4}$-minor. The matroid $U_{2,4}$ has the following representation:

$$
\begin{bmatrix}
1 & 0 & 1 & 1\\
0 & 1 & 1 & a
\end{bmatrix}
$$
where $a \not\in \{0,1\}$. This leaves two choices for $a$, that are related through a field automorphism. Hence $U_{2,4}$ has (up to equivalence) a unique representation over $\textrm{GF}(4)$. But $U_{2,4}$ is a stabilizer, so $M$ is uniquely representable over $\textrm{GF}(4)$ as well. $\square$

Lemma. The matroids $U_{2,5}$ and $U_{3,5}$ are stabilizers for the class of quinary matroids.

Lemma. The matroid $U_{2,4}$ is a stabilizer for the class of quinary matroids with no minor isomorphic to $U_{2,5}$ and $U_{3,5}$.

Corollary [OVW]. A 3-connected, quinary matroid has at most six inequivalent representations over $\textrm{GF}(5)$.

Proof. $U_{2,5}$ and $U_{3,5}$ have six inequivalent representations and are stabilizers. If $M$ does not have such a minor, then either $M$ is regular (and thus uniquely representable over any field) or $M$ has a $U_{2,4}$-minor, which is has three inequivalent representations. $\square$

References

  • [BL] Tom Brylawski and Dean Lucas, Uniquely representable combinatorial geometries. In Teorie Combinatorie (proc. 1973 internat. colloq.) pp. 83-104 (1976).
  • [GGK] Jim Geelen, Bert Gerards, Ajai Kapoor, The excluded minors for $\textrm{GF}(4)$-representable matroids. J. Combin. Th. Ser. B, Vol. 79, pp. 247-299 (2000).
  • [Kah] Jeff Kahn, On the uniqueness of matroid representations over $\textrm{GF}(4)$. Bull. London Math. Soc. Vol. 20, pp. 5–10 (1988).
  • [OVW] James Oxley, Dirk Vertigan, Geoff Whittle, On inequivalent representations of matroids over finite fields.J. Combin. Theory Ser. B. Vol. 67, pp. 325–343 (1996).
  • [Oxley] James Oxley, Matroid Theory, 2nd edition. Oxford University Press (2011).
  • [Whi] Geoff Whittle, Stabilizers of classes of representable matroids. J. Combin. Theory Ser. B, Vol. 77, pp. 39–72 (1999).