With those three examples to guide us, we can now define the budget matroids.
We’re going to call them ‘matroids’ right from the start, even though we shan’t get around to proving that the matroid axioms are satisfied until Section 5.3.
Let b =b1+ · · · +bk be some partition of a nonnegative integer b, called the total budget, into k nonnegative parts, called the column budgets. For reasons that
4.2. THE DEFINITION 43 we discuss in Exercise 4.4-3 and Section 5.1, we require that at least two of the column budgets be positive, which implies that k ≥ 2. We are going to define a budget matroid associated with this partition, which we shall denote by Bb1,...,bk. A representation of the matroid Bb1,...,bkinvolves bk points in projective(b−1)-space, which we think of as organized into a b-by-k matrix:
Note that we have written the jthcolumn budget at the head of the jth column, as we did in the three examples above. The column budgets constrain the locations of the points in two ways:
• The b points in the jth column are constrained to lie in a common flat of dimension bj.
• For each of the b b
1...bk
possible ways of choosing one point from each row so that, for all j in [1. .k], precisely bj points are taken from the jth column, the b points so chosen are mutually incident — that is, rather than spanning the entire ambient(b−1)-space, they lie in a common hyperplane.
Let X be a subset of the matrix(Ei j)of points that contains precisely one point from each row and contains, for each j in [1 . .k], precisely bj points from the jthcolumn. We shall call such a set X perfect, on the grounds that it meets each column budget perfectly. Note that the multinomial coefficient b b
1...bk
counts the perfect sets. The second rule above requires that, in any representation of the bud-get matroid Bb1,...,bk, every perfect set of points must be mutually incident. Hence, in the budget matroid Bb1,...,bk itself, every perfect sets of elements is going to be dependent.
To define the matroid Bb1,...,bk, we need a set of elements and a rule for inde-pendence. The ground set of the matroid Bb1,...,bk is the set of entries of a b-by-k matrix
which we shall refer to as the ground matrix. We define a subset X of the ground matrix e to be perfect when X contains precisely one element from each row and, for each j , contains precisely bj elements from the jthcolumn. We define a subset X of e to be independent when it satisfies the following three rules:
44 CHAPTER 4. THE BUDGET MATROIDS Ambient Rule X contains at most b elements overall;
Column Rule X contains at most bj +1 of the b elements in the jth column, for each j in [1. .k]; and
Perfect Rule X has no perfect subsets.
What is the effect of these three rules?
When talking about the Pappus configuration B1,1,1 and the complete quadri-lateral B2,1, we implicitly assumed that all of the points involved lay in a common plane. In the general case, we want all of the points in any representation of the budget matroid Bb1,...,bk to lie in an ambient projective space of dimension b−1 and hence of rank b. The Ambient Rule guarantees this by making any set with more than b elements dependent.
Furthermore, we want the b points in the jth column to lie in a common flat of dimension bj and hence of rank bj +1. The Column Rule guarantees this in an analogous way.
As for the Perfect Rule, if a set X has a proper subset that is perfect, then X must contain more than b elements, so X also violates the Ambient Rule. Thus, in the presence of Ambient Rule, the effect of the Perfect Rule is just to make the perfect sets themselves dependent, as we intended to do.
The special case of a zero column budget deserves comment. For one thing, if bj =0, then the Column Rule forces all of the elements of the jthcolumn to lie in a common 0-flat — that is, to coincide, say at some point Zj. But more importantly, none of the elements of the jthcolumn belong to any perfect set. Hence, the point Zj does not participate in any incidences that involve any other points; instead, it lies in the ambient space in general position with respect to the other points of the representation. Exercise 4.2-2 shows that zero column budgets are not of much geometric interest.
Exercise 4.2-1 Construct a representation of the budget matroid B1,1. [Answer: Map the four elements of the ground matrix to the points
1 1
P Q
Q P
,
where P and Q are any two distinct points along a line.]
Exercise 4.2-2 Construct a representation of the budget matroid B2,1,0,0,...,0in the projective plane, where there are, say, a thousand columns whose budgets are zero.
Note that you wouldn’t succeed if the projective plane were built over a small finite field.
4.2. THE DEFINITION 45 [Answer: Map the six elements
e11 e12
e21 e22 e31 e32
of the first two columns of the ground matrix to the six vertices
2 1
P1 A1
P2 A2 P3 A3
of some complete quadrilateral. For j >2, map all three elements e1 j, e2 j, and e3 j of the jthcolumn to a common point Zj in the plane, where the points Z3through Z1002are chosen so that
• none of them lies on the line determined by any two vertices of the quadrilat-eral (neither on a line of the quadrilatquadrilat-eral itself nor on one of its diagonals),
• none of the lines determined by any two of them passes through any vertex of the quadrilateral, and
• no three of them are collinear.]
Exercise 4.2-3 Recall that the rank of a matroid is the common cardinality of all of its maximal independent sets. Show that the rank of the budget matroid Bb1,...,bk is the total budget b :=b1+ · · · +bk.
[Hint: The Ambient Rule guarantees that the rank is at most b. To show that it is at least b, it suffices to construct some independent set with b elements. One choice is to take the top bj elements from the jthcolumn, for all j in [1. .k]. Note that the resulting set is not perfect, because we have required that at least two of the column budgets be positive.]
Exercise 4.2-4 Recall that a circuit in a matroid is a minimal dependent set. Show that the circuits of the budget matroid Bb1,...,bk are precisely
perfect circuits the perfect sets,
column circuits the subsets — if any — of the jth column of size bj +2, and ambient circuits those sets of size b+1 that do not include any perfect circuit or
any column circuit as a subset.
Exercise 4.2-5 A matroid is called simple when all of its dependent sets have size at least 3; that is, there are no circuits of size 1 or size 2 — no loops and no parallel pairs. Which budget matroids are simple?
[Answer: No budget matroid has any loops. A budget matroid has parallel pairs either when its total budget is 2 or when some column budget is 0.]
46 CHAPTER 4. THE BUDGET MATROIDS