Partition matroid

In mathematics, a partition matroid or partitional matroid is a matroid that is a direct sum of uniform matroids.^[1] It is defined over a base set in which the elements are partitioned into different categories. For each category, there is a capacity constraint - a maximum number of allowed elements from this category. The independent sets of a partition matroid are exactly the sets in which, for each category, the number of elements from this category is at most the category capacity.

Let ${\displaystyle C_{i}}$ be a collection of disjoint sets ("categories"). Let ${\displaystyle d_{i}}$ be integers with ${\displaystyle 0\leq d_{i}\leq |C_{i}|}$ ("capacities"). Define a subset ${\displaystyle I\subset \bigcup _{i}C_{i}}$ to be "independent" when, for every index ${\displaystyle i}$, ${\displaystyle |I\cap C_{i}|\leq d_{i}}$. The sets satisfying this condition form the independent sets of a matroid, called a partition matroid.

The sets ${\displaystyle C_{i}}$ are called the categories or the blocks of the partition matroid.

A basis of the partition matroid is a set whose intersection with every block ${\displaystyle C_{i}}$ has size exactly ${\displaystyle d_{i}}$. A circuit of the matroid is a subset of a single block ${\displaystyle C_{i}}$ with size exactly ${\displaystyle d_{i}+1}$. The rank of the matroid is ${\displaystyle \sum d_{i}}$.^[2]

Every uniform matroid ${\displaystyle U{}_{n}^{r}}$ is a partition matroid, with a single block ${\displaystyle C_{1}}$ of ${\displaystyle n}$ elements and with ${\displaystyle d_{1}=r}$. Every partition matroid is the direct sum of a collection of uniform matroids, one for each of its blocks.

In some publications, the notion of a partition matroid is defined more restrictively, with every ${\displaystyle d_{i}=1}$. The partitions that obey this more restrictive definition are the transversal matroids of the family of disjoint sets given by their blocks.^[3]

As with the uniform matroids they are formed from, the dual matroid of a partition matroid is also a partition matroid, and every minor of a partition matroid is also a partition matroid. Direct sums of partition matroids are partition matroids as well.

A maximum matching in a graph is a set of edges that is as large as possible subject to the condition that no two edges share an endpoint. In a bipartite graph with bipartition ${\displaystyle (U,V)}$, the sets of edges satisfying the condition that no two edges share an endpoint in ${\displaystyle U}$ are the independent sets of a partition matroid with one block per vertex in ${\displaystyle U}$ and with each of the numbers ${\displaystyle d_{i}}$ equal to one. The sets of edges satisfying the condition that no two edges share an endpoint in ${\displaystyle V}$ are the independent sets of a second partition matroid. Therefore, the bipartite maximum matching problem can be represented as a matroid intersection of these two matroids.^[4]

More generally the matchings of a graph may be represented as an intersection of two matroids if and only if every odd cycle in the graph is a triangle containing two or more degree-two vertices.^[5]

A clique complex is a family of sets of vertices of a graph ${\displaystyle G}$ that induce complete subgraphs of ${\displaystyle G}$. A clique complex forms a matroid if and only if ${\displaystyle G}$ is a complete multipartite graph, and in this case the resulting matroid is a partition matroid. The clique complexes are exactly the set systems that can be formed as intersections of families of partition matroids for which every ${\displaystyle d_{i}=1}$.^[6]

The number of distinct partition matroids that can be defined over a set of ${\displaystyle n}$ labeled elements, for ${\displaystyle n=0,1,2,\dots }$, is

1, 2, 5, 16, 62, 276, 1377, 7596, 45789, 298626, 2090910, ... (sequence A005387 in the OEIS).

The exponential generating function of this sequence is ${\displaystyle f(x)=\exp(e^{x}(x-1)+2x+1)}$.^[7]