Characterizing matchings as the intersection of matroids

Abstract

This paper deals with the problem of representing the matching independence system in a graph as the intersection of finitely many matroids. After characterizing the graphs for which the matching independence system is the intersection of two matroids, we study the function µ(G), which is the minimum number of matroids that need to be intersected in order to obtain the set of matchings on a graph G, and examine the maximal value, µ(n), for graphs with n vertices. We describe an integer programming formulation for deciding whether µ(G) ≤ k. Using combinatorial arguments, we prove that µ(n) ∈ Ω(log log n). On the other hand, we establish that µ(n) ∈ O(log n/ log log n). Finally, we prove that µ(n) = 4 for n = 5, . . . , 12, and sketch a proof of µ(n)=5 for n = 13, 14, 15.

Matroid representation of clique complexes

“…Formerly, Fekete et al [9] investigated the same problem for matching complexes, and they characterized a graph whose matching complex is the intersection of k matroids, for every natural number k. Since the matching complexes form a subclass of the class of clique complexes, we can observe that some of their results can be derived from our theorems as corollaries.…”

“…In Section 6, we study a graph itself as an independence system and relate it to our theorem. In Section 7, we deduce some results by Fekete et al [9] from our theorems. We conclude with Section 8.…”

“…Lemma 2.2 (see Faigle [8], Fekete et al [9] and Korte and Vygen [18]). Every independence system can be represented as the intersection of a finite number of matroids on the same ground set.…”

“…Note that Fekete et al [9] gave a characterization of the graphs whose matching complexes can be represented as the intersections of two matroids. So the theorem in this section is a generalization of their result.…”

Persistency and matroid intersection

“…However, the number of intersecting matroids forming this IS was not known until recently. In Fekete et al 2003, it is shown that the minimum number of matroids that need to be intersected in order to obtain a set of matchings in a graph G is of at most O(log|V |/loglog|V |). Combining this with our result implies that the PPP on matching is polynomially solvable.…”