Matroids

Abstract

One of the primary goals of pure mathematics is to identify common patterns that occur in disparate circumstances, and to create unifying abstractions which identify commonalities and provide a useful framework for further theorems. For example the pattern of an associative operation with inverses and an identity occurs frequently, and gives rise to the notion of an abstract group. On top of the basic axioms of a group, a vast theoretical framework can be built up, investigating the classification of groups, their internal structure, and the relationships and operations on groups. Matroids similarly provide a useful linking abstraction. They were first discovered independently by Hassler Whitney [10] and B. L. van der Waerden in the mid 1930’s [11]. Whitney had developed a notion of independence and rank in the context of graph theory, and noted similarities with the concepts of linear independence and dimension from linear algebra. By identifying the properties of abstract ‘independence’ which made these commonalities occur, he introduced the concept of a matroid, whose definition has proven immensely fruitful. Similarly, van der Waerden was interested in generalizing the notion of ‘independence’ from the examples of linear independence and algebraic independence. Shortly after the initial work by Whitney and van der Waerden, Birkhoff [2] noted that matroids were connected with a certain type of semimodular lattice that he had been studying. Thus, matroids provide a link between graph theory, linear algebra, transcendence theory, and semimodular lattices. Several decades later, Jack Edmonds noted the importance of matroids for the field of combinatorial optimization. This connection is due to two fundamental breakthroughs. First of all, Edmonds and a number of other researchers discovered a new type of matroid arising from the combinatorial theory of transversals. Second, Rado and Edmonds noted that matroids were intrinsically connected with the notion of a greedy algorithm (more historical details are in [11] and [3]). These developments have made matroids a mainstay of the field of combinatorial optimization.