Transformation and decomposition of clutters into matroids

A clutter is a family of mutually incomparable sets. The set of circuits of a matroid, its set of bases, and its set of hyperplanes are examples of clutters arising from matroids. In this paper we address the question of determining which are the matroidal clutters that best approximate an arbitrary clutter Λ. For this, we first define two orders under which to compare clutters, which give a total of four possibilities for approximating Λ (i.e., above or below with respect to each order); in fact, we actually consider the problem of approximating Λ with clutters from any collection of clutters Σ, not necessarily arising from matroids. We show that, under some mild conditions, there is a finite non-empty set of clutters from Σ that are the closest to Λ and, moreover, that Λ is uniquely determined by them, in the sense that it can be recovered using a suitable clutter operation. We then particularize these results to the case where Σ is a collection of matroidal clutters and give algorithmic procedures to compute these clutters.