On combinatorial properties of binary spaces

Abstract

A binary clutter is the family of inclusionwise minimal supports of vectors of affine spaces over GF(2). Binary clutters generalize various objects studied in Combinatorial Optimization, such as paths, Chinese Postman Tours, multiflows and one-sided circuits on surfaces. The present work establishes connections among three matroids associated with binary clutters, and between any of them and the binary clutter. These connections are then used to compare well-known classes of binary clutters; to provide polynomial algorithms which either confrom the membership in subclasses, or provide a forbidden clutter-minor; to reformulate and generalize a celebrated conjecture of Seymour on ideal binary clutters in terms of multiflows in matroids, and to exhibit new cases of its validity.