Intersecting Restrictions in Clutters

Abstract

A clutter is intersecting if the members do not have a common element yet every two members intersect. It has been conjectured that for clutters without an intersecting minor, total primal integrality and total dual integrality of the corresponding set covering linear system must be equivalent. In this paper, we provide a polynomial characterization of clutters without an intersecting minor. One important class of intersecting clutters comes from projective planes, namely the deltas, while another comes from graphs, namely the blockers of extended odd holes. Using similar techniques, we provide a polynomial algorithm for finding a delta or the blocker of an extended odd hole minor in a given clutter. This result is quite surprising as the same problem is NP-hard if the input were the blocker instead of the clutter.