Relaxations of the Maximum Flow Minimum Cut Property for Ideal Clutters

Given a family of sets, a covering problem consists of finding a minimum cost collection of elements that hits every set. This objective can always be bound by the maximum number of disjoint sets in the family, we refer to this as the covering dual, since when we allow covers to be fractional and relax the notion of disjoint sets, the natural Linear Programming (LP) formulations become duals and the optimal objective values of the two LPs match. A consequence of the Edmonds-Giles theorem about Totally Dual Integral systems is that if we assume the covering dual always has an optimal integer solution for every cost function, then we can always find an optimal integer cover. The converse does not hold in general, but a still standing conjecture from the mid-1970s states that the existence of an optimal integer cover for every cost function implies the existence of a 1/4-integer optimal solution to the dual for every cost function. In this thesis we discuss weaker versions of the conjecture and build tools allowing us to approach them