Minimal implying subset preserving (total) implied integrality

Determine, for a given rational polyhedron P ⊆ R^N and index sets S,T ⊆ N such that x_T is (totally) implied integer by x_S for P, the minimum cardinality of a subset S′ ⊆ S for which x_T remains (totally) implied integer by x_{S′} for P.

Background

The paper introduces implied integrality and total implied integrality and develops tools to deduce when integrality of some variables enforces integrality of others. In applications, minimizing the number of variables that must be treated as integral is valuable for presolve and decomposition.

This question seeks to quantify the redundancy within an implying set S by asking for the smallest subset S′ ⊆ S that still suffices to imply integrality of x_T, for both the plain and total versions of implied integrality.