Complexity of Poset-Induced Blocker-Set Instances

Determine the computational complexity of the structured minimum-hitting-set instances formed by the blocker sets induced by finite posets, where the blocker sets are defined by B_P(a,b) = (Pred_P(a) × Succ_P(b)) \setminus \preceq_P for ordered incomparable pairs (a,b).

Background

The paper characterizes the open-world teaching number of a finite poset P as the sum of the number of strict cover relations and the minimum size β(P) of a set of negative ordered-pair labels that intersects every blocker set B_P(a,b) associated with an ordered incomparable pair. Consequently, computing β(P) is a minimum hitting-set problem over a structured family of subsets determined by the poset's predecessor and successor relations.

Although minimum hitting set is NP-hard for arbitrary set systems, the paper does not establish whether the special blocker-set instances arising from finite posets are computationally tractable, NP-hard, or have some intermediate complexity. Determining this complexity would clarify the algorithmic difficulty of computing exact open-world teaching numbers.