Constructive algorithm for checking EP-pairs for R-minor-free targets

Develop a constructive algorithm that, given a minor antichain R and an input antichain Z, decides whether (excl(R), excl(Z)) is an Erdős–Pósa pair by explicitly constructing the finite parametric obstruction sets and witnesses (rather than relying on non-constructive existence), assuming knowledge of R.

Background

The authors define a decision procedure (CHECKING EP-PAIRS FOR R-MINOR FREE TARGET) that can be solved in almost-linear time via minor-checking calls when the finite obstruction sets are known. However, the presented approach is non-constructive, depending on existence of finite sets derived from their theory.

They highlight that turning this into a constructive algorithm—producing explicit obstructions and witnesses—remains open even when R is known.