Collapse of the Polynomial Hierarchy

Determine whether the polynomial hierarchy collapses; that is, ascertain whether there exists a finite level k such that PH equals Σ_k^p (and hence Σ_i^p = Σ_k^p for all i ≥ k), or whether the hierarchy is strictly infinite.

Background

The thesis discusses min-max optimization problems and notes many are complete for higher levels of the polynomial hierarchy, such as Σ2^p and Σ3^p, via the SSP framework. Their hardness relative to NP hinges on the broader structural question of whether the polynomial hierarchy (PH) collapses.

If PH does not collapse below Σ2^p, then the Σ2^p-complete problems identified (e.g., certain network interdiction and min-max regret variants) are strictly harder than NP-complete problems. Thus, settling the collapse question directly impacts how these problems are positioned in the landscape of computational complexity.