XP complexity of min-max-min MSO problems

Determine whether problems of the form \(\exists X (|X| \le k_x \land \forall Y (|Y| \le k_y \to \exists Z (|Z| \le k_z \land \phi(X,Y,Z))))\) belong to XP when parameterized by the clique-width of the input graph and the MSO formula \(\phi\).

Background

The paper establishes fixed-parameter tractable model-checking results for MSO extended with minimum- and maximum-value predicates on graphs of bounded clique-width and treewidth. It contrasts these results with the greater difficulty of MSO extensions involving cardinality comparisons.

In the conclusion, the authors compare the known XP status of two-level quantified problems of the form ∃X ∀Y\exists X\,\forall Y with the unresolved status of three-level ∃X ∀Y ∃Z\exists X\,\forall Y\,\exists Z problems. The open issue is whether the additional existential optimization layer can still be handled within XP under parameterization by clique-width and the formula, or whether such problems are harder. The question is presented as part of the broader complexity of min-max-min-type MSO problems.

References

Another interesting open question is the complexity of $\min$-$\max$-$\min$-type $\MSO$ problems. By the result of $\LinE\MSO$, we can compute $\exists X (|X| \le k \land \phi(X))$ in FPT time, and we also know that $\exists X (|X| \le k_x \land \forall Y (|Y| \le k_y \to \phi(X,Y)))$ type problems are XP, when parameterized by the clique-width of the input graph and an MSO formula $\phi$. However, to the best of our knowledge, it is not known whether problems of the form $\exists X (|X| \le k_x \land \forall Y (|Y| \le k_y \to \exists Z (|Z| \le k_z \land \phi(X,Y,Z))))$ are in XP or not when parameterized by the clique-width of the input graph and $\phi$.

— Quantifying over Optimal MSO-Definable Sets on Graphs of Bounded Clique-Width  (2608.20175 - Gima, 20 Aug 2026) in Section 1, Conclusion