Quasipolynomial-time approximation scheme for MIS on planar-induced-minor-free graphs

Establish that, for every planar graph H, Max Independent Set admits a quasipolynomial-time approximation scheme on graphs excluding H as an induced minor.

Background

The paper proves that Max Independent Set is NP-hard on graphs excluding the 5 × 5 grid as an induced minor, thereby refuting polynomial- and quasipolynomial-time exact-solving conjectures for this class of graph restrictions. It then turns to approximation algorithms, where the corresponding approximation conjecture is explicitly stated as not being refuted by the paper.

The conjecture would require a quasipolynomial-time approximation scheme for Max Independent Set for every fixed planar excluded induced minor H. The paper notes that this conjecture follows from a structural conjecture of Gartland and Lokshtanov, so progress on the structural statement could resolve the approximation problem.

References

The following conjecture is not refuted by the present work.

\begin{conjecture}\label{conj:qptas} For every planar graph~$H$, Max Independent Set admits a~quasipolynomial-time approximation scheme (QPTAS) on graphs excluding $H$ as an induced minor. \end{conjecture}

— Max Independent Set Remains NP-hard when Excluding a Planar Induced Minor  (2609.11285 - Bonnet et al., 10 Sep 2026) in Conjecture 1, Section 1, Introduction (subparagraph “Future work”)

A~more adventurous (but still open) conjecture is the strengthening of~\cref{conj:qptas} to a~polynomial-time approximation scheme (PTAS).

— Max Independent Set Remains NP-hard when Excluding a Planar Induced Minor  (2609.11285 - Bonnet et al., 10 Sep 2026) in Final paragraph of Section 1, Introduction, immediately before Section 2