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QCSP(K3) on Bounded Treedepth

Determine whether the Quantified Constraint Satisfaction Problem over the 3-clique, QCSP(K3), is polynomial-time solvable on instances whose primal graphs of the quantifier-free part have bounded treedepth.

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Background

The authors show that QCSP(K3) is Pspace-complete on bounded pathwidth classes, but they observe a divergence for certain problems that become tractable on bounded treedepth while remaining hard on bounded pathwidth. They ask whether QCSP(K3) exhibits this behavior.

Establishing the complexity of QCSP(K3) on bounded treedepth would be a major step toward understanding quantified coloring and related games under strong structural constraints, and the authors identify it as the principal open problem arising from their work.

References

Whether $QCSP(K_3)$ is another such example is an open question. That is, we do not know if $QCSP(K_3)$ is polynomial-time for graph classes of bounded treedepth. This is the major open problem arising from our work.

— Graph Homomorphism, Monotone Classes and Bounded Pathwidth (2403.00497 - Eagling-Vose et al., 1 Mar 2024) in Conclusions (Section 8)