Dmitriy Zhuk

Dmitriy Zhuk, Barnaby Martin, Michal Wrona

We prove that QCSP$(\mathbb{N};x=y\rightarrow y=z)$ is PSpace-complete, settling a question open for more than ten years. This completes the complexity classification for the QCSP over equality languages as a trichotomy between Logspace, NP-complete and PSpace-complete. We additionally settle the classification for bounded alternation QCSP$(Γ)$, for $Γ$ an equality language. Such problems are either in Logspace, NP-complete, co-NP-complete or rise in complexity in the Polynomial Hierarchy.

Dmitriy Zhuk

A natural strengthening of an algorithm for the (promise) constraint satisfaction problem is its singleton version: we first fix a constraint to a tuple from the constraint relation, then run the algorithm, and remove the tuple from the constraint if the answer is negative. We characterize the power of the singleton versions of standard universal algorithms for the (promise) CSP over a fixed template in terms of the existence of a minion homomorphism. Using the Hales-Jewett theorem, we show that for finite relational structures this minion condition is equivalent to the existence of polymorphisms with certain symmetries, which we call palette symmetric polymorphisms. By proving the existence of such polymorphisms we establish that the singleton version of the BLP+AIP algorithm solves all (multi-sorted) tractable CSPs over domains of size at most 7. We further show that already for domain size 8 there exists a relational structure arising from the dihedral group $D_4$ that does not admit palette symmetric polymorphisms and cannot be solved by singleton BLP+AIP. By providing concrete CSP templates, we illustrate the limitations of linear programming, the power of the singleton versions, and the elegance of palette symmetric polymorphisms. Surprisingly, we also identify a concrete temporal relational structure whose CSP can be solved by a singleton algorithm after sampling, yet it does not admit palette symmetric polymorphisms. This highlights that the Hales-Jewett argument applies only to finite structures. Nevertheless, we introduce generalized palette polymorphisms for every tractable temporal relational structure and show that they yield a rounding procedure implying that the corresponding singleton algorithm solves its CSP after sampling.

Libor Barto, Maximilian Hadek, Dmitriy Zhuk

We develop a unified framework to characterize the power of higher-level algorithms for the constraint satisfaction problem (CSP), such as $k$-consistency, the Sherali-Adams LP hierarchy, and the affine IP hierarchy. As a result, solvability of a fixed-template CSP or, more generally, a Promise CSP by a given level is shown to depend only on the polymorphism minion of the template. Similarly, we obtain a minion-theoretic description of $k$-consistency reductions between Promise CSPs. We introduce a new hierarchy of SDP-like vector relaxations with vectors over $\mathbb Z_{p}$ in which orthogonality is imposed on $k$-tuples of vectors. Surprisingly, this relaxation turns out to be equivalent to the $k$-th level of the AIP-$\mathbb{Z}_p$ relaxation. We show that it solves the CSP of the dihedral group $\mathbf{D}_4$, the smallest CSP that fools the singleton BLP+AIP algorithm. Using this vector representation, we further show that the $p$-th level of the $\mathbb{Z}_p$ relaxation solves linear equations modulo $p^2$.