Hybrid (V)CSPs and algebraic reductions.
Rustem Takhanov · arXiv (Cornell University) · 2015
Constraint Satisfaction Problem (CSP) can be stated as computing a homomorphism $\mbox{$\bR \rightarrow \bGamma$}$ between two relational structures, e.g. between two directed graphs. Recently, the {\em hybrid} setting, where both sides are restricted simultaneously, attracted some attention. It assumes that the right side structure $\bGamma$ is fixed and $\bR$ belongs to a class of relational structures $\mathcal{H}$ (called a {\em structural restriction}) that is, additionally, {\em closed under inverse homomorphisms}. The key tool that connects hybrid CSPs with fixed-template CSPs is a construction called a lifted language, namely a multi-sorted language $\bGamma_{\bR}$ that can be constructed from an input $\bR$. The tractability of a language $\bGamma_{\bR}$ for any input $\bR\in\mathcal{H}$ is a necessary condition for tractability of the hybrid problem. First we investigate the case when the last property is not only necessary, but also is sufficient. It turns out that in the latter case, if Bulatov-Jeavons-Krokhin characterization of tractable constraint languages is correct, a structural restriction $\mathcal{H}$ is tractable if and only if it consists of structures that can be homomorphically mapped to some fixed finite relational structure $\bGamma'$ (that depends only on $\bGamma$). In the second part we generalize the construction of $\bGamma'$ and introduce a finite structure $\bGamma^{\mathfrak{B}}$, indexed by some set of finite algebras $\mathfrak{B}$. We prove that under some natural conditions on $\mathfrak{B}$, $\textsc{CSP}(\bGamma)$ is polynomial-time Turing reducible to $\textsc{CSP}(\bGamma^{\mathfrak{B}})$ and some polymorphisms of $\bGamma$ have analogs in $\pol(\bGamma^{\mathfrak{B}})$. This construction introduce a new set of algorithms for fixed-template CSPs and we suggest it as a tool to approach Feder-Vardi dichotomy conjecture.