[Paper Review] Hybrid (V)CSPs and algebraic reductions.

Rustem Takhanov|arXiv (Cornell University)|Jun 22, 2015

Advanced Graph Theory Research26 references3 citations

TL;DR

This paper introduces a generalized lifted language construction, $\bGamma^{\mathfrak{B}}$, indexed by finite algebras $\mathfrak{B}$, to reduce fixed-template CSPs to simpler instances. It proves polynomial-time Turing reducibility of $\textsc{CSP}(\bGamma)$ to $\textsc{CSP}(\bGamma^{\mathfrak{B}})$ under natural conditions, linking polymorphisms and offering a new tool for tackling the Feder-Vardi dichotomy conjecture.

ABSTRACT

Constraint Satisfaction Problem (CSP) can be stated as computing a homomorphism $\mbox{$\bR ightarrow \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 $\mathfrak{B}$, $ extsc{CSP}(\bGamma)$ is polynomial-time Turing reducible to $ extsc{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.

Motivation & Objective

• To investigate when the tractability of lifted languages $\bGamma_{\bR}$ is both necessary and sufficient for hybrid CSP tractability under structural restrictions.
• To generalize the construction of a finite structure $\bGamma'$ to a broader class of structures $\bGamma^{\mathfrak{B}}$ indexed by finite algebras $\mathfrak{B}$.
• To establish a polynomial-time Turing reduction from $\textsc{CSP}(\bGamma)$ to $\textsc{CSP}(\bGamma^{\mathfrak{B}})$ under natural conditions on $\mathfrak{B}$.
• To show that polymorphisms of $\bGamma$ have analogs in $\pol(\bGamma^{\mathfrak{B}})$, preserving structural and algebraic properties.
• To propose $\bGamma^{\mathfrak{B}}$ as a novel algorithmic tool for advancing the Feder-Vardi dichotomy conjecture in CSP theory.

Proposed method

• Construct a multi-sorted lifted language $\bGamma_{\bR}$ from an input structure $\bR$ in a structural restriction $\mathcal{H}$ closed under inverse homomorphisms.
• Introduce a finite relational structure $\bGamma^{\mathfrak{B}}$ indexed by a set of finite algebras $\mathfrak{B}$, generalizing the earlier $\bGamma'$ construction.
• Establish a polynomial-time Turing reduction from $\textsc{CSP}(\bGamma)$ to $\textsc{CSP}(\bGamma^{\mathfrak{B}})$ under natural conditions on $\mathfrak{B}$, such as closure under certain term operations.
• Demonstrate that polymorphisms of $\bGamma$ correspond to polymorphisms in $\pol(\bGamma^{\mathfrak{B}})$, preserving algebraic invariants.
• Use the lifted structure $\bGamma^{\mathfrak{B}}$ to simulate and reduce complex CSP instances to simpler ones, enabling algorithmic exploration.
• Leverage the Bulatov-Jeavons-Krokhin characterization to link structural restrictions $\mathcal{H}$ to homomorphic mappings into a fixed $\bGamma'$, under the assumption of the characterization's correctness.

Experimental results

Research questions

• RQ1 Under what conditions is the tractability of lifted languages $\bGamma_{\bR}$ both necessary and sufficient for the tractability of hybrid CSPs?
• RQ2 Can the construction of $\bGamma'$ be generalized beyond finite relational structures to include algebraic indexing via finite algebras $\mathfrak{B}$?
• RQ3 Is $\textsc{CSP}(\bGamma)$ polynomial-time Turing reducible to $\textsc{CSP}(\bGamma^{\mathfrak{B}})$ for appropriately chosen $\mathfrak{B}$?
• RQ4 Do polymorphisms of $\bGamma$ have identifiable analogs in $\pol(\bGamma^{\mathfrak{B}})$, and how does this preserve algebraic structure?
• RQ5 Can the $\bGamma^{\mathfrak{B}}$ construction serve as a viable pathway toward resolving the Feder-Vardi dichotomy conjecture?

Key findings

• The tractability of a structural restriction $\mathcal{H}$ in the hybrid CSP setting is equivalent to all structures in $\mathcal{H}$ being homomorphically mappable to a fixed finite relational structure $\bGamma'$, assuming the Bulatov-Jeavons-Krokhin characterization is correct.
• The construction of $\bGamma^{\mathfrak{B}}$ generalizes $\bGamma'$ by indexing it over a set of finite algebras $\mathfrak{B}$, enabling broader algebraic manipulation.
• Under natural conditions on $\mathfrak{B}$, $\textsc{CSP}(\bGamma)$ is polynomial-time Turing reducible to $\textsc{CSP}(\bGamma^{\mathfrak{B}})$, enabling algorithmic reduction of complex CSPs.
• Polymorphisms of $\bGamma$ have corresponding analogs in $\pol(\bGamma^{\mathfrak{B}})$, preserving essential algebraic properties across the reduction.
• The $\bGamma^{\mathfrak{B}}$ construction provides a new framework for exploring fixed-template CSPs and may serve as a tool to approach the Feder-Vardi dichotomy conjecture.
• The lifted language $\bGamma_{\bR}$ remains a necessary condition for hybrid CSP tractability, and the paper identifies conditions under which it is also sufficient.