On the complexity of $\mathbb H$-coloring for special oriented trees

TL;DR

This paper proves the CSP dichotomy for a class of special oriented trees, showing each tractable case has bounded width and can be solved by local consistency methods, using algebraic tools.

Contribution

It confirms the CSP dichotomy for special oriented trees and demonstrates that all tractable cases have bounded width, solvable by local consistency checking.

Findings

Confirmed CSP dichotomy for special oriented trees

All tractable special trees have bounded width

Tractability is characterized via algebraic tools

Abstract

For a fixed digraph $\mathbb H$, the $\mathbb H$-coloring problem is the problem of deciding whether a given input digraph $\mathbb G$ admits a homomorphism to $\mathbb H$. The CSP dichotomy conjecture of Feder and Vardi is equivalent to proving that, for any $\mathbb H$, the $\mathbb H$-coloring problem is in in P or NP-complete. We confirm this dichotomy for a certain class of oriented trees, which we call special trees (generalizing earlier results on special triads and polyads). Moreover, we prove that every tractable special oriented tree has bounded width, i.e., the corresponding $\mathbb H$-coloring problem is solvable by local consistency checking. Our proof relies on recent algebraic tools, namely characterization of congruence meet-semidistributivity via pointing operations and absorption theory.