CSP DICHOTOMY FOR SPECIAL POLYADS

Abstract

For a digraph ℍ, the Constraint Satisfaction Problem with template ℍ, or CSP(ℍ), is the problem of deciding whether a given input digraph 𝔾 admits a homomorphism to ℍ. The CSP dichotomy conjecture of Feder and Vardi states that for any digraph ℍ, CSP(ℍ) is either in P or NP-complete. Barto, Kozik, Maróti and Niven (Proc. Amer. Math. Soc.137 (2009) 2921–2934) confirmed the conjecture for a class of oriented trees called special triads. We generalize this result, establishing the dichotomy for a class of oriented trees which we call special polyads. We prove that every tractable special polyad has bounded width and provide the description of special polyads of width 1. We also construct a tractable special polyad which neither has width 1 nor admits any near-unanimity polymorphism.