ZCnk

ZCnk is a term used in theoretical discussions of constraint networks and combinatorial optimization to denote a class of synthetic, locally constrained systems. In this article, ZCnk is treated as a hypothetical construct: an n-node network in which each node assumes a discrete state and is influenced only by interactions with up to k neighboring nodes. The "zero-cost" descriptor refers to the absence of a single, global objective function embedded in the local constraints, rather than to the computational cost of solving instances.

Formally, a ZCnk instance consists of a graph G = (V,E) with |V| = n, a finite state domain D, and a family of local constraint relations {R_v ⊆ D^{N(v)}} defined for each vertex v ∈ V, where N(v) is the neighborhood of v with |N(v)| ≤ k. An assignment a: V → D is feasible if for every v, a|_{N(v)} ∈ R_v. There is no objective value assigned to configurations within ZCnk; instead, research focuses on feasibility, solution count, and structural properties of the constraint network.

Purpose and use: ZCnk is used as a stylized model to study algorithmic behavior and phase transitions in constraint satisfaction problems, to benchmark local-search and belief-propagation methods, and to explore how network topology and local constraint width (k) influence tractability. It serves as a sandbox for comparing heuristics and for illustrating theoretical results about constraint density and solvability.

Origins and status: The term ZCnk does not represent a standard, widely adopted notion. It appears in informal discussions and tutorials as a compact label for a class of locally constrained, zero-objective networks. As such, interpretations of ZCnk may vary across communities, and there is no unified formalism established in the literature.