Sensitive instances of the Constraint Satisfaction Problem

We investigate the impact of modifying the constraining relations of a Constraint Satisfaction Problem (CSP) instance, with a fixed template, on the set of solutions of the instance. More precisely we investigate sensitive instances: an instance of the CSP is called sensitive, if removing any tuple from any constraining relation invalidates some solution of the instance. Equivalently, one could require that every tuple from any one of its constraints extends to a solution of the instance. Clearly, any non-trivial template has instances which are not sensitive. Therefore we follow the direction proposed (in the context of strict width) by Feder and Vardi (SICOMP 1999) and require that only the instances produced by a local consistency checking algorithm are sensitive. In the language of the algebraic approach to the CSP we show that a finite idempotent algebra $\mathbf{A}$ has a $k+2$ variable near unanimity term operation if and only if any instance that results from running the $(k, k+1)$-consistency algorithm on an instance over $\mathbf{A}^2$ is sensitive. A version of our result, without idempotency but with the sensitivity condition holding in a variety of algebras, settles a question posed by G. Bergman about systems of projections of algebras that arise from some subalgebra of a finite product of algebras. Our results hold for infinite (albeit in the case of $\mathbf{A}$ idempotent) algebras as well and exhibit a surprising similarity to the strict width $k$ condition proposed by Feder and Vardi. Both conditions can be characterized by the existence of a near unanimity operation, but the arities of the operations differ by 1.