Many combinatorial search problems can be expressed as ''constraint sa
tisfaction problems'' and this class of problems is known to be NP-com
plete in general. In this paper, we investigate the subclasses that ar
ise from restricting the possible constraint types. We first show that
any set of constraints that does not give rise to an NP-complete clas
s of problems must satisfy a certain type of algebraic closure conditi
on. We then investigate all the different possible forms of this algeb
raic closure property, and establish which of these are sufficient to
ensure tractability. As examples, we show that all known classes of tr
actable constraints over finite domains can be characterized by such a
n algebraic closure property. Finally, we describe a simple computatio
nal procedure that can be used to determine the closure properties of
a given set of constraints. This procedure involves solving a particul
ar constraint satisfaction problem, which we call an ''indicator probl
em.''