  In the recent literature of Artificial Intelligence, an intensive research
effort has been spent, for various algebras of qualitative relations used in
the representation of temporal and spatial knowledge, on the problem of
classifying the computational complexity of reasoning problems for subsets of
algebras. The main purpose of these researches is to describe a restricted set
of maximal tractable subalgebras, ideally in an exhaustive fashion with respect
to the hosting algebras. In this paper we introduce a novel algebra for
reasoning about Spatial Congruence, show that the satisfiability problem in the
spatial algebra MC-4 is NP-complete, and present a complete classification of
tractability in the algebra, based on the individuation of three maximal
tractable subclasses, one containing the basic relations. The three algebras
are formed by 14, 10 and 9 relations out of 16 which form the full algebra.
