  In a minimal binary constraint network, every tuple of a constraint relation
can be extended to a solution. The tractability or intractability of computing
a solution to such a minimal network was a long standing open question. Dechter
conjectured this computation problem to be NP-hard. We prove this conjecture.
We also prove a conjecture by Dechter and Pearl stating that for k\geq2 it is
NP-hard to decide whether a single constraint can be decomposed into an
equivalent k-ary constraint network. We show that this holds even in case of
bi-valued constraints where k\geq3, which proves another conjecture of Dechter
and Pearl. Finally, we establish the tractability frontier for this problem
with respect to the domain cardinality and the parameter k.
