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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=2 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=3, 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.