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We resolve an open question by [3]: the exponential complexity of deciding whether a k-CNF has a solution is the same as that of deciding whether it has exactly one solution, both when it is promised and when it is not promised that the input formula has a solution. We also show that this has the same exponential complexity as deciding whether a given variable is backbone (i.e. forced to a particular value), given the promise that there is a solution. We show similar results for True Quantified Boolean Formulas in k-CNF, k-Hitting Set (and therefore Vertex Cover), k-Hypergraph Independent Set (and therefore Independent Set), Max-k-SAT, Min-k-SAT, and 0-1 Integer Programming with inequalities and k-wide constraints. [ABSTRACT FROM AUTHOR]