k-SAT Is No Harder Than Decision-Unique-k-SAT

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.