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Is there any other standard FP^NP-complete problem other than the Traveling Salesman Problem? For instance, in the canonical propositional logic?

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TSP is actually OptP-complete and so are many other optimization problems, including Knapsack, MAX-SAT, 0-1 Integer Linear Programming. This class and proof technique is largely the work of Mark Krentel; orig paper STC'86. The journal version of the paper has more details, including the fact that OptP is subclass of FPNP.

LEX-MIN-SAT, finding the lexicographically smallest satisfying assignment, is OptP-complete under metric reductions. There's also an analogue of Schaefer's dichotomy theorem for LEX-MIN-SAT proved by Reith and Vollmer http://arxiv.org/pdf/cs/9809116.pdf

I think is almost a research-level, or at least graduate-level question, by the way.

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    $\begingroup$ Note that research level questions are fine on CS.SE as well. $\endgroup$ – Juho Feb 9 '15 at 14:05

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