Uniqueness in quadratic and hyperbolic 0-1 programming problems

We analyze the question of deciding whether a quadratic or a hyperbolic 0-1 programming instance has a unique optimal solution. Both uniqueness questions are known to be NP-hard, but are unlikely to be contained in the class NP. We precisely pinpoint their computational complexity by showing that they both are complete for the complexity class {\mbox{${\mathrm{\Delta }}_2$P}.