  We investigate the complexity of three optimization problems in Boolean
propositional logic related to information theory: Given a conjunctive formula
over a set of relations, find a satisfying assignment with minimal Hamming
distance to a given assignment that satisfies the formula
($\mathsf{NeareastOtherSolution}$, $\mathsf{NOSol}$) or that does not need to
satisfy it ($\mathsf{NearestSolution}$, $\mathsf{NSol}$). The third problem
asks for two satisfying assignments with a minimal Hamming distance among all
such assignments ($\mathsf{MinSolutionDistance}$, $\mathsf{MSD}$).
  For all three problems we give complete classifications with respect to the
relations admitted in the formula. We give polynomial time algorithms for
several classes of constraint languages. For all other cases we prove hardness
or completeness regarding APX, APX, NPO, or equivalence to well-known hard
optimization problems.
