  We study the complexity of local search for the Boolean constraint
satisfaction problem (CSP), in the following form: given a CSP instance, that
is, a collection of constraints, and a solution to it, the question is whether
there is a better (lighter, i.e., having strictly less Hamming weight) solution
within a given distance from the initial solution. We classify the complexity,
both classical and parameterized, of such problems by a Schaefer-style
dichotomy result, that is, with a restricted set of allowed types of
constraints. Our results show that there is a considerable amount of such
problems that are NP-hard, but fixed-parameter tractable when parameterized by
the distance.
