On the hardness of losing weight

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.