  The natural generalization of the Boolean satisfiability problem to
optimization problems is the task of determining the maximum number of clauses
that can simultaneously be satisfied in a propositional formula in conjunctive
normal form. In the weighted maximum satisfiability problem each clause has a
positive weight and one seeks an assignment of maximum weight. The literature
almost solely considers the case of positive weights. While the general case of
the problem is only restricted slightly by this constraint, many special cases
become trivial in the absence of negative weights. In this work we study the
problem with negative weights and observe that the problem becomes
computationally harder - which we formalize from a parameterized perspective in
the sense that various variations of the problem become W[1]-hard if negative
weights are present.
  Allowing negative weights also introduces new variants of the problem:
Instead of maximizing the sum of weights of satisfied clauses, we can maximize
the absolute value of that sum. This turns out to be surprisingly expressive
even restricted to monotone formulas in disjunctive normal form with at most
two literals per clause. In contrast to the versions without the absolute
value, however, we prove that these variants are fixed-parameter tractable. As
technical contribution we present a kernelization for an auxiliary problem on
hypergraphs in which we seek, given an edge-weighted hypergraph, an induced
subgraph that maximizes the absolute value of the sum of edge-weights.
