Parameterized shifted combinatorial optimization

Shifted combinatorial optimization is a new nonlinear optimization framework broadly extending standard combinatorial optimization, involving the choice of several feasible solutions simultaneously. This framework captures well studied and diverse problems, from sharing and partitioning to so-called vulnerability problems. In particular, every standard combinatorial optimization problem has its shifted counterpart, typically harder. Already with explicitly given input set SCO may be NP -hard. Here we initiate a study of the parameterized complexity of this framework. First we show that SCO over an explicitly given set parameterized by its cardinality may be in XP, FPT or P, depending on the objective function. Second, we study SCO over sets definable in MSO logic (which includes, e.g., the well known MSO-partitioning problems). Our main results are that SCO over MSO definable sets is in XP parameterized by the MSO formula and treewidth (or clique-width) of the input graph, and W[1] -hard even under further severe restrictions.