Parameterized Shifted Combinatorial Optimization

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

💡 Research Summary

The paper initiates a systematic study of the parameterized complexity of Shifted Combinatorial Optimization (SCO), a nonlinear extension of standard combinatorial optimization where one must select r feasible solutions simultaneously and then evaluate a linear objective after “shifting” the resulting matrix so that each row is non‑increasing. The authors consider two main settings.

First, they assume the feasible set S ⊆{0,1}ⁿ is given explicitly as a list S={s₁,…,s_m}. The parameter is m=|S|. By counting how many columns of a solution matrix equal each s_k, any feasible matrix can be represented by a vector (r₁,…,r_m) with ∑r_k=r. The original problem reduces to a nonlinear integer program over the simplex (4). The shape of the objective function f(r₁,…,r_m) depends on the structure of the cost matrix c.

• If c is arbitrary, f has no convexity/concavity guarantees. The authors show that enumerating all O(r^{m‑1}) feasible vectors yields an XP algorithm, but the problem is W