# Parameterized Shifted Combinatorial Optimization

## TL;DR

This paper explores the parameterized complexity of shifted combinatorial optimization, a nonlinear extension of standard problems, revealing varying computational complexities depending on problem parameters and logical definability.

## Contribution

It introduces the parameterized complexity analysis of shifted combinatorial optimization, including results on MSO definable sets and graph parameters like treewidth.

## Key findings

- Shifted problems can be NP-hard even with explicit input sets.
- Shifted MSO problems are in XP with respect to MSO formula and graph parameters.
- Shifted MSO problems are W[1]-hard under certain restrictions.

## Abstract

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.