On the Complexity of a Matching Problem with Asymmetric Weights

TL;DR

This paper investigates the computational complexity of orienting graphs to maximize structural controllability, revealing that a related problem is polynomially solvable while its asymmetric weighted version is NP-complete and APX-hard.

Contribution

It introduces the Orientation Control Matching problem and its asymmetric weighted variant, proving the latter's NP-completeness and APX-hardness, highlighting their computational difficulty.

Findings

OCM is polynomially solvable.

AOCM is NP-complete.

AOCM is APX-hard.

Abstract

We present complexity results regarding a matching-type problem related to structural controllability of dynamical systems modelled on graphs. Controllability of a dynamical system is the ability to choose certain inputs in order to drive the system from any given state to any desired state; a graph is said to be structurally controllable if it represents the structure of a controllable system. We define the Orientation Control Matching problem (OCM) to be the problem of orienting an undirected graph in a manner that maximizes its structural controllability. A generalized version, the Asymmetric Orientation Control Matching problem (AOCM), allows for asymmetric weights on the possible directions of each edge. These problems are closely related to 2-matchings, disjoint path covers, and disjoint cycle covers. We prove using reductions that OCM is polynomially solvable, while AOCM is much harder; we show that it is NP-complete as well as APX-hard.