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arXiv · 1307.4420

On the Complexity of a Matching Problem with Asymmetric Weights

Lily Briggs
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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.

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Lily Briggs. 2013-07-19. On the Complexity of a Matching Problem with Asymmetric Weights. https://arxiv.org/abs/1307.4420

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