On the Complexity of a Matching Problem with Asymmetric Weights

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.

💡 Research Summary

The paper investigates two graph‑orientation problems that arise from the study of structural controllability of linear dynamical systems. A system is structurally controllable if there exists a choice of edge weights that makes the underlying directed network controllable; the number of external inputs required equals the number of unmatched vertices in a maximum control matching. A control matching in a directed graph is a set of arcs such that each vertex is incident to at most one incoming and at most one outgoing arc. Finding a maximum control matching in a given directed graph is polynomial‑time solvable because it is equivalent to a 1‑matching in the bipartite representation of the digraph.

Orientation Control Matching (OCM).
Given an undirected graph G, OCM asks for an orientation of its edges that maximizes the size of a maximum control matching in the resulting digraph. When all edge orientations have the same benefit (uniform weight), the problem reduces to finding a simple 2‑matching: a set of edges where each vertex is incident to at most two selected edges. A simple 2‑matching can be interpreted as a collection of node‑disjoint paths and cycles covering all vertices. Edmonds showed that a maximum simple 2‑matching can be found in polynomial time via a polyhedral description. Consequently, OCM belongs to P.

Asymmetric Orientation Control Matching (AOCM).
AOCM generalizes OCM by assigning two possibly different weights w(u→v) and w(v→u) to each undirected edge {u,v}. The goal is to choose an orientation (i.e., pick exactly one direction for each edge) and then a control matching that maximizes the total weight of the selected arcs. The authors first prove that AOCM is in NP. To establish NP‑hardness they reduce from the 3‑Cycle Cover problem (3‑DCC), which asks whether a directed graph can be partitioned into vertex‑disjoint directed cycles of length at least three covering all vertices. Given a 3‑DCC instance G, they construct a symmetric digraph G′ by adding the missing reverse arcs and assign weight 1 to the original arcs and weight 0 to the added ones. An orientation of G′ that admits a control matching of total weight |V| exists iff G has a 3‑cycle cover. Hence AOCM is NP‑complete.

APX‑hardness.
To show that AOCM does not admit a PTAS (unless P = NP), the paper gives an L‑reduction from Maximum Independent Set on cubic graphs (Max‑E3‑Ind‑Set), a known APX‑hard problem. For each vertex u of the cubic graph G they create a gadget consisting of three “edge‑arcs” (one for each incident edge) and two “node‑arcs” that connect these edge‑arcs in a chain. All non‑zero arcs receive weight 1; zero‑weight arcs are added to make the digraph symmetric. In any feasible orientation set σ, the maximum control matching M(σ) can contain at most three weight‑1 arcs incident to a vertex u, and the exact number depends on how many of u’s edge‑arcs are selected. By a careful case analysis (0, 1, 2, 3 edge‑arcs), the authors prove that the total weight v(σ) satisfies

 v(σ) ≤ 2n + |Vσ³|,

where n is the number of vertices of G and Vσ³ denotes the set of vertices whose three edge‑arcs all belong to M(σ). In an optimal orientation σ* the equality holds, and |Vσ*³| equals the size of a maximum independent set of the original cubic graph. Therefore the optimal AOCM value is tightly linked to the optimal independent‑set value. The reduction satisfies the two L‑reduction conditions with constant factors α and β, establishing that AOCM is APX‑hard.

Implications.
The results draw a sharp contrast between the uniform‑weight version (OCM) and the asymmetric version (AOCM). While OCM can be solved efficiently via 2‑matching algorithms, introducing direction‑dependent weights makes the problem computationally intractable: exact optimization is NP‑complete and even constant‑factor approximation is unlikely unless P = NP. This has practical relevance for network design problems where the cost or benefit of orienting a link depends on direction (e.g., power flow, traffic routing, or sensor placement). The paper also showcases how classic graph‑theoretic problems—2‑matching, cycle cover, and independent set—serve as powerful tools for proving hardness of newer combinatorial optimization problems.