The b-matching problem is a well-known generalization of the classical matching problem with various applications in operations research and computer science. Given an undirected graph, each vertex v has a capacity b(v), indicating the maximum number of times it can be matched, while edges can also be used multiple times. The problem is solvable in polynomial time and has many real-world applications.
In some of them, a feasible matching must exactly satisfy the capacities b(v), leading to the so-called perfect b-matching problem. Typically, the capacities b(v) are assumed to be fixed and known. However, in practice, these capacities often face uncertainties, such as worker availability or customer demand fluctuations.
This paper analyses a robust variant of both the b-matching and perfect b-matching problems, accounting for such capacity uncertainties, termed the Directed Robust b-Matching Problem. We study the computational complexity of this problem across different classes of graphs, providing insights into its tractability for potential applications.