On the Complexity of Nucleolus Computation for Bipartite b-Matching Games.

We explore the complexity of nucleolus computation in b-matching games on bipartite graphs. We show that computing the nucleolus of a simple b-matching game is \(\mathcal {NP}\)-hard when \(b\equiv 3\) even on bipartite graphs of maximum degree 7. We complement this with partial positive results in the special case where b values are bounded by 2. In particular, we describe an efficient algorithm when a constant number of vertices satisfy \(b_v = 2\) as well as an efficient algorithm for computing the non-simple b-matching nucleolus when \(b\equiv 2\).