On the Complexity of Nucleolus Computation for Bipartite b-Matching Games
Jochen Koenemann, Justin Toth, Felix Zhou
TL;DR
This work studies the computational complexity of the nucleolus in bipartite $b$-matching games. It proves that computing the nucleolus for simple bipartite $b$-matching games is $\\mathcal{NP}$-hard, already for graphs with maximum degree $7$, via a reduction from a variant called Two from Cubic Subgraph using a gadget construction and the Kopelowitz framework. The paper also identifies two polynomial-time tractable regimes when $b\leq 2$: (i) a simple case with $b_v=2$ on one side and only $k$ vertices with $b_v=2$ on the other, and (ii) the non-simple $2$-matching case with $b\equiv 2$, leveraging dual LP simplifications and structural properties of maximum $b$-matchings. These results delineate the boundary between tractable and intractable nucleolus computations in bipartite $b$-matching games and guide future algorithmic approaches and extensions to broader graph classes. The findings rely on the Kopelowitz Scheme to connect LP-based characterizations with combinatorial gadget constructions, yielding both hardness and constructive polynomial-time algorithms.
Abstract
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 NP-hard 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 = 2.