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

TL;DR

This paper investigates the computational complexity of finding the nucleolus in bipartite b-matching games, revealing NP-hardness in general but providing efficient algorithms for specific bounded cases.

Contribution

It proves NP-hardness of nucleolus computation for simple b-matching games on bipartite graphs and offers efficient algorithms for cases where b values are limited to 2.

Findings

NP-hardness of nucleolus computation for simple b-matching games.

Efficient algorithms for cases with bounded b values, especially b=2.

Partial positive results for special cases with limited b-value vertices.

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.