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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 N P-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 bv = 2 as well as an efficient algorithm for computing the non-simple b-matching nucleolus when b ≡ 2. * We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC). Cette recherche a été financée par le Conseil de recherches en sciences naturelles et en génie du Canada (CRSNG).

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We consider the Steiner tree problem in quasi-bipartite graphs, where no two Steiner vertices are connected by an edge. For this class of instances, we present an efficient algorithm to exactly solve the so called directed component relaxation (DCR), a specific form of hypergraphic LP relaxation that was instrumental in the recent break-through result by Byrka et al. [2]. Our algorithm hinges on an efficiently computable map from extreme points of the bidirected cut relaxation to feasible solutions of (DCR). As a consequence, together with [2] we immediately obtain an efficient 73/60-approximation for quasi-bipartite Steiner tree instances. We also present a particularly simple (BCR)-based random sampling algorithm that achieves a performance guarantee slightly better than 77/60.