Matchings and games on networks
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We investigate computational aspects of popular solution concepts for different models of network games. In chapter 3 we study balanced solutions for network bargaining games with general capacities, where agents can participate in a fixed but arbitrary number of contracts. We fully characterize the existence of balanced solutions and provide the first polynomial time algorithm for their computation. Our methods use a new idea of reducing an instance with general capacities to an instance with unit capacities defined on an auxiliary graph. In chapter 4 we propose a generalization of the classical stable marriage problem. In our model the preferences on one side of the partition are given in terms of arbitrary binary relations, that need not be transitive nor acyclic. This generalization is practically well-motivated, and as we show, encompasses the well studied hard variant of stable marriage where preferences are allowed to have ties and to be incomplete. Our main result shows that deciding the existence of a stable matching in our model is NP-complete. We then use our model to study a long standing open problem about cyclic 3D stable matchings. In particular, we prove that deciding whether a fixed 2D perfect matching can be extended to a 3D stable matching is NP-complete. In chapter 5 we study a long standing open problem of whether the nucleolus of matching games can be computed efficiently. Our approach follows previous techniques that rely on obtaining a polynomial sized characterization of the least core as both the initial and crucial step in establishing an efficient algorithm. As a preliminary result, we introduce a generalisation of the least core and show that for node-weighted matching games existing polynomial sized characterizations of the least core can be extended to this generalised version. We then use this result to show that for a certain class games with general weights one can identify a node-weighted subgraph such that the least core of the original matching game is equal to the generalised least core of this subgraph. This allows us to obtain a polynomial time algorithm for computing the nucleolus for this class of matching games with general weights.