Stability in one-sided matching markets

The stable roommates problem may be unsolvable for sorne instances, therefore we study a relaxation, when it is allowed to form groups of any size (the stable partition problem). Two extensions of preferences over individuals to preferences over sets are suggested. For the first one, derived from the most prefered member of a set, it is shown that a stable partition always existis if the original preferences are strict and a simple algorithm for its computation is derived. This algorithm turns out to be strategy proof. The second extension, based on the least prefered member of a set, produces solutions very similar to those for the stable roornmates problem

The Kidney Exchange Game

The most effective treatment for kidney failure that is currently known is transplantation. As the number of cadaveric donors is not sufficient and kidneys from living donors are often not suitable for immunological reasons, there are attempts to organize exchanges between patient-donor pairs. In this paper we model this situation as a cooperative game and propose some algorithms for finding a solution

Chore division on a graph

The paper considers fair allocation of indivisible nondisposable items that generate disutility (chores). We assume that these items are placed in the vertices of a graph and each agent's share has to form a connected subgraph of this graph. Although a similar model has been investigated before for goods, we show that the goods and chores settings are inherently different. In particular, it is impossible to derive the solution of the chores instance from the solution of its naturally associated fair division instance. We consider three common fair division solution concepts, namely proportionality, envy-freeness and equitability, and two individual disutility aggregation functions: additive and maximum based. We show that deciding the existence of a fair allocation is hard even if the underlying graph is a path or a star. We also present some efficiently solvable special cases for these graph topologies

The exchange-stable marriage problem

In this paper we consider instances of stable matching problems, namely the classical stable marriage (SM) and stable roommates (SR) problems and their variants. In such instances we consider a stability criterion that has recently been proposed, that of <i>exchange-stability</i>. In particular, we prove that ESM — the problem of deciding, given an SM instance, whether an exchange-stable matching exists — is NP-complete. This result is in marked contrast with Gale and Shapley's classical linear-time algorithm for finding a stable matching in an instance of SM. We also extend the result for ESM to the SR case. Finally, we study some variants of ESM under weaker forms of exchange-stability, presenting both polynomial-time solvability and NP-completeness results for the corresponding existence questions

Fair Division of a Graph

We consider fair allocation of indivisible items under an additional constraint: there is an undirected graph describing the relationship between the items, and each agent's share must form a connected subgraph of this graph. This framework captures, e.g., fair allocation of land plots, where the graph describes the accessibility relation among the plots. We focus on agents that have additive utilities for the items, and consider several common fair division solution concepts, such as proportionality, envy-freeness and maximin share guarantee. While finding good allocations according to these solution concepts is computationally hard in general, we design efficient algorithms for special cases where the underlying graph has simple structure, and/or the number of agents -or, less restrictively, the number of agent types- is small. In particular, despite non-existence results in the general case, we prove that for acyclic graphs a maximin share allocation always exists and can be found efficiently.Comment: 9 pages, long version of accepted IJCAI-17 pape

Strong regularity of matrices in a discrete bounded bottleneck algebra

AbstractThe results concerning strong regularity of matrices over bottleneck algebras are reviewed. We extend the known conditions to the discrete bounded case and modify the known algorithms for testing strong regularity

Pareto optimality in the kidney exchange problem

summary:To overcome the shortage of cadaveric kidneys available for transplantation, several countries organize systematic kidney exchange programs. The kidney exchange problem can be modelled as a cooperative game between incompatible patient-donor pairs whose solutions are permutations of players representing cyclic donations. We show that the problems to decide whether a given permutation is not (weakly) Pareto optimal are NP-complete

Soluble approximation of linear systems in max-plus algebra

summary:We propose an efficient method for finding a Chebyshev-best soluble approximation to an insoluble system of linear equations over max-plus algebra

Pareto optimal matchings in many-to-many markets with ties

We consider Pareto optimal matchings (POMs) in a many-to-many market of applicants and courses where applicants have preferences, which may include ties, over individual courses and lexicographic preferences over sets of courses. Since this is the most general setting examined so far in the literature, our work unifies and generalizes several known results. Specifically, we characterize POMs and introduce the Generalized Serial Dictatorship Mechanism with Ties (GSDT) that effectively handles ties via properties of network flows. We show that GSDT can generate all POMs using different priority orderings over the applicants, but it satisfies truthfulness only for certain such orderings. This shortcoming is not specific to our mechanism; we show that any mechanism generating all POMs in our setting is prone to strategic manipulation. This is in contrast to the one-to-one case (with or without ties), for which truthful mechanisms generating all POMs do exist