favorite6If G is a star then the problem to decide the existence of a valid allocation where each agent gets a bundle with disutility 0 can be solved in polynomial time.
favorite14Later in this paper we shall see that even for paths the problems to decide whether a valid allocation exists such that the disutility of each agent equals 0 is NP-hard.
favorite16First, observe that if in a fair division instance there are more agents than items, no proportional and envy-free allocation can exist, which is not necessarily the case with chores.
favorite14By contrast, if the underlying graph is a star, we propose an efficient algorithm, based on bipartite matching techniques, to decide whether a valid allocation exists such that each agent has disutility 0.
favorite2We consider three common fair division solution concepts, namely proportionality, envy-freeness and equitability, and two individual disutility aggregation functions: additive and maximum based.
favorite1It remains to observe that a minimum-weight perfect matching can be computed in polynomial time [see e.g. If the underlying graph is a path and all players are of the same type, then a simple greedy algorithm finds a proportional allocation (or reports that none exists) in linear time: we build connected pieces one by one, by moving along the path from left to right and adding vertices to the current piece until its value to a player reaches 1/n; at this point we start building a new piece.
favorite291We prove a strong positive result for our setting: an MMS allocation always exists if the underlying graph is a tree, and can be computed efficiently.
favorite7Recently, several papers have studied the concept of the maximin share guarantee (MMS) [Budish, 2011], which captures a desirable property of allocations that is easy to achieve for divisible items via cut-and-choose protocols.
favorite29In particular, both proportional and complete envy-free allocations can be found efficiently when the graph is a path and agents can be classified into a small number of types, where agents are said to have the same type when they have the same preferences over items.1 If we assume that not just the number of player types, but the actual number of players is small, we obtain an efficient algorithm for finding proportional allocations on arbitrary trees.
favorite1Our contribution We propose a framework for fair division under connectivity constraints, and investigate the complexity of finding good allocations in this framework according to three well-studied solution concepts: proportionality, envyfreeness (in conjunction with completeness), and maximin share guarantee.
favorite1He defined three different types of coalitions with respect to a given matching such that the existence of either means that a subset of applicants can trade among themselves (possibly using some exposed course) ensuring that, at the end, no one is worse off and at least one applicant is better off.
favorite5We provide a characterization of POMs in this setting, leading to a polynomial-time algorithm for testing whether a given matching is Pareto optimal.
favorite21This work provides a characterization of POMs assuming that the preferences of applicants over sets of courses are obtained from their (strict) preferences over individual courses in a lexicographic manner.
favorite1In  the authors provide a characterization of POMs in the case of strict preferences and utilize it in order to construct polynomial-time algorithms for checking whether a given matching is a POM and for finding a POM of maximum size.
favorite99Institute of Mathematics, Faculty of Science, P.J. Abstract 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.