We study the fair allocation of indivisible goods under the assumption that the goods form an undirected graph and each agent must receive a connected subgraph. Our focus is on well-studied fairness notions including envy-freeness and maximin share fairness. We establish graph-specific maximin share guarantees, which are tight for large classes of graphs in the case of two agents and for paths and stars in the general case. Unlike in previous work, our guarantees are with respect to the complete-graph maximin share, which allows us to compare possible guarantees for different graphs. For instance, we show that for biconnected graphs it is possible to obtain at least @math of the maximin share, while for the remaining graphs the guarantee is at most @math . In addition, we determine the optimal relaxation of envy-freeness that can be obtained with each graph for two agents, and characterize the set of trees and complete bipartite graphs that always admit an allocation satisfying envy-freeness up to one good (EF1) for three agents. Our work demonstrates several applications of graph-theoretical tools and concepts to fair division problems.
Fair allocation of indivisible goods has received considerable attention from the research community, especially in the last few years. We refer to surveys by @cite_7 , @cite_10 , and @cite_14 for an overview of recent developments in the area.
@cite_14 @cite_10 @cite_7
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The fair division of indivisible goods has long been an important topic in economics and, more recently, computer science. We investigate the existence of envyfree allocations of indivisible goods, that is, allocations where each player values her own allocated set of goods at least as highly as any other player's allocated set of goods. Under additive valuations, we show that even when the number of goods is larger than the number of agents by a linear fraction, envy-free allocations are unlikely to exist.We then show that when the number of goods is larger by a logarithmic factor, such allocations exist with high probability. We support these results experimentally and show that the asymptotic behavior of the theory holds even when the number of goods and agents is quite small. We demonstrate that there is a sharp phase transition from nonexistence to existence of envy-free allocations, and that on average the computational problem is hardest at that transition. We generalize the classic problem of fairly allocating indivisible goods to the problem of fair public decision making, in which a decision must be made on several social issues simultaneously, and, unlike the classic setting, a decision can provide positive utility to multiple players. We extend the popular fairness notion of proportionality (which is not guaranteeable) to our more general setting, and introduce three novel relaxations --- proportionality up to one issue, round robin share, and pessimistic proportional share --- that are also interesting in the classic goods allocation setting. We show that the Maximum Nash Welfare solution, which is known to satisfy appealing fairness properties in the classic setting, satisfies or approximates all three relaxations in our framework. We also provide polynomial time algorithms and hardness results for finding allocations satisfying these axioms, with or without insisting on Pareto optimality. 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. We study the problem of fair allocation for indivisible goods. We use the the maxmin share paradigm introduced by Budish as a measure for fairness. Procacciafirst (EC'14) were first to investigate this fundamental problem in the additive setting. In contrast to what real-world experiments suggest, they show that a maxmin guarantee (1- @math allocation) is not always possible even when the number of agents is limited to 3. While the existence of an approximation solution (e.g. a @math - @math allocation) is quite straightforward, improving the guarantee becomes subtler for larger constants. Procaccia provide a proof for existence of a @math - @math allocation and leave the question open for better guarantees. Our main contribution is an answer to the above question. We improve the result of ! to a @math factor in the additive setting. The main idea for our @math - @math allocation method is clustering the agents. To this end, we introduce three notions and techniques, namely reducibility, matching allocation, and cycle-envy-freeness, and prove the approximation guarantee of our algorithm via non-trivial applications of these techniques. Our analysis involves coloring and double counting arguments that might be of independent interest. One major shortcoming of the current studies on fair allocation is the additivity assumption on the valuations. We alleviate this by extending our results to the case of submodular, fractionally subadditive, and subadditive settings. More precisely, we give constant approximation guarantees for submodular and XOS agents, and a logarithmic approximation for the case of subadditive agents. Furthermore, we complement our results by providing close upper bounds for each class of valuation functions. Finally, we present algorithms to find such allocations for additive, submodular, and XOS settings in polynomial time.