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... We follow the above line of research and restrict our attention to envy-free orientations in graphs -a model that has attracted significant attention since [CFKS23]. Here, agents are the vertices in a graph G = (V, E) and goods are the edges so that each agent is only interested in the goods incident to her, which are called relevant items of the agent. ...
... An allocation is said to be an orientation if every item is allocated to one of the two incident agents. Christodoulou et al. [CFKS23] introduced this model and proved that deciding the existence of an envy-free up to any item (EFX) orientation is NP-hard. An allocation is EFX if no agent envies another agent's bundle after (hypothetically) removing any item in it. ...
... Zeng and Mehta [ZM24] characterized the structure of the graphs so that it is guaranteed that an EFX orientation exists. Deligkas et al. [DEGK24] strengthened the result of [CFKS23] by proving that even with 8 agents, the problem remains NP-hard when the graph has parallel edges. However, how to use subsidy to compensate agents so that the agents are EF has not been studied before our work. ...
We study a fair division problem in (multi)graphs where n agents (vertices) are pairwise connected by items (edges), and each agent is only interested in its incident items. We consider how to allocate items to incident agents in an envy-free manner, i.e., envy-free orientations, while minimizing the overall payment, i.e., subsidy. We first prove that computing an envy-free orientation with the minimum subsidy is NP-hard, even when the graph is simple and the agents have bi-valued additive valuations. We then bound the worst-case subsidy. We prove that for any multigraph (i.e., allowing parallel edges) and monotone valuations where the marginal value of each good is at most \1 for each agent, \1 each (a total subsidy of , where n is the number of agents) is sufficient. This is one of the few cases where linear subsidy is known to be necessary and sufficient to guarantee envy-freeness when agents have monotone valuations. When the valuations are additive (while the graph may contain parallel edges) and when the graph is simple (while the valuations may be monotone), we improve the bound to n/2 and , respectively. Moreover, these two bounds are tight.
... The existence or not of EFX allocations is a major open problem in Fair Division, and there are only positive results for special cases. Christodoulou et al. [19] introduced a restriction on the agents' valuations according to a graph structure: the vertices correspond to agents and the edges to goods, and each vertex/agent has zero marginal value (or in other words, they are indifferent) for the edges/goods that are not adjacent to them. The existence of EFX allocations has been shown for simple graphs with general monotone valuations [19], and for multigraphs for restricted additive valuations [29]. ...
... Christodoulou et al. [19] introduced a restriction on the agents' valuations according to a graph structure: the vertices correspond to agents and the edges to goods, and each vertex/agent has zero marginal value (or in other words, they are indifferent) for the edges/goods that are not adjacent to them. The existence of EFX allocations has been shown for simple graphs with general monotone valuations [19], and for multigraphs for restricted additive valuations [29]. In this work, we push the state-of-the-art further, and show that the EFX allocations always exists in multigraphs and general monotone valuations if any of the following three conditions hold: either (a) the multigraph is bipartite, or (b) each agent has at most ⌈ n 4 ⌉ − 1 neighbors, where n is the total number of agents, or (c) the shortest cycle with non-parallel edges has length at least 6. ...
... All those works consider high restrictions and resemblance on the agents valuations. Towards broadening our understanding for the case of multiple agents and goods, Christodoulou et al. [19] introduced a setting that is related to our work, where the valuations are defined based on a graph: given a graph, the agents correspond to the vertices of the graph, and the goods to the edges. Then, each agent is indifferent for the goods/edges that are not adjacent to them. ...
We study the problem of "fairly" dividing indivisible goods to several agents that have valuation set functions over the sets of goods. As fair we consider the allocations that are envy-free up to any good (EFX), i.e., no agent envies any proper subset of the goods given to any other agent. The existence or not of EFX allocations is a major open problem in Fair Division, and there are only positive results for special cases. [George Christodoulou, Amos Fiat, Elias Koutsoupias, Alkmini Sgouritsa 2023] introduced a restriction on the agents' valuations according to a graph structure: the vertices correspond to agents and the edges to goods, and each vertex/agent has zero marginal value (or in other words, they are indifferent) for the edges/goods that are not adjacent to them. The existence of EFX allocations has been shown for simple graphs with general monotone valuations [George Christodoulou, Amos Fiat, Elias Koutsoupias, Alkmini Sgouritsa 2023], and for multigraphs for restricted additive valuations [Alireza Kaviani, Masoud Seddighin, Amir Mohammad Shahrezaei 2024]. In this work, we push the state-of-the-art further, and show that the EFX allocations always exists in multigraphs and general monotone valuations if any of the following three conditions hold: either (a) the multigraph is bipartite, or (b) each agent has at most neighbors, where n is the total number of agents, or (c) the shortest cycle with non-parallel edges has length at least 6.
... Given the difficulties in establishing the existence of EFX allocations with multiple agents, researchers study special cases of structured valuations. One such class of structured valuations is graphical valuations when the instance can be represented as a graph [CFKS23]. Here the agents correspond to the vertices of the graph, and each item is an edge e = {u, v}. e restriction is that each item has nonzero marginal value for only the agents at its end-points, and has zero value for all other agents. ...
... Further, the graph is simple, and thus there is at most one good valued by every pair of agents. For this case, and agents with monotone valuations over incident edges, Christodolou et al. [CFKS23] show that EFX allocations exist, marking one of the rare cases where EFX allocations are shown to exist with multiple agents. ...
... A natural question with graphical valuations is if an EFX orientation exists-an EFX allocation where goods are only allocated to agents at their endpoints. Christodolou et al. show that an EFX orientation may not exist, and it is NP-hard to determine if a given instance has an EFX orientation [CFKS23]. Zeng and Mehta characterize graphs that permit EFX orientations [ZM24]. ...
The existence of EFX allocations is one of the most significant open questions in fair division. Recent work by Christodolou, Fiat, Koutsoupias, and Sgouritsa ("Fair allocation in graphs", EC 2023) establishes the existence of EFX allocations for graphical valuations, when agents are vertices in a graph, items are edges, and each item has zero value for all agents other than those at its end-points. Thus in this setting, each good has non-zero value for at most two agents, and there is at most one good valued by any pair of agents. This marks one of the few cases when an exact and complete EFX allocation is known to exist for arbitrary agents. In this work, we extend these results to multi-graphs, when each pair of vertices can have more than one edge between them. The existence of EFX allocations in multi-graphs is a natural open question given their existence in simple graphs. We show that EFX allocations exist, and can be computed in polynomial time, for agents with cancellable valuations in the following cases: (i) bipartite multi-graphs, (ii) multi-trees with monotone valuations, and (iii) multi-graphs with girth , where t is the chromatic number of the multi-graph. The existence in multi-cycles follows from (i) and (iii).
... A question of interest is understanding for what scenarios or valuation classes, EFX allocations are guaranteed to exist. To this effect, in this work, we consider a recently introduced model by Christodoulou et al. [2023] where agent valuations are represented via a graph (or a multi-graph). Here, every item can have a positive value for at most two agents and that item appears as an edge between these two agents, i.e., an item (edge) is valued at zero for all agents that are not its endpoints. ...
... Another example is when football matches in the New Champions League Format must be played on the home grounds of either of the participating teams, and other teams do not care for it. Christodoulou et al. [2023] proved that EFX allocations exist for all simple graphs, i.e., where any two agent vertices share at most one edge item. A natural question, therefore, arises as to what happens when we consider classes of multi-graphs, i.e., where two agents can have multiple relevant items in common? ...
... In this work, we answer the above question raised by Christodoulou et al. [2023] and study fair division instances where agent valuations are represented via bipartite multi-graphs and provide a complete picture of EFX allocations for additive valuations. Note that bipartite multi-graphs is a huge class consisting of multi-trees, multi-cycles with even length, and planer multi-graphs with faces of even length, to name a few. ...
We consider the fundamental problem of fairly allocating a set of indivisible items among agents having valuations that are represented by a multi-graph -- here, agents appear as the vertices and items as the edges between them and each vertex (agent) only values the set of its incident edges (items). The goal is to find a fair, i.e., envy-free up to any item (EFX) allocation. This model has recently been introduced by Christodoulou et al. (EC'23) where they show that EFX allocations always exist on simple graphs for monotone valuations, i.e., where any two agents can share at most one edge (item). A natural question arises as to what happens when we go beyond simple graphs and study various classes of multi-graphs? We answer the above question affirmatively for the valuation class of bipartite multi-graphs and multi-cycles. Our main positive result is that EFX allocations always exist on bipartite multi-graphs for agents with additive valuations, thereby joining in the few sets of scenarios where EFX allocations are known to exist for an arbitrary number of agents. We also show that EFX allocations can be computed in polynomial time if the number of edges between any two agents in the given bipartite multi-graph is a constant. Next, we study EFX orientations (i.e., allocations where every item is allocated to one of its two endpoint agents) and give a complete picture of when they exist for bipartite multi-graphs dependent on two parameters -- the number of edges shared between any two agents and the diameter of the graph. Finally, we prove that it is NP-complete to determine whether a given fair division instance on a bipartite multi-graph admits an EFX orientation.
... We consider the complexity of finding envy-free allocations for the class of graphical valuations. Graphical valuations were introduced by Christodoulou et al. [14] as a structured class of valuations that admit allocations that are envy-free up to any item(EFX). These are valuations where every item is valued by two agents, lending a (simple) graph structure to the utilities, where the agents are vertices and are adjacent if and only if they value a (unique) common item. ...
... Motivated by these issues, we focus on a recently introduced class of structured valuations, called graphical valuations introduced by Christodoulou et al. [14] as a class of valuations that admit allocations that are envy-free up to any item (EFX) 1 . These are valuations where every item is valued by exactly two agents, lending a (simple) graph structure to the utilities, where the agents are associated with vertices and items with edges. ...
... Such valuations may arise in scenarios where agents only value the items that are geographically closer. For instance, in real estate allocation, potential buyers might only be interested in properties within a certain distance from their workplace or amenities; employees might value office spaces closer to their teams and likewise [14]. ...
We consider the complexity of finding envy-free allocations for the class of graphical valuations. Graphical valuations were introduced by Christodoulou et. al.(2023) as a structured class of valuations that admit allocations that are envy-free up to any item (EFX). These are valuations where every item is valued by two agents, lending a (simple) graph structure to the utilities, where the agents are vertices and are adjacent if and only if they value a (unique) common item. Finding envy-free allocations for general valuations is known to be computationally intractable even for very special cases: in particular, even for binary valuations, and even for identical valuations with two agents. We show that, for binary graphical valuations, the existence of envy-free allocations can be determined in polynomial time. In contrast, we also show that allowing for even slightly more general utilities {0, 1, d} leads to intractability even for graphical valuations. This motivates other approaches to tractability, and to that end, we exhibit the fixed-parameter tractability of the problem parameterized by the vertex cover number of the graph when the number of distinct utilities is bounded. We also show that, all graphical instances that admit EF allocations also admit one that is non-wasteful. Since EFX allocations are possibly wasteful, we also address the question of determining the price of fairness of EFX allocations. We show that the price of EFX with respect to utilitarian welfare is one for binary utilities, but can be arbitrarily large {0, 1, d} valuations. We also show the hardness of deciding the existence of an EFX allocation which is also welfare-maximizing and of finding a welfare-maximizing allocation within the set of EFX allocations.
... Both of these algorithms achieve an optimal number of transfers (at most m/n, where m and n are the number of items and agents respectively) for identical additive valuations. Next, we consider a valuation class with graphical structure, introduced by Christodoulou et al. [CFKS23], where each item is valued by at most two agents, and hence can be seen as an edge between these two agents in a graph. Here, we consider EF1 orientations on (multi)graphs -allocations in which each item is allocated to an agent who values it. ...
... 2. We consider a class of graphical valuations introduced by Christodoulou et al. [CFKS23], where every item is positively valued by at most two agents, and hence can be represented as an edge between them. Moreover, let the initial near-EF1 allocation have the property that each item is allocated to an agent who values it positively (known as an orientation, studied in [ZM24,DEGK24]). ...
... As mentioned earlier, the class of graphical valuations, in which every item is valued by at most two agents and hence can be seen as an edge between them, was introduced in [CFKS23] in the context of envy-freeness up to any item (EFX) allocations. They proved the existence of EFX allocations on simple graphs. ...
We study the classic problem of fairly dividing a set of indivisible items among a set of agents and consider the popular fairness notion of envy-freeness up to one item (EF1). While in reality, the set of agents and items may vary, previous works have studied static settings, where no change can occur in the system. We initiate and develop a formal model to understand fair division under the variable input setting: here, there is an EF1 allocation that gets disrupted because of the loss/deletion of an item, or the arrival of a new agent, resulting in a near-EF1 allocation. The objective is to perform a sequence of transfers of items between agents to regain EF1 fairness by traversing only via near-EF1 allocations. We refer to this as the EF1-Restoration problem. In this work, we present algorithms for the above problem when agents have identical monotone valuations, and items are either all goods or all chores. Both of these algorithms achieve an optimal number of transfers (at most m/n, where m and n are the number of items and agents respectively) for identical additive valuations. Next, we consider a valuation class with graphical structure, introduced by Christodoulou et al. (EC'23), where each item is valued by at most two agents, and hence can be seen as an edge between these two agents in a graph. Here, we consider EF1 orientations on (multi)graphs - allocations in which each item is allocated to an agent who values it. While considering EF1 orientations on multi-graphs with additive binary valuations, we present an optimal algorithm for the EF1-Restoration problem. Finally, for monotone binary valuations, we show that the problem of deciding whether EF1-Restoration is possible is PSPACE-complete.
... For example, EFX allocations of goods are known to exist if the agents have identical utility functions or if the agents have identical ranking of the goods [PR20], if the agents have lexicographic utility functions [HSVX23], or if the number of agents is at most 3 [CGM20, AAC + 22]. Recently, Christodoulou et al. [CFKS23] introduced a new setting in which instances of the fair division problem are represented as graphs. Specifically, they represented agents as vertices and goods as edges, and imposed the requirement that a good can only have positive utility to an agent if the edge representing the good is incident with the vertex representing the agent. ...
... Although finding EFX orientations is desirable, it is more difficult than finding EFX allocations. Indeed, Christodoulou et al. [CFKS23] demonstrated this by showing that although EFX allocations always exist for simple graphs, deciding whether EFX orientations exist is NP-complete. This leads to the question of whether there are any classes of graphs for which the existence of an EFX orientation can be decided efficiently. ...
... In this section, we prove Theorem 1. Our proof works by adapting the idea due to Christodoulou et al. [CFKS23] to the setting of multigraphs. Similarly to their proof, we also find a reduction from the NPcomplete problem CircuitSAT by constructing a bi-valued symmetric multigraph. ...
We study the fair division of multigraphs with self-loops. In this setting, vertices represent agents and edges represent goods, and a good provides positive utility to an agent only if it is incident with the agent. Whereas previous research has so far only considered simple graphs, we consider the general setting of multigraphs, specifically focusing on the case in which each edge has equal utility to both incident agents, and edges have one of two possible utilities . In contrast with the case of simple graphs for which bipartiteness implies the existence of an EFX orientation, we show that deciding whether a symmetric multigraph G of multiplicity admits an EFX orientation is NP-complete even if G is bipartite, , and G contains a structure called a non-trivial odd multitree. Moreover, we show that non-trivial odd multitrees are a forbidden structure in the sense that even very simple non-trivial odd multitrees can fail to admit EFX orientations, and multigraphs that do not contain non-trivial odd multitrees always admit EFX orientations.
... The value of an agent for every other good, non-incident to their vertex, was zero. Christodoulou et al. (2023) studied the existence and complexity of finding EFX allocations and EFX orientations. An orientation is an allocation where every agent gets only edges adjacent to them, i.e., every edge is "oriented" towards the incident agent that gets it. ...
... Our main result is to prove that an EF1 orientation always exists when the valuations of the agents are monotone! In fact, we prove our result for a more general model than the one from Christodoulou et al. (2023), where instead of graphs, we consider hypergraphs, i.e., the goods now correspond to hyperedges. In other words, each agent has a subset of goods that are relevant to them. ...
... They treat orientations as a special case of their problem and show that deciding if an EFX orientation exists is NP-complete. The paper by Payan et al. (2023) also studies graph restrictions but these are subtly different to that of Christodoulou et al. (2023). In this model, edges are not items but instead, they are where EFX (/other fairness notions) must apply, intuitively this aims to capture a model where we want envy freeness between an agent and some of their neighbors. ...
We study the problem of finding fair allocations -- EF1 and EFX -- of indivisible goods with orientations. In an orientation, every agent gets items from their own predetermined set. For EF1, we show that EF1 orientations always exist when agents have monotone valuations, via a pseudopolynomial-time algorithm. This surprisingly positive result is the main contribution of our paper. We complement this result with a comprehensive set of scenarios where our algorithm, or a slight modification of it, finds an EF1 orientation in polynomial time. For EFX, we focus on the recently proposed graph instances, where every agent corresponds to a vertex on a graph and their allowed set of items consists of the edges incident to their vertex. It was shown that finding an EFX orientation is NP-complete in general. We prove that it remains intractable even when the graph has a vertex cover of size 8, or when we have a multigraph with only 10 vertices. We essentially match these strong negative results with a fixed-parameter tractable algorithm that is virtually the best someone could hope for.
... Recently, Christodoulou et al. [2023] studied EFX allocations on graphs where vertices correspond to agents and edges correspond to indivisible goods. An agent (vertex) is only interested in the goods (edges) that are incident to her and all other edges have zero marginal values to her. ...
... Thus, each good is liked by exactly two agents in their setting. As motivated in Christodoulou et al. [2023], a direct application of this setting is the allocation of geographical resources, for instance, natural resources among countries on the boundaries, working offices among research groups, and public areas among communities in a region, etc. Christodoulou et al. [2023] proved that EFX allocations always exist and can be computed in polynomial time for arbitrary graphs. Remarkably, this is one more rare case with more than three agents for which an EFX allocation is guaranteed to exist. ...
... Thus, each good is liked by exactly two agents in their setting. As motivated in Christodoulou et al. [2023], a direct application of this setting is the allocation of geographical resources, for instance, natural resources among countries on the boundaries, working offices among research groups, and public areas among communities in a region, etc. Christodoulou et al. [2023] proved that EFX allocations always exist and can be computed in polynomial time for arbitrary graphs. Remarkably, this is one more rare case with more than three agents for which an EFX allocation is guaranteed to exist. ...
We study envy-free up to any item (EFX) allocations on graphs where vertices and edges represent agents and items respectively. An agent is only interested in items that are incident to her and all other items have zero marginal values to her. Christodoulou et al. [EC, 2023] first proposed this setting and studied the case of goods. We extend this setting to the case of mixed manna where an item may be liked or disliked by its endpoint agents. In our problem, an agent has an arbitrary valuation over her incident items such that the items she likes have non-negative marginal values to her and those she dislikes have non-positive marginal values. We provide a complete study of the four notions of EFX for mixed manna in the literature, which differ by whether the removed item can have zero marginal value. We prove that an allocation that satisfies the notion of EFX where the virtually-removed item could always have zero marginal value may not exist and determining its existence is NP-complete, while one that satisfies any of the other three notions always exists and can be computed in polynomial time. We also prove that an orientation (i.e., a special allocation where each edge must be allocated to one of its endpoint agents) that satisfies any of the four notions may not exist, and determining its existence is NP-complete.
... In this paper, we present a collection of symmetrical results on the existence of EFX notion and its approximate variations. These results pertain to two seemingly distinct valuation settings: the restricted additive valuations and (p, q)-bounded valuations recently introduced by Christodoulou et al. [27]. In a (p, q)-bonuded instance, each good holds relevance (i.e., has a non-zero marginal value) for at most p agents, and any pair of agents share at most q common relevant goods. ...
... In a (p, q)-bonuded instance, each good holds relevance (i.e., has a non-zero marginal value) for at most p agents, and any pair of agents share at most q common relevant goods. The only known guarantees on (p, q)-bounded valuations is that (2, 1)-bounded instances always admit EFX allocations (EC'22) [27]. Here we show that instances with (∞, 1)bounded valuations always admit EF2X allocations, and EFX allocations with at most ⌊n/2⌋ − 1 discarded goods. ...
... For a comprehensive overview of these results, refer to Section 4. In recent advances related to EFX allocations, our focus has been drawn to two separate studies that provide EFX guarantees in distinct scenarios. The first study, conducted by Christodoulou et al. [27], introduces a family of valuation functions where each good holds relevance (i.e., has a nonzero marginal value) for at most p agents, and any pair of agents share at most q common relevant goods. There are several real-word motivations for this setting. ...
The existence of allocations stands as one of the main challenges in discrete fair division. In this paper, we present a collection of symmetrical results on the existence of notion and its approximate variations. These results pertain to two seemingly distinct valuation settings: the restricted additive valuations and (p,q)-bounded valuations recently introduced by Christodoulou \textit{et al.} \cite{christodoulou2023fair}. In a (p,q)-bonuded instance, each good holds relevance (i.e., has a non-zero marginal value) for at most p agents, and any pair of agents share at most q common relevant goods. The only known guarantees on (p,q)-bounded valuations is that (2,1)-bounded instances always admit allocations (EC'22) \cite{christodoulou2023fair}. Here we show that instances with -bounded valuations always admit allocations, and allocations with at most discarded goods. These results mirror the existing results for the restricted additive setting \cite{akrami2023efx}. Moreover, we present allocation algorithms for both the restricted additive and -bounded settings. The symmetry of these results suggests that these valuations exhibit symmetric structures. Building on this observation, we conjectured that the -bounded and restricted additive setting might admit guarantee. Intriguingly, our investigation confirms this conjecture. We propose a rather complex allocation algorithm for restricted additive valuations when p=2 and .
... However, in these works, the social network restricts the communication between agents and does not encode their preferences as in our case. A different line of work on fair division also considered the presence of graphs (Bouveret et al. 2017;Christodoulou et al. 2023;Zhou et al. 2024); however, here, the edges correspond to items and do not encode the preferences. ...
In the recently introduced model of fair partitioning of friends, there is a set of agents located on the vertices of an underlying graph that indicates the friendships between the agents. The task is to partition the graph into k balanced-sized groups, keeping in mind that the value of an agent for a group is equal to the number of edges they have in that group. The goal is to construct partitions that are "fair", i.e., no agent would like to replace an agent in a different group. We generalize the standard model by considering utilities for the agents that are beyond binary and additive. Having this as our foundation, our contribution is threefold: (a) we adapt several fairness notions that have been developed in the fair division literature to our setting; (b) we give several existence guarantees supported by polynomial-time algorithms; (c) we initiate the study of the computational (and parameterized) complexity of the model and provide an almost complete landscape of the (in)tractability frontier for our fairness concepts.
... EFX allocations have been shown to exist for any number of agents when these agents have identical valuations (Plaut and Roughgarden 2020), lexicographic preferences (Hosseini et al. 2021), additive valuations with at most two types of goods (Gorantla, Marwaha, and Velusamy 2023), as well as binary valuations (Halpern et al. 2020), subsequently extended to bi-valued valuations (Amanatidis et al. 2021). Christodoulou et al. (2023) proved EFX allocations exist when the agent's valuations are captured by a graph. ...
We study the fair allocation of indivisible goods among a group of agents, aiming to limit the envy between any two agents. The central open problem in this literature, which has proven to be extremely challenging, is regarding the existence of an EFX allocation, i.e., an allocation such that any envy from some agent i toward another agent j would vanish if we were to remove any single good from the bundle allocated to j. Prior work has shown that when the agents’ valuations are additive, which has been the main focus of prior works, an EFX allocation is guaranteed to exist for all instances involving up to three agents. Subsequent work extended this guarantee to more general valuations, like nice-cancelable and MMS-feasible. However, the existence of EFX allocations for instances involving four agents remains open, even for additive valuations. We contribute to this literature by focusing on EF2X, a relaxation of EFX which requires that any envy toward some agent would vanish if any two of the goods allocated to that agent were to be removed. Our main result shows that EF2X allocations exist for any instance with four agents, even for the class of cancelable valuations, which is more general than additive. Our proof is constructive, proposing an algorithm that computes such an allocation in pseudo-polynomial time. Furthermore, for instances involving three agents we provide an algorithm that computes an EF2X allocation in polynomial time, in contrast to EFX for which the fastest known algorithm for three agents is only pseudo-polynomial.
... Several works also explored fair division on graphs [10,13,14,15,18,21,22,46,48,61]. The closest model to ours is the one where we have a graph over items, each agent has certain utility for every item, and the goal is not only to find a fair allocation, but each bundle must additionally form a disjoint and connected sub-graphs. ...
Distributing services, goods, and tasks in the gig economy heavily relies upon on-demand workers (aka agents), leading to new challenges varying from logistics optimization to the ethical treatment of gig workers. We focus on fair and efficient distribution of delivery tasks -- placed on the vertices of a graph -- among a fixed set of agents. We consider the fairness notion of minimax share (MMS), which aims to minimize the maximum (submodular) cost among agents and is particularly appealing in applications without monetary transfers. We propose a novel efficiency notion -- namely non-wastefulness -- that is desirable in a wide range of scenarios and, more importantly, does not suffer from computational barriers. Specifically, given a distribution of tasks, we can, in polynomial time, i) verify whether the distribution is non-wasteful and ii) turn it into an equivalent non-wasteful distribution. Moreover, we investigate several fixed-parameter tractable and polynomial-time algorithms and paint a complete picture of the (parameterized) complexity of finding fair and efficient distributions of tasks with respect to both the structure of the topology and natural restrictions of the input. Finally, we highlight how our findings shed light on computational aspects of other well-studied fairness notions, such as envy-freeness and its relaxations.
... However, in these works, the social network restricts the communication between agents and does not encode their preferences as in our case. A different line of work on fair division also considered the presence of graphs [16,26,56,69]; however, here, the edges correspond to items and do not encode the preferences. ...
In the recently introduced model of fair partitioning of friends, there is a set of agents located on the vertices of an underlying graph that indicates the friendships between the agents. The task is to partition the graph into k balanced-sized groups, keeping in mind that the value of an agent for a group equals the number of edges they have in that group. The goal is to construct partitions that are "fair", i.e., no agent would like to replace an agent in a different group. We generalize the standard model by considering utilities for the agents that are beyond binary and additive. Having this as our foundation, our contribution is threefold (a) we adapt several fairness notions that have been developed in the fair division literature to our setting; (b) we give several existence guarantees supported by polynomial-time algorithms; (c) we initiate the study of the computational (and parameterized) complexity of the model and provide an almost complete landscape of the (in)tractability frontier for our fairness concepts.
... In a very recent independent work done in parallel with our work, HV et al. [2024] proved EFX allocations exist for any number of agents when there are at most three types of additive valuations. [Christodoulou et al., 2023] studied EFX allocations on graphs, and they proved that EFX allocations always exist for arbitrary graphs. Another line of research has aimed to achieve multiplicative approximations of EFX. ...
The two standard fairness notions in the resource allocation literature are proportionality and envy-freeness. If there are n agents competing for the available resources, then proportionality requires that each agent receives at least a 1/n fraction of their total value for the set of resources. On the other hand, envy-freeness requires that each agent weakly prefers the resources allocated to them over those allocated to any other agent. Each of these notions has its own benefits, but it is well known that neither one of the two is always achievable when the resources being allocated are indivisible. As a result, a lot of work has focused on satisfying fairness notions that relax either proportionality or envy-freeness. In this paper, we focus on MXS (a relaxation of proportionality) and EFL (a relaxation of envy-freeness). Each of these notions was previously shown to be achievable on its own [Barman et al.,2018, Caragiannis et al., 2023], and our main result is an algorithm that computes allocations that simultaneously satisfy both, combining the benefits of approximate proportionality and approximate envy-freeness. In fact, we prove this for any instance involving agents with valuation functions that are restricted MMS-feasible, which are more general than additive valuations. Also, since every EFL allocation directly satisfies other well-studied fairness notions like EF1, 1/2-EFX, 1/2-GMMS, and 2/3-PMMS, and every MXS allocation satisfies 4/7-MMS, the allocations returned by our algorithm simultaneously satisfy a wide variety of fairness notions and are, therefore, universally fair [Amanatidis et al., 2020].
... EFX allocations have been shown to exist for any number of agents when these agents have valuations that are identical [Plaut and Roughgarden, 2020], lexicographic preferences [Hosseini et al., 2021], additive valuations with at most two types of goods [Gorantla et al., 2023], as well as binary valuations [Halpern et al., 2020], which was subsequently extended to bi-valued valuations [Amanatidis et al., 2021]. [Christodoulou et al., 2023] studied EFX allocations on graphs, and they proved that EFX allocations always exist for arbitrary graphs. ...
We study the fair allocation of indivisible goods among a group of agents, aiming to limit the envy between any two agents. The central open problem in this literature, which has proven to be extremely challenging, is regarding the existence of an EFX allocation, i.e., an allocation such that any envy from some agent i toward another agent j would vanish if we were to remove any single good from the bundle allocated to j. When the agents' valuations are additive, which has been the main focus of prior works, Chaudhury et al. [2024] showed that an EFX allocation is guaranteed to exist for all instances involving up to three agents. Subsequently, Berger et al. [2022] extended this guarantee to nice-cancelable valuations and Akrami et al. [2023] to MMS-feasible valuations. However, the existence of EFX allocations for instances involving four agents remains open, even for additive valuations. We contribute to this literature by focusing on EF2X, a relaxation of EFX which requires that any envy toward some agent vanishes if any two of the goods allocated to that agent were to be removed. Our main result shows that EF2X allocations are guaranteed to exist for any instance with four agents, even for the class of cancelable valuations, which is more general than additive. Our proof is constructive, proposing an algorithm that computes such an allocation in pseudopolynomial time. Furthermore, for instances involving three agents we provide an algorithm that computes an EF2X allocation in polynomial time, in contrast to EFX, for which the fastest known algorithm for three agents is only pseudopolynomial.
... In a breakthrough result, Chaudhury et al. (2020) showed that complete, exact EFX allocations always exist for three agents under additive utilities, later improved to more general valuations Berger et al. (2022); Akrami et al. (2023a). Other works considered a limited number of items or agents sharing one of two possible valuation types Amanatidis et al. (2020); Mahara (2023Mahara ( , 2024, while a major line of work focused on restrictions on the valuation space, prompting a series of works on dichotomous valuations Amanatidis et al. (2021); Babaioff et al. (2021a); Benabbou et al. (2021); Halpern et al. (2020), lexicographic preferences Hosseini et al. (2021), graph instances Christodoulou et al. (2023), and allocations of multisets Gorantla et al. (2023). Recent work has shown that EFX allocations might not exist under general monotone valuations when the items to be allocated are chores Christoforidis and Santorinaios (2024). ...
We study the problem of fairly allocating indivisible goods among agents which are equipped with {\em leveled} valuation functions. Such preferences, that have been studied before in economics and fair division literature, capture a simple and intuitive economic behavior; larger bundles are always preferred to smaller ones. We provide a fine-grained analysis for various subclasses of leveled valuations focusing on two extensively studied notions of fairness, (approximate) MMS and EFX. In particular, we present a general positive result, showing the existence of 2/3-MMS allocations under valuations that are both leveled and submodular. We also show how some of our ideas can be used beyond the class of leveled valuations; for the case of two submodular (not necessarily leveled) agents we show that there always exists a 2/3-MMS allocation, complementing a recent impossibility result. Then, we switch to the case of subadditive and fractionally subadditive leveled agents, where we are able to show tight (lower and upper) bounds of 1/2 on the approximation factor of MMS. Moreover, we show the existence of exact EFX allocations under general leveled valuations via a simple protocol that in addition satisfies several natural economic properties. Finally, we take a mechanism design approach and we propose protocols that are both truthful and approximately fair under leveled valuations.
We study truthful mechanisms for allocation problems in graphs, both for the minimization (i.e., scheduling) and maximization (i.e., auctions) setting. The minimization problem is a special case of the well-studied unrelated machines scheduling problem, in which every given task can be executed only by two pre-specified machines in the case of graphs or a given subset of machines in the case of hypergraphs. This corresponds to a multigraph whose nodes are the machines and its hyperedges are the tasks. This class of problems belongs to multidimensional mechanism design, for which there are no known general mechanisms other than the VCG and its generalization to affine minimizers. We propose a new class of truthful mechanisms that have significantly better performance than affine minimizers in many settings. Specifically, we provide upper and lower bounds for truthful mechanisms for general multigraphs, as well as special classes of graphs such as stars, trees, planar graphs, k -degenerate graphs, and graphs of a given treewidth. We also consider the objective of minimizing or maximizing the L p -norm of the values of the players, a generalization of the makespan minimization that corresponds to p = ∞, and extend the results to any p > 0.
We study the problem of allocating a set of indivisible goods to a set of agents with additive valuation functions, aiming to achieve approximate envy-freeness up to any good (-EFX). The state-of-the-art results on the problem include that (exact) EFX allocations exist when (a) there are at most three agents, or (b) the agents' valuation functions can take at most two values, or (c) the agents' valuation functions can be represented via a graph. For -EFX, it is known that a 0.618-EFX allocation exists for any number of agents with additive valuation functions. In this paper, we show that 2/3-EFX allocations exist when (a) there are at most \emph{seven agents}, (b) the agents' valuation functions can take at most \emph{three values}, or (c) the agents' valuation functions can be represented via a \emph{multigraph}. Our results can be interpreted in two ways. First, by relaxing the notion of EFX to 2/3-EFX, we obtain existence results for strict generalizations of the settings for which exact EFX allocations are known to exist. Secondly, by imposing restrictions on the setting, we manage to beat the barrier of 0.618 and achieve an approximation guarantee of 2/3. Therefore, our results push the \emph{frontier} of existence and computation of approximate EFX allocations, and provide insights into the challenges of settling the existence of exact EFX allocations.
We study the problem of fairly allocating a set of chores to a group of agents. The existence of envy-free up to any item (EFX) allocations is a long-standing open question for both goods and chores. We resolve this question by providing a negative answer for the latter, presenting a simple construction that admits no EFX solutions for allocating six items to three agents equipped with superadditive cost functions, thus proving a separation result between goods and bads. In fact, we uncover a deeper insight, showing that the instance has unbounded approximation ratio. Moreover, we show that deciding whether an EFX allocation exists is NP-complete. On the positive side, we establish the existence of EFX allocations under general monotone cost functions when the number of items is at most n+2. We then shift our attention to additive cost functions. We employ a general framework in order to improve the approximation guarantees in the well-studied case of three additive agents, and provide several conditional approximation bounds that leverage ordinal information.
We consider the problem of allocating a set on indivisible items to players with private preferences in an efficient and fair way. We focus on valuations that have dichotomous marginals, in which the added value of any item to a set is either 0 or 1, and aim to design truthful allocation mechanisms (without money) that maximize welfare and are fair. For the case that players have submodular valuations with dichotomous marginals, we design such a deterministic truthful allocation mechanism. The allocation output by our mechanism is Lorenz dominating, and consequently satisfies many desired fairness properties, such as being envy-free up to any item (EFX), and maximizing the Nash Social Welfare (NSW). We then show that our mechanism with random priorities is envy-free ex-ante, while having all the above properties ex-post. Furthermore, we present several impossibility results precluding similar results for the larger class of XOS valuations.
We study the problem of fairly allocating a set of indivisible goods to a set of n agents. Envy-freeness up to any good (EFX) criterion (which requires that no agent prefers the bundle of another agent after the removal of any single good) is known to be a remarkable analogue of envy-freeness when the resource is a set of indivisible goods. In this paper, we investigate EFX for restricted additive valuations, that is, every good has a non-negative value, and every agent is interested in only some of the goods. We introduce a natural relaxation of EFX called EFkX which requires that no agent envies another agent after the removal of any k goods. Our main contribution is an algorithm that finds a complete (i.e., no good is discarded) EF2X allocation for restricted additive valuations. In our algorithm we devise new concepts, namely configuration and envy-elimination that might be of independent interest. We also use our new tools to find an EFX allocation for restricted additive valuations that discards at most n/2 -1 goods.
We study envy-free allocations of indivisible goods to agents in settings where each agent is unaware of the goods allocated to other agents. In particular, we propose the maximin aware (MMA) fairness measure, which guarantees that every agent, given the bundle allocated to her, is aware that she does not envy at least one other agent, even if she does not know how the other goods are distributed among other agents. We also introduce two of its relaxations, and discuss their egalitarian guarantee and existence. Finally, we present a polynomial-time algorithm, which computes an allocation that approximately satisfies MMA or its relaxations. Interestingly, the returned allocation is also 1/2-approximate EFX when all agents have sub- additive valuations, which improves the algorithm in [Plaut and Roughgarden, 2018].
How long does it take until economic agents converge to an equilibrium? By studying the complexity of the problem of computing a mixed Nash equilibrium in a game, we provide evidence that there are games in which convergence to such an equilibrium takes prohibitively long. Traditionally, computational problems fall into two classes: those that have a polynomial-time algorithm and those that are NP-hard. However, the concept of NP-hardness cannot be applied to the rare problems where "every instance has a solution"--for example, in the case of games Nash's theorem asserts that every game has a mixed equilibrium (now known as the Nash equilibrium, in honor of that result). We show that finding a Nash equilibrium is complete for a class of problems called PPAD, containing several other known hard problems; all problems in PPAD share the same style of proof that every instance has a solution.
Envy freeness is one of the most widely studied notions in fair division. Because envy-free allocations do not always exist when items are indivisible, several relaxations have been considered. Among them, possibly the most compelling notion is envy freeness up to any item (EFX). Informally speaking, EFX requires that no agent i envies another agent j after the removal of any item in j’s bundle. The existence of EFX allocations is a major open problem. We study the existence of EFX allocations when agents have general valuations. For general valuations, it is known that an EFX allocation always exists (i) when n = 2 or (ii) when all agents have identical valuations, where n is the number of agents. It is also known that an EFX allocation always exists when one can leave at most n − 1 items unallocated. We develop new techniques and extend some results of additive valuations to general valuations on the existence of EFX allocations. We show that an EFX allocation always exists (i) when all agents have one of two general valuations or (ii) when the number of items is at most n + 3. We also show that an EFX allocation always exists when one can leave at most n − 2 items unallocated. In addition to the positive results, we construct an instance with n = 3, in which an existing approach does not work.
Funding: This work was partially supported by Kyoto University and Toyota Motor Corporation [Joint Project “Advanced Mathematical Science for Mobility Society”].
Envy-freeness up to any good (EFX) provides a strong and intuitive guarantee of fairness in the allocation of indivisible goods. But whether such allocations always exist or whether they can be efficiently computed remains an important open question. We study the existence and computation of EFX in conjunction with various other economic properties under lexicographic preferences--a well-studied preference restriction model in artificial intelligence and economics. In sharp contrast to the known results for additive valuations, we not only prove the existence of EFX and Pareto optimal allocations, but in fact provide an algorithmic characterization of these two properties. We also characterize the mechanisms that are, in addition, strategyproof, non-bossy, and neutral. When the efficiency notion is strengthened to rank-maximality, we obtain non-existence and computational hardness results, and show that tractability can be restored when EFX is relaxed to another well-studied fairness notion called maximin share guarantee (MMS).
The existence of EFX allocations of goods is a major open problem in fair division, even for additive valuations. The current state of the art is that no setting where EFX allocations are impossible is known, and yet, existence results are known only for very restricted settings, such as: (i) agents with identical valuations, (ii) 2 agents, and (iii) 3 agents with additive valuations. It is also known that EFX exists if one can leave n-1 items unallocated, where n is the number of agents. We develop new techniques that allow us to push the boundaries of the enigmatic EFX problem beyond these known results, and (arguably) to simplify proofs of earlier results. Our main result is that every setting with 4 additive agents admits an EFX allocation that leaves at most a single item unallocated. Beyond our main result, we introduce a new class of valuations, termed nice cancelable, which includes additive, unit-demand, budget-additive and multiplicative valuations, among others. Using our new techniques, we show that both our results and previous results for additive valuations extend to nice cancelable valuations.
We consider the classic problem of fairly allocating indivisible goods among agents with additive valuation functions and explore the connection between two prominent fairness notions: maximum Nash welfare (MNW) and envy-freeness up to any good (EFX). We establish that an MNW allocation is always EFX as long as there are at most two possible values for the goods, whereas this implication is no longer true for three or more distinct values. As a notable consequence, this proves the existence of EFX allocations for these restricted valuation functions. While the efficient computation of an MNW allocation for two possible values remains an open problem, we present a novel algorithm for directly constructing EFX allocations in this setting. Finally, we study the question of whether an MNW allocation implies any EFX guarantee for general additive valuation functions under a natural new interpretation of approximate EFX allocations.
Several relaxations of envy-freeness, tailored to fair division in settings with indivisible goods, have been introduced within the last decade. Due to the lack of general existence results for most of these concepts, great attention has been paid to establishing approximation guarantees. In this work, we propose a simple algorithm that is universally fair in the sense that it returns allocations that have good approximation guarantees with respect to four such fairness notions at once. In particular, this is the first algorithm achieving a (ϕ−1)-approximation of envy-freeness up to any good () and a 2ϕ+2-approximation of groupwise maximin share fairness (), where ϕ is the golden ratio (ϕ≈1.618). The best known approximation factor, in polynomial time, for either one of these fairness notions prior to this work was 1/2. Moreover, the returned allocation achieves envy-freeness up to one good () and a 2/3-approximation of pairwise maximin share fairness (). While is our primary focus, we also exhibit how to fine-tune our algorithm and further improve the guarantees for or .
Finally, we show that —and thus and —allocations always exist when the number of goods does not exceed the number of agents by more than two.
The maximum Nash welfare (MNW) solution—which selects an allocation that maximizes the product of utilities—is known to provide outstanding fairness guarantees when allocating divisible goods. And while it seems to lose its luster when applied to indivisible goods, we show that, in fact, the MNW solution is strikingly fair even in that setting. In particular, we prove that it selects allocations that are envy-free up to one good—a compelling notion that is quite elusive when coupled with economic efficiency. We also establish that the MNW solution provides a good approximation to another popular (yet possibly infeasible) fairness property, the maximin share guarantee, in theory and—even more so—in practice. While finding the MNW solution is computationally hard, we develop a nontrivial implementation and demonstrate that it scales well on real data. These results establish MNW as a compelling solution for allocating indivisible goods and underlie its deployment on a popular fair-division website.
Several fairness concepts have been proposed recently in attempts to approximate envy-freeness in settings with indivisible goods. Among them, the concept of envy-freeness up to any item (EFX) is arguably the closest to envy-freeness. Unfortunately, EFX allocations are not known to exist except in a few special cases. We make significant progress in this direction. We show that for every instance with additive valuations, there is an EFX allocation of a subset of items with a Nash welfare that is at least half of the maximum possible Nash welfare for the original set of items. That is, after donating some items to a charity, one can distribute the remaining items in a fair way with high efficiency. This bound is proved to be best possible. Our proof is constructive and highlights the importance of maximum Nash welfare allocation. Starting with such an allocation, our algorithm decides which items to donate and redistributes the initial bundles to the agents, eventually obtaining an allocation with the claimed efficiency guarantee. The application of our algorithm to large markets, where the valuations of an agent for every item is relatively small, yields EFX with almost optimal Nash welfare. We also show that our algorithm can be modified to compute, in polynomial-time, EFX allocations that approximate optimal Nash welfare within a factor of at most 2ρ, using a ρ-approximate allocation on input instead of the maximum Nash welfare one.
The goal of fair division is to distribute resources among competing players in a "fair" way. Envy-freeness is the most extensively studied fairness notion in fair division. Envy-free allocations do not always exist with indivisible goods, motivating the study of relaxed versions of envy-freeness. We study the envy-freeness up to any good (EFX) property, which states that no player prefers the bundle of another player following the removal of any single good, and prove the first general results about this property. We use the leximin solution to show existence of EFX allocations in several contexts, sometimes in conjunction with Pareto optimality. For two players with valuations obeying a mild assumption, one of these results provides stronger guarantees than the currently deployed algorithm on Spliddit, a popular fair division website. Unfortunately, finding the leximin solution can require exponential time. We show that this is necessary by proving an exponential lower bound on the number of value queries needed to identify an EFX allocation, even for two players with identical valuations. We consider both additive and more general valuations, and our work suggests that there is a rich landscape of problems to explore in the fair division of indivisible goods with different classes of player valuations.
This paper proposes a new mechanism for combinatorial assignment—for example, assigning schedules of courses to students—based on an approximation to competitive equilibrium from equal incomes (CEEI) in which incomes are unequal but arbitrarily close together. The main technical result is an existence theorem for approximate CEEI. The mechanism is approximately efficient, satisfies two new criteria of outcome fairness, and is strategyproof in large markets. Its performance is explored on real data, and it is compared to alternatives from theory and practice: all other known mechanisms are either unfair ex post or manipulable even in large markets, and most are both manipulable and unfair.
We study the problem of fairly allocating a set of indivisible goods to a set of people from an algorithmic perspective. fair division has been a central topic in the economic literature and several concepts of fairness have been suggested. The criterion that we focus on is envy-freeness. In our model, a monotone utility function is associated with every player specifying the value of each subset of the goods for the player. An allocation is envy-free if every player prefers her own share than the share of any other player. When the goods are divisible, envy-free allocations always exist. In the presence of indivisibilities, we show that there exist allocations in which the envy is bounded by the maximum marginal utility, and present a simple algorithm for computing such allocations. We then look at the optimization problem of finding an allocation with minimum possible envy. In the general case the problem is not solvable or approximable in polynomial time unless P = NP. We consider natural special cases (e.g.additive utilities) which are closely related to a class of job scheduling problems. Approximation algorithms as well as inapproximability results are obtained. Finally we investigate the problem of designing truthful mechanisms for producing allocations with bounded envy.
We consider algorithmic problems in a distributed setting where the participants cannot be assumed to follow the algorithm but rather their own self-interest. As such participants, termed agents, are capable of manipulating the algorithm, the algorithm designer should ensure in advance that the agents' interests are best served by behaving correctly. Following notions from the field of mechanism design, we suggest a framework for studying such algorithms. In this model the algorithmic solution is adorned with payments to the participants and is termed a mechanism. The payments should be carefully chosen as to motivate all participants to act as the algorithm designer wishes. We apply the standard tools of mechanism design to algorithmic problems and in particular to the shortest path problem. Our main technical contribution concerns the study of a representative problem, task scheduling, for which the standard tools do not suffice. We present several theorems regarding this problem including an approximation mechanism, lower bounds and a randomized mechanism. We also suggest and motivate extensions to the basic model and prove improved upper bounds in the extended model. Many open problems are suggested as well. 1
We prove that Bimatrix, the problem of finding a Nash equilibrium in a two-player game, is complete for the complexity class PPAD (Polynomial Parity Argument, Directed version) introduced by Papadimitriou in 1991.
Our result, building upon the work of Daskalakis et al. [2006a] on the complexity of four-player Nash equilibria, settles a long standing open problem in algorithmic game theory. It also serves as a starting point for a series of results concerning the complexity of two-player Nash equilibria. In particular, we prove the following theorems:
—Bimatrix does not have a fully polynomial-time approximation scheme unless every problem in PPAD is solvable in polynomial time.
—The smoothed complexity of the classic Lemke-Howson algorithm and, in fact, of any algorithm for Bimatrix is not polynomial unless every problem in PPAD is solvable in randomized polynomial time.
Our results also have a complexity implication in mathematical economics:
—Arrow-Debreu market equilibria are PPAD -hard to compute.
2022 a. EFX Allocations: Simplifications and Improvements
- Hannaneh Akrami
- Noga Alon
- Jugal Bhaskar Ray Chaudhury
- Kurt Garg
- Ruta Mehlhorn
- Mehta
- Akrami Hannaneh
Benjamin Aram Berendsohn, Simona Boyadzhiyska, and László Kozma. 2022. Fixed-point cycles and EFX allocations
- Simona Benjamin Aram Berendsohn
- László Boyadzhiyska
- Kozma
- Berendsohn Benjamin Aram
Pranay Gorantla Kunal Marwaha and Santhoshini Velusamy. 2023. Fair allocation of a multiset of indivisible items
- Kunal Pranay Gorantla
- Santhoshini Marwaha
- Velusamy
Hervé Moulin , Alexandros A. Voudouris , and Xiaowei Wu. 2022. Fair Division of Indivisible Goods: A Survey
- Georgios Amanatidis
- Haris Aziz
- Georgios Birmpas
- Aris Filos-Ratsikas
- Bo Li
- Amanatidis Georgios