On the Complexity of Chore Division

We study the proportional chore division problem where a protocol wants to divide an undesirable object, called chore, among $n$ different players. The goal is to find an allocation such that the cost of the chore assigned to each player be at most $1/n$ of the total cost. This problem is the dual variant of the cake cutting problem in which we want to allocate a desirable object. Edmonds and Pruhs showed that any protocol for the proportional cake cutting must use at least $\Omega(n \log n)$ queries in the worst case, however, finding a lower bound for the proportional chore division remained an interesting open problem. We show that chore division and cake cutting problems are closely related to each other and provide an $\Omega(n \log n)$ lower bound for chore division

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

Fair Allocation based on Diminishing Differences

Ranking alternatives is a natural way for humans to explain their preferences. It is being used in many settings, such as school choice, course allocations and residency matches. In some cases, several `items' are given to each participant. Without having any information on the underlying cardinal utilities, arguing about fairness of allocation requires extending the ordinal item ranking to ordinal bundle ranking. The most commonly used such extension is stochastic dominance (SD), where a bundle X is preferred over a bundle Y if its score is better according to all additive score functions. SD is a very conservative extension, by which few allocations are necessarily fair while many allocations are possibly fair. We propose to make a natural assumption on the underlying cardinal utilities of the players, namely that the difference between two items at the top is larger than the difference between two items at the bottom. This assumption implies a preference extension which we call diminishing differences (DD), where X is preferred over Y if its score is better according to all additive score functions satisfying the DD assumption. We give a full characterization of allocations that are necessarily-proportional or possibly-proportional according to this assumption. Based on this characterization, we present a polynomial-time algorithm for finding a necessarily-DD-proportional allocation if it exists. Using simulations, we show that with high probability, a necessarily-proportional allocation does not exist but a necessarily-DD-proportional allocation exists, and moreover, that allocation is proportional according to the underlying cardinal utilities. We also consider chore allocation under the analogous condition --- increasing-differences.Comment: Revised version, based on very helpful suggestions of JAIR referees. Gaps in some proofs were filled, more experiments were done, and mor

On Approximate Envy-Freeness for Indivisible Chores and Mixed Resources

We study the fair allocation of undesirable indivisible items, or chores. While the case of desirable indivisible items (or goods) is extensively studied, with many results known for different notions of fairness, less is known about the fair division of chores. We study envy-free allocation of chores and make three contributions. First, we show that determining the existence of an envy-free allocation is NP-complete even in the simple case when agents have binary additive valuations. Second, we provide a polynomial-time algorithm for computing an allocation that satisfies envy-freeness up to one chore (EF1), correcting a claim in the existing literature. A modification of our algorithm can be used to compute an EF1 allocation for doubly monotone instances (where each agent can partition the set of items into objective goods and objective chores). Our third result applies to a mixed resources model consisting of indivisible items and a divisible, undesirable heterogeneous resource (i.e., a bad cake). We show that there always exists an allocation that satisfies envy-freeness for mixed resources (EFM) in this setting, complementing a recent result of Bei et al. [Bei et al., 2021] for indivisible goods and divisible cake

Almost Envy-Free Allocations of Indivisible Goods or Chores with Entitlements

We here address the problem of fairly allocating indivisible goods or chores to $n$ agents with weights that define their entitlement to the set of indivisible resources. Stemming from well-studied fairness concepts such as envy-freeness up to one good (EF1) and envy-freeness up to any good (EFX) for agents with equal entitlements, we present, in this study, the first set of impossibility results alongside algorithmic guarantees for fairness among agents with unequal entitlements. Within this paper, we expand the concept of envy-freeness up to any good or chore to the weighted context (WEFX and XWEF respectively), demonstrating that these allocations are not guaranteed to exist for two or three agents. Despite these negative results, we develop a WEFX procedure for two agents with integer weights, and furthermore, we devise an approximate WEFX procedure for two agents with normalized weights. We further present a polynomial-time algorithm that guarantees a weighted envy-free allocation up to one chore (1WEF) for any number of agents with additive cost functions. Our work underscores the heightened complexity of the weighted fair division problem when compared to its unweighted counterpart.Comment: Appears in the 38th AAAI Conference on Artificial Intelligence (AAAI), 202

Externalities in Chore Division

The chore division problem simulates the fair division of a heterogeneous, undesirable resource among several agents. In the fair division of chores, each agent only gets the disutility from its own piece. Agents may, however, also be concerned with the pieces given to other agents; these externalities naturally appear in fair division situations. We first demonstrate the generalization of the classical concepts of proportionality and envy-freeness while extending the classical model by taking externalities into account

Fair Interval Scheduling of Indivisible Chores

We study the problem of fairly assigning a set of discrete tasks (or chores) among a set of agents with additive valuations. Each chore is associated with a start and finish time, and each agent can perform at most one chore at any given time. The goal is to find a fair and efficient schedule of the chores, where fairness pertains to satisfying envy-freeness up to one chore (EF1) and efficiency pertains to maximality (i.e., no unallocated chore can be feasibly assigned to any agent). Our main result is a polynomial-time algorithm for computing an EF1 and maximal schedule for two agents under monotone valuations when the conflict constraints constitute an arbitrary interval graph. The algorithm uses a coloring technique in interval graphs that may be of independent interest. For an arbitrary number of agents, we provide an algorithm for finding a fair schedule under identical dichotomous valuations when the constraints constitute a path graph. We also show that stronger fairness and efficiency properties, including envy-freeness up to any chore (EFX) along with maximality and EF1 along with Pareto optimality, cannot be achieved