Algorithms for Competitive Division of Chores

We study the problem of allocating divisible bads (chores) among multiple agents with additive utilities, when money transfers are not allowed. The competitive rule is known to be the best mechanism for goods with additive utilities and was recently extended to chores by Bogomolnaia et al (2017). For both goods and chores, the rule produces Pareto optimal and envy-free allocations. In the case of goods, the outcome of the competitive rule can be easily computed. Competitive allocations solve the Eisenberg-Gale convex program; hence the outcome is unique and can be approximately found by standard gradient methods. An exact algorithm that runs in polynomial time in the number of agents and goods was given by Orlin. In the case of chores, the competitive rule does not solve any convex optimization problem; instead, competitive allocations correspond to local minima, local maxima, and saddle points of the Nash Social Welfare on the Pareto frontier of the set of feasible utilities. The rule becomes multivalued and none of the standard methods can be applied to compute its outcome. In this paper, we show that all the outcomes of the competitive rule for chores can be computed in strongly polynomial time if either the number of agents or the number of chores is fixed. The approach is based on a combination of three ideas: all consumption graphs of Pareto optimal allocations can be listed in polynomial time; for a given consumption graph, a candidate for a competitive allocation can be constructed via explicit formula; and a given allocation can be checked for being competitive using a maximum flow computation as in Devanur et al (2002). Our algorithm immediately gives an approximately-fair allocation of indivisible chores by the rounding technique of Barman and Krishnamurthy (2018).Comment: 38 pages, 4 figure

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 and Efficient Allocations under Subadditive Valuations

We study the problem of allocating a set of indivisible goods among agents with subadditive valuations in a fair and efficient manner. Envy-Freeness up to any good (EFX) is the most compelling notion of fairness in the context of indivisible goods. Although the existence of EFX is not known beyond the simple case of two agents with subadditive valuations, some good approximations of EFX are known to exist, namely $\tfrac{1}{2}$-EFX allocation and EFX allocations with bounded charity. Nash welfare (the geometric mean of agents' valuations) is one of the most commonly used measures of efficiency. In case of additive valuations, an allocation that maximizes Nash welfare also satisfies fairness properties like Envy-Free up to one good (EF1). Although there is substantial work on approximating Nash welfare when agents have additive valuations, very little is known when agents have subadditive valuations. In this paper, we design a polynomial-time algorithm that outputs an allocation that satisfies either of the two approximations of EFX as well as achieves an $\mathcal{O}(n)$ approximation to the Nash welfare. Our result also improves the current best-known approximation of $\mathcal{O}(n \log n)$ and $\mathcal{O}(m)$ to Nash welfare when agents have submodular and subadditive valuations, respectively. Furthermore, our technique also gives an $\mathcal{O}(n)$ approximation to a family of welfare measures, $p$-mean of valuations for $p\in (-\infty, 1]$, thereby also matching asymptotically the current best known approximation ratio for special cases like $p =-\infty$ while also retaining the fairness properties