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We study fair division of indivisible chores among n agents with additive disutility functions. Two well-studied fairness notions for indivisible items are envy-freeness up to one/any item (EF1/EFX) and the standard notion of economic efficiency is Pareto optimality (PO). There is a noticeable gap between the results known for both EF1 and EFX in the goods and chores settings. The case of chores turns out to be much more challenging. We reduce this gap by providing slightly relaxed versions of the known results on goods for the chores setting. Interestingly, our algorithms run in polynomial time, unlike their analogous versions in the goods setting.We introduce the concept of k surplus in the chores setting which means that up to k more chores are allocated to the agents and each of them is a copy of an original chore. We present a polynomial-time algorithm which gives EF1 and PO allocations with n-1 surplus.We relax the notion of EFX slightly and define tEFX which requires that the envy from agent i to agent j is removed upon the transfer of any chore from the i's bundle to j's bundle. We give a polynomial-time algorithm that in the chores case for 3 agents returns an allocation which is either proportional or tEFX. Note that proportionality is a very strong criterion in the case of indivisible items, and hence both notions we guarantee are desirable.

 

We study the fair and efficient allocation of a set of indivisible goods among agents, where each good has several copies, and each agent has an additively separable concave valuation function with a threshold. These valuations capture the property of diminishing marginal returns, and they are more general than the well-studied case of additive valuations. We present a polynomial-time algorithm that approximates the optimal Nash social welfare (NSW) up to a factor of e1/e ≈ 1.445. This matches with the state-of-the-art approximation factor for additive valuations. The computed allocation also satisfies the popular fairness guarantee of envy-freeness up to one good (EF1) up to a factor of 2 + ε. For instances without thresholds, it is also approximately Pareto-optimal. For instances satisfying a large market property, we show an improved approximation factor. Lastly, we show that the upper bounds on the optimal NSW introduced in Cole and Gkatzelis (2018) and Barman et al. (2018) have the same value. 

We consider the task of assigning indivisible goods to a set of agents in a fair manner. Our notion of fairness is Nash social welfare, i.e., the goal is to maximize the geometric mean of the utilities of the agents. Each good comes in multiple items or copies, and the utility of an agent diminishes as it receives more items of the same good. The utility of a bundle of items for an agent is the sum of the utilities of the items in the bundle. Each agent has a utility cap beyond which he does not value additional items. We give a polynomial time approximation algorithm that maximizes Nash social welfare up to a factor of e^{1/{e}} ~ 1.445. The computed allocation is Pareto-optimal and approximates envy-freeness up to one item up to a factor of 2 + epsilon.