Approximately EFX allocations for indivisible chores

In this paper, we study how to fairly allocate a set of m indivisible chores to a group of n agents, each of which has a general additive cost function on the items. Since envy-free (EF) allocations are not guaranteed to exist, we consider the notion of envy-freeness up to any item (EFX). In contrast to the fruitful results regarding the (approximation of) EFX allocations for goods, very little is known for the allocation of chores. Prior to our work, for the allocation of chores, it is known that EFX allocations always exist for two agents or general number of agents with identical ordering cost functions. For general instances, no non-trivial approximation result regarding EFX allocation is known. In this paper, we make progress in this direction by providing several polynomial time algorithms for the computation of EFX and approximately EFX allocations. We root show that for three agents we can always compute a (2 + 6) -approximation of EFX allocation. For n >= 4 agents, our algorithm always computes a (3n2 - n) -approximation. We also study the bi-valued instances, in which agents have at most two cost values on the chores. For three agents, we provide an algorithm for the computation of EFX allocations. For n >= 4 agents, we present algorithms for the computation of partial EFX allocations with at most (n - 1) unallocated items; and (n - 1) -approximation of EFX allocations.