On The Pursuit of EFX for Chores: Non-Existence and Approximations
Vasilis Christoforidis, Christodoulos Santorinaios
TL;DR
This paper investigates EFX fairness for chores, establishing a separation from goods by proving non-existence and hardness results even for small problem instances under general monotone and superadditive costs, and proving that deciding EFX existence is NP-hard. It then identifies a positive frontier where EFX allocations exist when the number of chores satisfies m \\le n+2 under general monotone costs, and develops an approximation framework for additive costs that improves EFX guarantees, including a tight 2-EFX result for the case of three additive agents. The main methods combine explicit hard instances, reductions from partition problems, and an Envy Cycle Elimination-based framework (TTECE) extended to chores with a ratio-maintenance analysis. The findings provide both separation results between goods and bads and practical approximation techniques, with implications for understanding the limits of fair division with indivisible chores and guiding future research on exact EFX existence and tighter additive-cost guarantees.
Abstract
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.