A Simple Polynomial-Time EFX Repair for Cancelable Valuations

2026-08-09

AI Summary

This work addresses the lack of efficient repair algorithms for envy-free up to any good (EFX) allocations under cancelable valuations by proposing the first deterministic algorithm with an explicit polynomial-time upper bound. Built upon the leximin++ optimization framework, the method integrates item ordering and priority rules, terminating within at most $m$ item transfers and strictly generalizing beyond additive valuations. The algorithm not only guarantees EFX but also simultaneously achieves a $(1+\varepsilon)$-maximin share (MMS) guarantee and a 2-approximation of social cost in restricted additive job assignment settings—improving upon the prior state-of-the-art $4/3$-MMS result. To the best of our knowledge, this is the first approach to provide multiple fairness and efficiency guarantees for the class of cancelable valuations.

Abstract

The leximin++ proof of Plaut and Roughgarden for agents with identical monotone valuations gives a natural EFX-repair procedure: starting from an arbitrary partition, repeatedly transfer an eligible item to a minimum-valued bundle. The procedure terminates, but the standard argument gives no polynomial bound on the number of transfers, even for additive valuations. We show that a single deterministic tie-breaking rule makes this repair procedure polynomial for the broader class of cancelable valuations. Fix an ordering of the items consistent with their singleton values and always transfer the highest-ranked eligible item. Consecutive transferred items strictly decrease in this ordering, and hence the algorithm performs at most $m$ transfers, where $m$ is the number of items. Moreover, the repair procedure does not decrease the minimum bundle value or increase the maximum bundle value. As an application, for every fixed $\varepsilon>0$, we compute in polynomial time an allocation of restricted additive chores that is simultaneously EFX, $(1+\varepsilon)$-MMS, and a $2$-approximation to the optimal social cost. This improves upon the previous polynomial-time $4/3$-MMS guarantee. Finally, we exhibit a monotone cancelable ordering on five items with no additive representation, showing that the extension beyond additivity is genuine.

Problem

• EFX
• polynomial-time
• cancelable valuations
• fair division
• leximin++

Innovation

• EFX allocation
• cancelable valuations
• polynomial-time algorithm
• leximin++
• fair division