Minimizing Cumulative Envy in Allocating a Sequence of Items

2026-10-07

AI Summary

This study addresses the irreversible allocation of sequential items by investigating how to minimize cumulative envy in order to quantify the magnitude and duration of unfairness. It introduces a novel metric, cumulative maximum envy, and establishes its strong NP-completeness as well as the non-existence of constant-factor approximations. To tackle this problem, the authors employ dynamic programming, fixed-parameter tractability analysis, and greedy algorithms. The work delineates theoretical boundaries for fair allocation along the temporal dimension, providing pseudo-polynomial solutions and a fully polynomial-time approximation scheme (FPTAS) for a constant number of agents. Furthermore, it achieves exact polynomial-time solutions under identical or binary valuation settings. For the sequential ordering variant, the proposed greedy algorithm attains approximation ratios of 3/2 and n/(n−1).

Abstract

We study temporal fair division with indivisible goods that arrive sequentially and must be allocated irrevocably. In contrast to the usual online model, we assume that valuations and future arrivals are known in advance, and ask how unfairness evolves during the process. We introduce \emph{cumulative maximum envy}: the sum, over all rounds, of the maximum pairwise envy at that round. Equivalently, this is the area under the worst-envy curve, and it captures both the magnitude and the duration of envy. For a fixed arrival order, we show that the corresponding decision problem is strongly NP-complete and that minimizing this objective admits no constant-factor approximation unless P = NP, even under identical valuations and even under binary valuations. We complement these hardness results with a dynamic program that gives pseudopolynomial-time solvability for a constant number of agents, polynomial-time algorithms in further restricted settings, and an FPTAS for fixed $n$ under identical integer valuations. We then study a sequencing variant where the algorithm may choose the arrival order. This variant remains NP-complete even for two agents with identical valuations; however, a simple greedy algorithm achieves a $3/2$-approximation for $n=2$ agents, an $n/(n-1)$-approximation for any number of agents, and an additive guarantee depending on the maximum value of any good.

Problem

• temporal fair division
• indivisible goods
• cumulative envy
• online allocation
• sequencing

Innovation

• Cumulative Maximum Envy
• Temporal Fair Division
• Online Allocation
• Dynamic Programming
• Approximation Algorithms