EF1-Constrained Nash Social Welfare with Identical Additive Valuations: Complexity, Guarantees, and Experiments

arXiv:2609.03846v1 Announce Type: cross Abstract: We study the allocation of indivisible goods among agents with identical additive valuations, focusing on envy-freeness up to one good (EF1) and Nash social welfare (NSW). Since every maximum-NSW allocation is EF1 under additive valuations, the associated threshold problem inherits the known strong NP-hardness of NSW maximization under identical additive valuations and is strongly NP-complete. We therefore focus on welfare guarantees satisfied by arbitrary EF1 allocations. Although every such allocation is known to achieve an e^{-1/e}-approximation to the unrestricted optimal NSW, we identify conditions yielding stronger guarantees. Under uniform valuations, every EF1 allocation is NSW-optimal. Under an arepsilon-small-item condition, every EF1 allocation achieves an explicit approximation ratio rho_n(arepsilon) satisfying rho_n(arepsilon) = 1-O(arepsilon^2) as arepsilono 0 for fixed n. We further consider the stronger sequential requirement that EF1 be maintained after every item assignment. For this setting, we propose PriorityNet, a deep reinforcement learning framework trained using Proximal Policy Optimization and equipped with prospective EF1 action masking. The mask restricts every decision to assignments that preserve EF1, thereby guaranteeing prefix-wise EF1 by construction without post-processing repair. Across 3,000 test instances in each of the offline and random-order online regimes (nin[2,20] and min[5,100]), PriorityNet attains mean normalized operatorname{NSW} values of 0.9911 and 0.9701, respectively. Relative to offline Longest Processing Time (LPT) and online least-valued-bundle baselines, it achieves instance-wise win-minus-loss rates of +27.10% and +17.87%, while matching the offline baseline's mean normalized welfare to four decimal places and modestly improving the online mean from 0.9694 to 0.9701.

Source: arXiv cs.LG | 2026-09-04

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