Resolving Envy by Adding Goods with Bounded Supply: A Type-Count Dichotomy and Two-Agent Hardness

2026-08-10

AI Summary

This study investigates the computational complexity of achieving envy-free allocations by adding items when item supplies are limited and agents face no individual budget constraints. Focusing on binary additive valuations, the work establishes a sharp dichotomy based on the number of added item types: a polynomial-time algorithm exists for a single item type, whereas the problem becomes NP-complete with just two item types—even weakly NP-complete for only two agents. The approach models envy relations via a system of difference constraints and employs the Bellman-Ford algorithm to compute a minimal feasible extension. Complexity reductions further delineate the hardness across multiple scenarios. This work fully resolves the open two-type case posed by Bentert et al. and establishes the first precise computational complexity boundary for this setting.

Abstract

We study envy elimination by adding goods (EEAG) when the additional pool has bounded supply and no separate budget bound. We establish a sharp type-count dichotomy for binary additive valuations. With one additional item type, EEAG is polynomial-time solvable for any number of agents. More generally, our algorithm permits arbitrary nonnegative integer per-copy values. The envy constraints form a system of difference constraints, and Bellman--Ford returns the componentwise least feasible extension. In contrast, with exactly two additional item types, EEAG is \textsf{NP}-complete even when both types have positive finite supply and the approvers of one type form a subset of the approvers of the other. This closes the two-type case left open by Bentert et al. Separately, we prove weak \textsf{NP}-completeness even for two agents with identical additive valuations, one initially endowed good, and a growing number of unit-supply item types. Thus, bounded-supply hardness appears both with two item-types and many agents and with two agents and many item-types.

Problem

• envy elimination
• bounded supply
• additive valuations
• NP-completeness
• item types

Innovation

• envy elimination
• bounded supply
• type-count dichotomy
• NP-completeness
• additive valuations