通过添加有限供应的商品来消除嫉妒：类型计数二分法与两智能体的难解性
Resolving Envy by Adding Goods with Bounded Supply: A Type-Count Dichotomy and Two-Agent Hardness
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中文总结 AI 辅助
该研究针对有限供应下通过添加商品消除嫉妒（EEAG）问题，建立了二元加性估值的类型计数二分法，证明其在两类型等情形下的NP完全性，明确了难解性的两类出现场景。
中文摘要 AI 辅助
我们研究了在额外商品池具有有限供应且无单独预算约束的情况下，通过添加商品消除嫉妒（EEAG）的问题。针对二元加性估值，我们建立了一个明确的类型计数二分法：当额外商品类型为1种时，无论智能体数量多少，EEAG都可在多项式时间内求解，更一般地，我们的算法允许任意非负整数的单份商品值；嫉妒约束构成一个差分约束系统，Bellman-Ford算法返回分量意义下最小的可行扩展。相比之下，当额外商品类型恰好为2种时，即使两种类型均具有正有限供应，且其中一种类型的认可者是另一种类型认可者的子集，EEAG仍是NP完全问题，这解决了Bentert等人留下的两类型情形的开放性问题。此外，我们证明即使对于具有相同加性估值、初始拥有1件商品且存在不断增加的单位供应商品类型的两个智能体，该问题也是弱NP完全问题。因此，有限供应下的难解性既出现在两商品类型与多智能体的情形中，也出现在两智能体与多商品类型的情形中。
英文摘要
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.
