🤖 AI Summary
研究了在负二分值下通过单位补贴分配不可分割的杂务问题，提出了一种能在多项式时间内计算出无嫉妒分配及补贴的方法。
📝 Abstract
We study the allocation of indivisible chores with subsidies under negative dichotomous valuations, where the marginal disutility of every chore is either zero or one. This is the chore analogue of the dichotomous goods model of Barman et al. (2022), for which a subsidy of at most one unit per agent is known to suffice for envy-freeness.
We show that the same optimal guarantee holds for chores, under no structural assumption beyond binary marginals. For every instance with negative dichotomous valuations, there exists a complete allocation $A$ and a subsidy vector $p\in\{0,1\}^n$ with $\sum_{i\in N} p_i\le n-1$ such that $(A,p)$ is envy-free. Moreover, the allocation $A$ is EF1 even before subsidies are paid. The bound is tight, and the allocation and subsidies can be computed in polynomial time in the value-oracle model. We further show that the unit-subsidy guarantee cannot, in general, be maintained if Pareto optimality is also required.
Our algorithm starts from an envy-free partial allocation produced by the algorithm of Tao et al. (2025), assigns the remaining chores to distinct agents in a tail strongly connected component of the terminal equality graph, and determines the subsidised agents through a backward closure in the associated equality graph.