We consider the following control problem on fair allocation of indivisiblegoods. Given a set $I$ of items and a set of agents, each having strict linearpreference over the items, we ask for a minimum subset of the items whosedeletion guarantees the existence of a proportional allocation in the remaininginstance; we call this problem Proportionality by Item Deletion (PID). Our mainresult is a polynomial-time algorithm that solves PID for three agents. Bycontrast, we prove that PID is computationally intractable when the number ofagents is unbounded, even if the number $k$ of item deletions allowed is small,since the problem turns out to be W[3]-hard with respect to the parameter $k$.Additionally, we provide some tight lower and upper bounds on the complexity ofPID when regarded as a function of $|I|$ and $k$.
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机译：我们考虑以下有关不可分割货物公平分配的控制问题。给定一组$ I $的项目和一组agent，每个项目对这些项目都具有严格的线性偏好，我们要求获得这些项目的最小子集，其删除可确保其余情况下按比例分配。我们称此问题为按项目删除（PID）的比例。我们的主要结果是多项式时间算法，该算法可以求解三个代理的PID。相比之下，我们证明了当代理数量不受限制时，即使在允许的项目删除数量$ k $很小的情况下，PID在计算上也是难处理的，因为相对于参数$ k而言，问题出在W [3]难$。此外，当将PID的复杂度视为$ | I | $和$ k $的函数时，我们提供了一些严格的上下限。
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