Obtaining a Proportional Allocation by Deleting Items

We consider the following control problem on fair allocation of indivisible goods. Given a set I of items and a set of agents, each having strict linear preferences over the items, we ask for a minimum subset of the items whose deletion guarantees the existence of a proportional allocation in the remaining instance; we call this problem Proportionality by Item Deletion (PID). Our main result is a polynomial-time algorithm that solves PID for three agents. By contrast, we prove that PID is computationally intractable when the number of agents is unbounded, even if the number k of item deletions allowed is small—we show that the problem is W[3]-hard with respect to the parameter k. Additionally, we provide some tight lower and upper bounds on the complexity of PID when regarded as a function of |I| and k. Considering the possibilities for approximation, we prove a strong inapproximability result for PID. Finally, we also study a variant of the problem where we are given an allocation π  in advance as part of the input, and our aim is to delete a minimum number of items such that π is proportional in the remainder; this variant turns out to be NP-hard for six agents, but polynomial-time solvable for two agents, and we show that it is W[2]-hard when parameterized by the number k of deletions.