Abstract
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 {mathsf {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 pi in advance as part of the input, and our aim is to delete a minimum number of items such that pi is proportional in the remainder; this variant turns out to be {{mathsf {N}}}{{mathsf {P}}}-hard for six agents, but polynomial-time solvable for two agents, and we show that it is mathsf {W[2]}-hard when parameterized by the number k of
Highlights
We consider a situation where a set I of indivisible items needs to be allocated to a set N of agents in a way that is perceived as fair
The computational study of such control problems was first proposed by Bartholdi, III et al [5] for voting systems; our paper follows the work of Aziz et al [4] who have recently initiated the systematic study of control problems in the area of fair division
We show that the problem of deciding whether there exist at most k items whose deletion allows for a proportional allocation is NP-complete, and that this problem is W[3]-hard with parameter k
Summary
We consider a situation where a set I of indivisible items needs to be allocated to a set N of agents in a way that is perceived as fair. Especially with a large number of items, it is unrealistic to assume that agents are able to assign a meaningful cardinal value to each of the items This may be due to lack of information, e.g., when agents need to declare preferences over items about which they have incomplete knowledge, or an unwillingness to associate a determined value for each item: in scenarios where the usefulness or virtue of an item cannot be measured by its monetary value (e.g., students ranking assignments, shared owners of a holiday home ranking time slots, heirs ranking family assets), people may find it much more convenient to express their preferences in an ordinal way, reducing their cognitive burden. We ask for the maximum number of items that can be allocated to the agents in a proportional way
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