# Obtaining a Proportional Allocation by Deleting Items

## TL;DR

This paper introduces a polynomial-time algorithm for ensuring proportional allocations by deleting items in three-agent scenarios, but proves the problem's computational hardness with more agents, highlighting its complexity bounds.

## Contribution

It presents the first polynomial-time solution for PID with three agents and establishes the problem's W[3]-hardness for multiple agents, revealing its computational limits.

## Key findings

- Polynomial-time algorithm for three agents
- W[3]-hardness for multiple agents
- Complexity bounds based on item set size and deletions

## 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 preference 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, 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 of PID when regarded as a function of $|I|$ and $k$.