# Computing an Approximately Optimal Agreeable Set of Items

## TL;DR

This paper develops efficient algorithms to find small, approximately optimal agreeable item subsets for multiple agents under various preference models, improving on prior worst-case bounds.

## Contribution

It introduces polynomial-time algorithms that approximate the smallest agreeable set within tight bounds across three preference models.

## Key findings

- Algorithms achieve near-optimal approximation ratios.
- Results apply to ordinal, value oracle, and additive utility models.
- The bounds are tight, indicating optimality of the algorithms.

## Abstract

We study the problem of finding a small subset of items that is \emph{agreeable} to all agents, meaning that all agents value the subset at least as much as its complement. Previous work has shown worst-case bounds, over all instances with a given number of agents and items, on the number of items that may need to be included in such a subset. Our goal in this paper is to efficiently compute an agreeable subset whose size approximates the size of the smallest agreeable subset for a given instance. We consider three well-known models for representing the preferences of the agents: ordinal preferences on single items, the value oracle model, and additive utilities. In each of these models, we establish virtually tight bounds on the approximation ratio that can be obtained by algorithms running in polynomial time.