# Approximate Core for Committee Selection via Multilinear Extension and Market Clearing

arXiv: https://arxiv.org/abs/2110.12499 · HTML: https://mlsanity.com/paper/2110.12499/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Oct 2021.

## Built on by (most cited first, 4 of 4)

- [Method of Equal Shares with Bounded Overspending](https://mlsanity.com/paper/2409.15005/) (2024)
- [A Linear Theory of Multi-Winner Voting](https://mlsanity.com/paper/2503.03082/) (2025)
- [Online Subset Selection using α-Core with no Augmented Regret](https://mlsanity.com/paper/2209.14222/) (2022)
- [A General Theory of Proportionality with Additive Utilities](https://mlsanity.com/paper/2602.08504/) (2026)

## Abstract (arXiv)

Motivated by civic problems such as participatory budgeting and multiwinner elections, we consider the problem of public good allocation: Given a set of indivisible projects (or candidates) of different sizes, and voters with different monotone utility functions over subsets of these candidates, the goal is to choose a budget-constrained subset of these candidates (or a committee) that provides fair utility to the voters. The notion of fairness we adopt is that of core stability from cooperative game theory: No subset of voters should be able to choose another blocking committee of proportionally smaller size that provides strictly larger utility to all voters that deviate. The core provides a strong notion of fairness, subsuming other notions that have been widely studied in computational social choice. It is well-known that an exact core need not exist even when utility functions of the voters are additive across candidates. We therefore relax the problem to allow approximation: Voters can only deviate to the blocking committee if after they choose any extra candidate (called an additament), their utility still increases by an α factor. If no blocking committee exists under this definition, we call this an α-core. Our main result is that an α-core, for α < 67.37, always exists when utilities of the voters are arbitrary monotone submodular functions, and this can be computed in polynomial time. This result improves to α < 9.27 for additive utilities, albeit without the polynomial time guarantee. Our results are a significant improvement over prior work that only shows logarithmic approximations for the case of additive utilities. We complement our results with a lower bound of α > 1.015 for submodular utilities, and a lower bound of any function in the number of voters and candidates for general monotone utilities.