arXiv subjects

Benjamin Cookson

Publications and source records attributed to Benjamin Cookson.

2 recordsLinked to original sources

arXiv2026

Constrained Fair Allocations via Partition Matroid Reductions

We study fair allocation of indivisible goods under additive valuations and matroid constraints. A challenging open question is whether a complete and feasible envy-free up to one good (EF1) allocation exists under every matroid that admits a complete and feasible allocation. The state-of-the-art result by Biswas and Barman [2018] positively resolves this question for partition matroids. Our first result positively resolves it for laminar matroids, which generalize partition matroids, when there are three agents. Our technique reduces this general existence question to finding an EF1 allocation satisfying a mild additional condition under a single finite-sized key laminar matroid, and we establish the required allocation by case analysis. We show that our technique somewhat extends to four agents, reducing the analogous problem to finding EF1 allocations under two finite-sized laminar matroids, although we are unable to establish their existence. We also use recent matroid decomposition results to establish EF1 existence under broader classes of matroids. Specifically, we show that EF1 allocations always exist under transversal matroids whenever a complete allocation is feasible, and obtain existence results for graphic matroids and gammoids under stronger assumptions.

cs.GT↗

arXiv2026

Optimally Selecting Representative Agents from a Metric Space

This paper studies the problem of proportionally fair clustering, where the goal is to select $k$ ``centers'' from a metric space that fairly represent a set of agents who also lie in the metric space. Specifically, we focus on finding a clustering satisfying a fairness property known as the Droop core. In the practical special case in which the set of feasible center locations contains every agent location, the previous best-known result guaranteed a $(1 + \sqrt{2})$-approximation of the Droop core, while the best-known lower bound was $2$. In this paper, we show that this lower bound is tight and that a clustering in the $2$-Droop core always exists. Further, we show that such a clustering can be achieved by only selecting centers from locations in the metric space where an agent resides. We establish this using Scarf's theorem guaranteeing a nonempty core for balanced non-transferable utility games. This result has several interesting corollaries. Most notably, it resolves the $β$-plurality problem of Aronov et al. [2021] for general metric spaces. The main result of this paper was generated by $\mathtt{ChatGPT}$-$\mathtt{5.6}$-$\mathtt{Sol}$ through a series of interactions with the authors. The authors of this paper verified the generated proof and rewrote it for clarity.

cs.GT↗