Weighted Equitability and Matroid-Constrained Discrepancy

2026-08-14

AI Summary

This study addresses weighted fair division and discrepancy theory under matroid constraints, proposing strongly polynomial-time algorithms grounded in local exchange theorems and constructive proofs. Key contributions include establishing weighted matroid partitionability and extending the Beck-Fiala framework to matroid settings, yielding logarithmic discrepancy bounds. The work achieves EF1 and additive approximation guarantees for fair allocation and scheduling optimization while refuting the weighted carpool conjecture. By systematically resolving fairness and discrepancy control challenges in constrained environments, this research provides novel theoretical foundations and efficient algorithmic tools for related combinatorial optimization problems.

Abstract

We prove weighted matroid equitability. Let $M=(E,\mathcal{I})$ be a matroid whose ground set can be partitioned into $k$ bases, and assign a nonnegative weight to every element. Then $E$ has a partition into $k$ bases such that the weights of any two bases differ by at most the largest element weight. We present two proofs based on the localized exchange theorem of Akrami, Liu, Raj, and Végh. The first is existential, while the second is constructive and leads to a strongly polynomial-time algorithm. As applications, we obtain an additive guarantee for matroid-constrained makespan minimization for identical machines and a strongly polynomial-time algorithm for finding an EF1 allocation under a matroid constraint and identical additive valuations. We further generalize the Beck--Fiala framework in discrepancy theory to settings with matroid constraints. Given a nonnegative matrix of column sparsity $Δ$, we show that a fractional basis can be rounded to a basis of no larger cost while preserving every row sum within additive error $2Δ$ times the largest matrix entry. Motivated by the $2$-sparse prefix Beck--Fiala conjecture, we formulate a conjecture on prefix-constrained matroid bases and prove a discrepancy bound $O(\log n)$. Finally, we give a counterexample to the weighted carpooling conjecture, thereby also disproving a conjecture by Morell and Skutella on single-source unsplittable flows with two-sided discrepancy bounds.

Problem

• Weighted Matroid Equitability
• Matroid-Constrained Discrepancy
• Beck-Fiala Framework
• Makespan Minimization
• EF1 Allocation

Innovation

• Weighted Matroid Equitability
• Matroid-Constrained Discrepancy
• Strongly Polynomial-Time Algorithm
• Beck-Fiala Framework
• EF1 Allocation