加权公平性与拟阵约束的偏差
Weighted Equitability and Matroid-Constrained Discrepancy
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中文总结 AI 辅助
本文证明加权拟阵公平性，基于局部交换定理给出两种证明并得到强多项式算法，推广 Beck–Fiala 框架，提出前缀约束拟阵基猜想并证偏差界，还给出加权拼车猜想等的反例。
中文摘要 AI 辅助
我们证明了加权拟阵公平性。设 $M=(E,\mathcal{I})$ 为一个拟阵，其基集可划分为 $k$ 个基，且为每个元素分配一个非负权重。那么 $E$ 存在一种划分为 $k$ 个基的方式，使得任意两个基的权重差不超过最大元素权重。我们基于 Akrami、Liu、Raj 和 Végh 的局部交换定理给出两种证明：第一种是存在性证明，第二种是构造性证明，可得到强多项式时间算法。作为应用，我们得到了相同机器下拟阵约束最大完工时间最小化的加性保证，以及在拟阵约束与相同加性估值下寻找 EF1 分配的强多项式时间算法。我们进一步将偏差理论中的 Beck–Fiala 框架推广到拟阵约束场景：给定列稀疏度为 $\Delta$ 的非负矩阵，我们证明可将分数基四舍五入为成本不更大的基，同时保持每行和的加性误差在 $2\Delta$ 乘以矩阵最大元素范围内。受 2-稀疏前缀 Beck–Fiala 猜想的启发，我们提出前缀约束拟阵基的猜想，并证明偏差界为 $O(\log n)$。最后，我们给出加权拼车猜想的反例，从而也反驳了 Morell 和 Skutella 关于带双边偏差界的单源不可拆分流的猜想。
英文摘要
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.
