发表机构
University of Illinois Urbana-Champaign; University of Southern Denmark; EPFL Lausanne; Stanford University; ETH Zürich（伊利诺伊大学厄巴纳-香槟分校; 南丹麦大学; 洛桑联邦理工学院; 斯坦福大学; 苏黎世联邦理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究二分图完美匹配与拟阵交基上的单调子模最大化，证明其与有向图子模定向近似等价，并给出多项式时间双准则近似，目标值达到最优的1/2减epsilon。
AI 中文摘要
受公平性应用和基础性问题的启发，我们考虑在公共地面集合 $E$ 上两个拟阵的交集中，最大化单调子模函数 $f\colon 2^E \rightarrow \mathbb{R}_+$ 在最大基数集合上的问题。一个重要的特例是二分图中的子模完美匹配。在本工作之前，其近似性一直缺乏理解，仅已知常数不可近似性，甚至不知道是否存在 $\frac{1}{o(\sqrt{|E|})}$ 的近似算法。即使允许轻微违反基数约束，也仅已知目标值有显著损失的 bicriteria 近似。在此，我们获得两个结果。首先，我们证明在常数因子内，该问题与有向图中的子模定向问题近似等价。这给出了拟多项式时间内的 $\Omega(1 / \log |E|)$ 近似，以及几乎匹配的困难性结果。其次，我们通过局部搜索框架获得了改进的多项式时间 bicriteria 近似。更精确地说，如果 $f(T^*)$ 是两个拟阵中大小至少为 $K$ 的公共独立集的最大子模值，我们找到一个公共独立集 $T$，使得 $|T| \geq (1 - \epsilon) K$ 且 $f(T) \geq (1/2 - \epsilon) f(T^*)$。相比之下，先前的工作仅保证在确保 $|T| \geq (1 - \epsilon) K$ 的同时，值达到 $\Omega(\epsilon) f(T^*)$。
英文摘要
Motivated by applications in fairness and foundational questions, we consider the problem of maximizing a monotone submodular function $f\colon 2^E \rightarrow \mathbb{R}_+$ over maximum cardinality sets in the intersection of two matroids on a common ground set $E$. An important special case is submodular perfect matching in bipartite graphs. Prior to this work, its approximability was poorly understood with only constant inapproximability known, despite not even a $\frac{1}{o(\sqrt{|E|})}$-approximation being known. Even when allowing to violate the cardinality constraint slightly, only a bicriteria approximation with a significant loss in the objective was known. Here, we obtain two results. First, we show that, within constant factors, the problem is approximation-equivalent to Submodular Orienteering in directed graphs. This yields an $Ω(1 / \log |E|)$-approximation in quasi-polynomial time together with an almost-matching hardness result. Second, we obtain an improved polynomial-time bicriteria approximation via a local search framework. More precisely, if $f(T^*)$ is the largest submodular value of a common independent set in both matroids of size at least $K$, we find a common independent set $T$ such that $|T| \geq (1 - ε) K$ and $f(T) \geq (1/2 - ε) f(T^*)$. In contrast, previous work only guarantees a value of $Ω(ε) f(T^*)$ while ensuring that $|T| \geq (1 - ε) K$.
