We consider the problem of finding an independent set of maximum weight simultaneously contained in $k$ matroids over a common ground set. This $k$-matroid intersection problem appears naturally in many contexts, for example in generalizing graph and hypergraph matching problems. In this paper, we provide a $(k+1)/(2 \ln 2)$-approximation algorithm for the weighted $k$-matroid intersection problem. This is the first improvement over the longstanding $(k-1)$-guarantee of Lee, Sviridenko and Vondr\'ak (2009). Along the way, we also give the first improvement over greedy for the more general weighted matroid $k$-parity problem. Our key innovation lies in a randomized reduction in which we solve almost unweighted instances iteratively. This perspective allows us to use insights from the unweighted problem for which Lee, Sviridenko, and Vondr\'ak have designed a $k/2$-approximation algorithm. We analyze this procedure by constructing refined matroid exchanges and leveraging randomness to avoid bad local minima.

翻译：我们考虑在公共基础集上的$k$个拟阵中同时寻找最大权独立集的问题。这一$k$-拟阵交问题自然出现在许多场景中，例如在图与超图匹配问题的推广中。本文针对加权$k$-拟阵交问题提出了一个$(k+1)/(2 \ln 2)$近似算法。这是对Lee、Sviridenko与Vondr\'ak（2009）长期保持的$(k-1)$保证率的首次改进。在此过程中，我们还为更一般的加权拟阵$k$-奇偶问题给出了超越贪心算法的首次改进。我们的核心创新在于一种随机归约方法，通过迭代求解几乎未加权的实例。这一视角使我们能够利用未加权问题的研究洞见——Lee、Sviridenko与Vondr\'ak曾为此设计了$k/2$近似算法。我们通过构建精细的拟阵交换结构并利用随机性规避局部最劣解来分析该过程。