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重独立集的问题。这一$k$-拟阵交问题自然出现在许多场景中，例如在图匹配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$(k+1)/(2 \ln 2)$近似算法。这是对Lee、Sviridenko与Vondr\'ak（2009）提出的长期存在的$(k-1)$近似保证的首次改进。在研究过程中，我们还对更一般的加权拟阵$k$-奇偶问题给出了优于贪心算法的首次改进。我们的核心创新在于一种随机化归约方法，通过迭代求解几乎无权的实例。这一视角使我们能够利用Lee、Sviridenko与Vondr\'ak针对无权问题设计的$k/2$近似算法的思想。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，对该过程进行了理论分析。