发表机构
University of Electronic Science and Technology of China（电子科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文将加权拟阵交和匹配拟阵的线性规划整数间隙上界分别改进为 k-1+1/k 和 p-1+1/p，并给出确定性算法，解决了相关猜想的一部分。
AI 中文摘要
加权 $k$-拟阵交问题要求在一个公共基集上的每个 $k$ 个拟阵中均独立的集合中寻找最大权重集合。自然的线性规划松弛在 $k$ 个拟阵独立集多面体的交集上优化。据猜想，该线性规划的整数间隙至多为 $k-1$。该猜想在 $k\le3$ 时已知成立，但对于 $k\ge4$，此前最佳的一般上界为 $k$。我们将此界改进为 $k-1+1/k$。更一般地，我们证明 $p$-匹配拟阵的自然线性规划的整数间隙至多为 $p-1+1/p$，并给出一个达到相同因子的确定性线性规划相关算法。匹配拟阵扩展解决了 Lee、Sviridenko 和 Vondrák 猜想中的 $p$-匹配拟阵部分；当存在 $p-1$ 阶射影平面时，射影平面给出显式的紧实例。
英文摘要
The weighted $k$-matroid intersection problem asks for a maximum-weight set that is independent in each of $k$ matroids on a common ground set. The natural LP relaxation optimizes over the intersection of the $k$ matroid independent set polytopes. It is conjectured that this LP has integrality gap at most $k-1$. The conjecture is known for $k\le3$, but for $k\ge4$ the best general upper bound was $k$. We improve this bound to $k-1+1/k$. More generally, we prove that the natural LP of a $p$-matchoid has integrality gap at most $p-1+1/p$, with a deterministic LP-relative algorithm attaining the same factor. The matchoid extension resolves the $p$-matchoid part of a conjecture of Lee, Sviridenko, and Vondrák; projective planes give explicit tight instances whenever one of order $p-1$ exists.
