Matroids are a fundamental object of study in combinatorial optimization. Three closely related and important problems involving matroids are maximizing the size of the union of $k$ independent sets (that is, $k$-fold matroid union), computing $k$ disjoint bases (a.k.a. matroid base packing), and covering the elements by $k$ bases (a.k.a. matroid base covering). These problems generalize naturally to integral and real-valued capacities on the elements. This work develops faster exact and/or approximation problems for these and some other closely related problems such as optimal reinforcement and matroid membership. We obtain improved running times both for general matroids in the independence oracle model and for the graphic matroid. The main thrust of our improvements comes from developing a faster and unifying push-relabel algorithm for the integer-capacitated versions of these problems, building on previous work by Frank and Mikl\'os [FM12]. We then build on this algorithm in two directions. First we develop a faster augmenting path subroutine for $k$-fold matroid union that, when appended to an approximation version of the push-relabel algorithm, gives a faster exact algorithm for some parameters of $k$. In particular we obtain a subquadratic-query running time in the uncapacitated setting for the three basic problems listed above. We also obtain faster approximation algorithms for these problems with real-valued capacities by reducing to small integral capacities via randomized rounding. To this end, we develop a new randomized rounding technique for base covering problems in matroids that may also be of independent interest.

翻译：拟阵是组合优化中的基本研究对象。涉及拟阵的三个密切相关且重要的问题包括：最大化 $k$ 个独立集并集的大小（即 $k$ 重拟阵并集）、计算 $k$ 个不相交基（即拟阵基打包）、以及用 $k$ 个基覆盖所有元素（即拟阵基覆盖）。这些问题可自然推广到元素上的整数容量和实数值容量情形。本文针对这些问题及其他密切相关的优化问题（如最优强化和拟阵成员判定）开发了更快的精确和/或近似算法。我们改进了独立预言机模型下一般拟阵和图拟阵的求解时间。改进的核心在于：基于 Frank 和 Miklós [FM12] 的前期工作，为整数容量版本的这些问题开发了一个更快的统一推送重标号算法。随后从两个方向扩展该算法：首先，为 $k$ 重拟阵并集开发一个更快的增广路子程序，将其附加到推送重标号算法的近似版本后，可在某些 $k$ 参数下获得更快的精确算法。特别地，针对上述三个基本问题的无容量情形，我们实现了次二次查询时间的运行复杂度。对于实数值容量情形，通过随机舍入将问题简化为小整数容量情形，从而获得更快的近似算法。为此，我们提出了一种新的拟阵基覆盖问题随机舍入技术，该技术可能具有独立的研究价值。