Finding the maximum number of disjoint spanning trees in a given graph is a well-studied problem with several applications and connections. The Tutte-Nash-Williams theorem provides a min-max relation for this problem which also extends to disjoint bases in a matroid and leads to efficient algorithms. Several other packing problems such as element disjoint Steiner trees, disjoint set covers, and disjoint dominating sets are NP-Hard but admit an $O(\log n)$-approximation. C\u{a}linescu, Chekuri, and Vondr\'ak viewed all these packing problems as packing bases of a polymatroid and provided a unified perspective. Motivated by applications in wireless networks, recent works have studied the problem of packing set covers in the online model. The online model poses new challenges for packing problems. In particular, it is not clear how to pack a maximum number of disjoint spanning trees in a graph when edges arrive online. Motivated by these applications and theoretical considerations, we formulate an online model for packing bases of a polymatroid, and describe a randomized algorithm with a polylogarithmic competitive ratio. Our algorithm is based on interesting connections to the notion of quotients of a polymatroid that has recently seen applications in polymatroid sparsification. We generalize the previously known result for the online disjoint set cover problem and also address several other packing problems in a unified fashion. For the special case of packing disjoint spanning trees in a graph (or a hypergraph) whose edges arrive online, we provide an alternative to our general algorithm that is simpler and faster while achieving the same poly-logarithmic competitive ratio.

翻译：在给定图中寻找最大数量的不相交生成树是一个经过深入研究的问题，具有多种应用和联系。Tutte-Nash-Williams定理为该问题提供了一个最小-最大关系，该关系也推广到拟阵中的不相交基，并引出了高效算法。其他几个包装问题，如元素不相交Steiner树、不相交集合覆盖和不相交支配集，是NP难的，但允许$O(\log n)$近似。Călinescu、Chekuri和Vondrák将所有这些问题视为多拟阵基的包装，并提供了一个统一的视角。受无线网络应用的启发，近期研究探讨了在线模型中包装集合覆盖的问题。在线模型为包装问题带来了新的挑战。特别是，当边在线到达时，如何在图中包装最大数量的不相交生成树尚不明确。受这些应用和理论考虑的驱动，我们为包装多拟阵基建立了一个在线模型，并描述了一种具有多对数竞争比的随机算法。我们的算法基于与多拟阵商概念的有趣联系，该概念最近在多拟阵稀疏化中得到了应用。我们推广了先前已知的在线不相交集合覆盖问题的结果，并以统一的方式处理了其他几个包装问题。对于边在线到达的图（或超图）中包装不相交生成树的特殊情况，我们提供了一种替代我们通用算法的方案，该方案更简单、更快，同时实现了相同的多对数竞争比。