The problem of covering the ground set of two matroids by a minimum number of common independent sets is notoriously hard even in very restricted settings, i.e.\ when the goal is to decide if two common independent sets suffice or not. Nevertheless, as the problem generalizes several long-standing open questions, identifying tractable cases is of particular interest. Strongly base orderable matroids form a class for which a basis-exchange condition that is much stronger than the standard axiom is met. As a result, several problems that are open for arbitrary matroids can be solved for this class. In particular, Davies and McDiarmid showed that if both matroids are strongly base orderable, then the covering number of their intersection coincides with the maximum of their covering numbers. Motivated by their result, we propose relaxations of strongly base orderability in two directions. First we weaken the basis-exchange condition, which leads to the definition of a new, complete class of matroids with distinguished algorithmic properties. Second, we introduce the notion of covering the circuits of a matroid by a graph, and consider the cases when the graph is ought to be 2-regular or a path. We give an extensive list of results explaining how the proposed relaxations compare to existing conjectures and theorems on coverings by common independent sets.

翻译：覆盖两个拟阵基集的最小公共独立集问题即使在非常受限的设置下（例如判断两个公共独立集是否足够）也以困难著称。然而，由于该问题泛化了数个长期悬而未决的开放问题，识别可处理情形具有特殊意义。强基序拟阵构成一类满足远强于标准公理的基交换条件的拟阵类，因此若干对任意拟阵仍为开放问题的问题可在此类中求解。特别地，Davies与McDiarmid指出：若两个拟阵均为强基序拟阵，则其交的覆盖数等于两者覆盖数的最大值。受此结果启发，我们从两个方向提出强基序性的松弛方案：首先弱化基交换条件，由此定义具有显著算法性质的全新拟阵完备类；其次引入通过图覆盖拟阵回路的概念，并特别考虑该图应为2-正则图或路径的情形。我们给出详尽的结论清单，阐明所提出的松弛方案如何与现有关于公共独立集覆盖的猜想及定理相互关联。