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The problem of covering the ground set of two matroids by a minimum number ofcommon 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 orderablematroids form a class for which a basis-exchange condition that is muchstronger than the standard axiom is met. As a result, several problems that areopen for arbitrary matroids can be solved for this class. In particular, Daviesand McDiarmid showed that if both matroids are strongly base orderable, thenthe covering number of their intersection coincides with the maximum of theircovering numbers.  Motivated by their result, we propose relaxations of strongly baseorderability in two directions. First we weaken the basis-exchange condition,which leads to the definition of a new, complete class of matroids withdistinguished algorithmic properties. Second, we introduce the notion ofcovering the circuits of a matroid by a graph, and consider the cases when thegraph is ought to be 2-regular or a path. We give an extensive list of resultsexplaining how the proposed relaxations compare to existing conjectures andtheorems on coverings by common independent sets.

Partitioning into common independent sets via relaxing strongly base orderability