Reducing Matroid Optimization to Basis Search

In combinatorial optimization, matroids provide one of the most elegant structures for algorithm design. This is perhaps best identified by the Edmonds-Rado theorem relating the success of the simple greedy algorithm to the anatomy of the optimal basis of a matroid [Edm71; Rad57]. As a response, much energy has been devoted to understanding a matroid's favorable computational properties. Yet surprisingly, not much is understood where parallel algorithm design is concerned. Specifically, while prior work has investigated the task of finding an arbitrary basis in parallel computing settings [KUW88], the more complex task of finding the optimal basis remains unexplored. We initiate this study by reexamining Bor\r{u}vka's minimum weight spanning tree algorithm in the language of matroid theory, identifying a new characterization of the optimal basis by way of a matroid's cocircuits as a result. Furthermore, we then combine such insights with special properties of binary matroids to reduce optimization in a binary matroid to the simpler task of search for an arbitrary basis, with only logarithmic asymptotic overhead. Consequentially, we are able to compose our reduction with a known basis search method of [KUW88] to obtain a novel algorithm for finding the optimal basis of a binary matroid with only sublinearly many adaptive rounds of queries to an independence oracle. To the authors' knowledge, this is the first parallel algorithm for matroid optimization to outperform the greedy algorithm in terms of adaptive complexity, for any class of matroid not represented by a graph.