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.

翻译：在组合优化中，拟阵为算法设计提供了最为优雅的结构之一。这或许最能通过Edmonds-Rado定理得以体现，该定理将简单贪心算法的成功与拟阵最优基的结构联系起来[Edm71; Rad57]。作为回应，大量研究致力于理解拟阵良好的计算性质。然而令人惊讶的是，在并行算法设计方面却鲜有深入理解。具体而言，虽然先前工作已探讨在并行计算环境中寻找任意基的任务[KUW88]，但寻找最优基这一更为复杂的任务仍未得到探索。我们通过用拟阵理论的语言重新审视Borůvka最小生成树算法来启动这项研究，并由此通过拟阵的余圈获得了最优基的新刻画。此外，我们进一步将这些洞见与二元拟阵的特殊性质相结合，将二元拟阵中的优化问题归约为更简单的任意基搜索任务，且仅产生对数级渐近开销。因此，我们能够将所提出的归约方法与[KUW88]已知的基搜索方法相结合，得到一种新颖算法，用于寻找二元拟阵的最优基，且仅需对独立性预言机进行亚线性次数的自适应轮次查询。据作者所知，这是首个在自适应复杂度方面超越贪心算法的拟阵优化并行算法，适用于任何非图表示的拟阵类。