Matroids provide one of the most elegant structures for algorithm design. This is 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 computational properties. Yet, less is understood where parallel algorithms are concerned. In response, we initiate the study of parallel matroid optimization in the adaptive complexity model [BS18]. First, we reexamine Bor\r{u}vka's classical minimum weight spanning tree algorithm [Bor26b; Bor26a] in the abstract language of matroid theory, and identify a new certificate of optimality for the basis of any matroid as a result. In particular, a basis is optimal if and only if it contains the points of minimum weight in every circuit of the dual matroid. Hence, we can witnesses whether any specific point belongs to the optimal basis via a test for local optimality in a circuit of the dual matroid, thereby revealing a general design paradigm towards parallel matroid optimization. To instantiate this paradigm, we use the special structure of a binary matroid to identify an optimization scheme with low adaptivity. Here, our key technical step is reducing optimization to the simpler task of basis search in the binary matroid, using only logarithmic overhead of adaptive rounds of queries to independence oracles. Consequentially, we compose our reduction with the parallel basis search method of [KUW88] to obtain an algorithm for finding the optimal basis of a binary matroid terminating in sublinearly many adaptive rounds of queries to an independence oracle. To the authors' knowledge, this is the first algorithm for matroid optimization to outperform the greedy algorithm in terms of adaptive complexity in the independence query model without assuming the matroid is encoded by a graph.

翻译：拟阵为算法设计提供了最优雅的结构之一。这一点通过Edmonds-Rado定理得到了最佳体现，该定理将简单贪心算法的成功与拟阵最优基的解剖结构联系起来[Edm71; Rad57]。作为回应，大量研究致力于理解拟阵的计算特性。然而，在并行算法方面，人们的理解仍较为有限。为此，我们在自适应复杂度模型[BS18]中开创了并行拟阵优化的研究。首先，我们以拟阵理论的抽象语言重新审视Borůvka的经典最小权重生成树算法[Bor26b; Bor26a]，并由此识别出任意拟阵基的最优性新判据。具体而言，一个基是最优的，当且仅当它包含对偶拟阵每个圈中的最小权重点。因此，我们可以通过对偶拟阵中圈的局部最优性测试，来验证任意特定点是否属于最优基，从而揭示出通向并行拟阵优化的通用设计范式。为了实例化这一范式，我们利用二元拟阵的特殊结构，提出了一种具有低自适应性的优化方案。在此，我们的关键技术步骤是将优化问题约简为二元拟阵中更简单的基搜索任务，且仅需对数级别的自适应轮次查询独立性预言机。因此，我们将该约简与[KUW88]的并行基搜索方法相结合，得到了一种寻找二元拟阵最优基的算法，该算法在对独立性预言机进行亚线性数量级的自适应轮次查询后终止。据作者所知，这是在独立性查询模型中，首个在自适应复杂度方面超越贪心算法的拟阵优化算法，且无需假设拟阵由图编码。