Distributed maximization of a submodular function in the MapReduce model has received much attention, culminating in two frameworks that allow a centralized algorithm to be run in the MR setting without loss of approximation, as long as the centralized algorithm satisfies a certain consistency property - which had only been shown to be satisfied by the standard greedy and continous greedy algorithms. A separate line of work has studied parallelizability of submodular maximization in the adaptive complexity model, where each thread may have access to the entire ground set. For the size-constrained maximization of a monotone and submodular function, we show that several sublinearly adaptive algorithms satisfy the consistency property required to work in the MR setting, which yields highly practical parallelizable and distributed algorithms. Also, we develop the first linear-time distributed algorithm for this problem with constant MR rounds. Finally, we provide a method to increase the maximum cardinality constraint for MR algorithms at the cost of additional MR rounds.

翻译：在MapReduce模型下分布式最大化子模函数的研究已引起广泛关注，最终形成了两个框架。这些框架允许在MR（MapReduce）设置中运行集中式算法而不损失近似性能，前提是该集中式算法满足特定的一致性属性——目前仅标准贪心算法和连续贪心算法被证明具备该属性。另一项独立工作研究了自适应复杂度模型下子模最大化的可并行性（该模型中各线程可访问完整全集）。针对单调子模函数在规模约束下的最大化问题，我们证明若干亚线性自适应算法满足MR设置所需的一致性属性，从而催生出高度实用的可并行化分布式算法。此外，我们首次提出该问题在恒定MR轮次下的线性时间分布式算法。最后，我们提供了一种方法，可通过增加MR轮次为代价来提升MR算法的最大基数约束。