We study stochastic graph optimization problems in a novel distributed setting. As in the standard centralized setting, a random subgraph $G^*$ of a known base graph $G$ is realized by including each edge $e$ independently with a known probability $p_e$, and we must solve an optimization problem on $G^*$ despite uncertainty about its edges. In the standard setting, to cope with this uncertainty, the algorithm can query any edge of $G$ to learn if the edge exists in $G^*$, and its complexity is the number of queried edges. The distributed setting incorporates uncertainty in a natural manner, by having each vertex know only about its own edges in $G^*$ (and only communicate over them), and the complexity is measured by the number of synchronous communication rounds. We establish that distributed stochastic algorithms can be drastically faster than their non-stochastic counterparts and overcome known lower bounds, by showing fast distributed approximation algorithms for maximum matching, minimum vertex cover, and minimum dominating set.

翻译：我们研究一种新的分布式设置下的随机图优化问题。与标准的集中式设置类似，已知基础图 $G$ 通过独立地以已知概率 $p_e$ 包含每条边 $e$ 来生成随机子图 $G^*$，并且我们必须在 $G^*$ 上解决一个优化问题，尽管其边存在不确定性。在标准设置中，为应对这种不确定性，算法可以查询 $G$ 中的任意边以了解该边是否存在于 $G^*$ 中，其复杂度为查询的边数。分布式设置以自然的方式引入不确定性：每个顶点仅知道 $G^*$ 中与其相连的边（并且仅通过这些边进行通信），复杂度通过同步通信轮数来衡量。我们证明，分布式随机算法可以比其非随机算法快得多，并克服已知的下界，通过展示最大匹配、最小顶点覆盖和最小支配集的快速分布式近似算法。