We consider the classical problem of maximizing a monotone submodular function subject to a cardinality constraint, which, due to its numerous applications, has recently been studied in various computational models. We consider a clean multi-player model that lies between the offline and streaming model, and study it under the aspect of one-way communication complexity. Our model captures the streaming setting (by considering a large number of players), and, in addition, two player approximation results for it translate into the robust setting. We present tight one-way communication complexity results for our model, which, due to the above-mentioned connections, have multiple implications in the data stream and robust setting. Even for just two players, a prior information-theoretic hardness result implies that no approximation factor above $1/2$ can be achieved in our model, if only queries to feasible sets are allowed. We show that the possibility of querying infeasible sets can actually be exploited to beat this bound, by presenting a tight $2/3$-approximation taking exponential time, and an efficient $0.514$-approximation. To the best of our knowledge, this is the first example where querying a submodular function on infeasible sets leads to provably better results. Through the above-mentioned link to the robust setting, both of these algorithms improve on the current state-of-the-art for robust submodular maximization, showing that approximation factors beyond $1/2$ are possible. Moreover, exploiting the link of our model to streaming, we settle the approximability for streaming algorithms by presenting a tight $1/2+\varepsilon$ hardness result, based on the construction of a new family of coverage functions. This improves on a prior $1-1/e+\varepsilon$ hardness and matches, up to an arbitrarily small margin, the best known approximation algorithm.

中文翻译：

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子模块最大化的单向通信复杂性与流和鲁棒性的应用

我们考虑最大化受基数约束的单调子模函数的经典问题，由于其众多应用，最近在各种计算模型中进行了研究。我们考虑了一种介于离线模型和流模型之间的干净的多人模型，并在单向通信复杂性方面对其进行了研究。我们的模型捕获了流媒体设置（通过考虑大量玩家），此外，两个玩家的近似结果转化为稳健的设置。我们为我们的模型提供了紧密的单向通信复杂性结果，由于上述连接，它对数据流和稳健设置有多重影响。即使只有两名球员，先验信息理论硬度结果表明，如果只允许对可行集的查询，则在我们的模型中无法实现高于 $1/2$ 的近似因子。我们展示了查询不可行集合的可能性实际上可以被利用来克服这个界限，通过呈现一个严格的 $2/3$-近似值，需要指数时间，以及一个有效的 $0.514$-近似值。据我们所知，这是第一个在不可行集合上查询子模块函数导致可证明更好结果的示例。通过上述与鲁棒设置的链接，这两种算法都改进了当前最先进的鲁棒子模最大化，表明超过 $1/2$ 的近似因子是可能的。此外，利用我们的模型与流媒体的联系，我们基于新的覆盖函数系列的构建，通过呈现严格的 $1/2+\varepsilon$ 硬度结果来解决流算法的近似性。这改进了先前的 $1-1/e+\varepsilon$ 硬度，并匹配最有名的近似算法，直到任意小的边距。