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, i.e., sets respecting the cardinality constraint, 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 (non-streaming) 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 + ε hardness result, based on the construction of a new family of coverage functions. This improves on a prior 0.586 hardness and matches, up to an arbitrarily small margin, the best known approximation algorithm.

我们考虑最大化受基数约束约束的单调子模函数的经典问题，由于其众多应用，最近已在各种计算模型中进行了研究。我们考虑了一种介于离线和流媒体模型之间的干净的多人游戏模型，并在单向通信复杂性方面对其进行了研究。我们的模型捕获流媒体设置（通过考虑大量玩家），此外，它的两个玩家近似结果转化为稳健设置。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，该模型在数据流和稳健设置中具有多重含义。

即使对于只有两个玩家，先验信息论硬度结果也意味着在我们的模型中无法实现超过 1/2 的近似因子，如果只允许查询可行集，即尊重基数约束的集。我们展示了查询不可行集的可能性实际上可以被利用来突破这个界限，通过提出一个采用指数时间的紧密 2/3 近似和一个有效的 0.514 近似。据我们所知，这是第一个在不可行集上查询子模函数可以得到更好结果的例子。通过上述链接到（非流式）鲁棒设置，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。而且，利用我们的模型与流媒体的联系，我们基于新的覆盖函数族的构造，通过呈现紧密的 1/2 + ε 硬度结果来确定流媒体算法的近似性。这改进了先前的 0.586 硬度和匹配，直到任意小的余量，最知名的近似算法。