The maximization of submodular functions is an NP-Hard problem for certain subclasses of functions, for which a simple greedy algorithm has been shown to guarantee a solution whose quality is within 1/2 of the optimal. When this algorithm is implemented in a distributed way, agents sequentially make decisions based on the decisions of all previous agents. This work explores how limited access to the decisions of previous agents affects the quality of the solution of the greedy algorithm. Specifically, we provide tight upper and lower bounds on how well the algorithm performs, as a function of the information available to each agent. Intuitively, the results show that performance roughly degrades proportionally to the size of the largest group of agents that make decisions independently. Additionally, we consider the case where a system designer is given a set of agents and a global limit on the amount of information that can be accessed. Our results show that the best designs partition the agents into equally sized sets and allow agents to access the decisions of all previous agents within the same set.

中文翻译：

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信息在贪婪的子模最大化中的影响

对于某些函数子类，子模函数的最大化是一个NP-Hard问题，针对该问题，已显示出一种简单的贪心算法可保证其质量在最优值的1/2以内。当以分布式方式实施此算法时，座席将根据所有先前座席的决策顺序进行决策。这项工作探讨了对先前代理决策的有限访问如何影响贪婪算法的解决方案质量。具体来说，我们根据每个代理可用的信息，对算法的性能提供了严格的上限和下限。直观地，结果表明，性能随独立做出决定的最大一组代理的大小成比例地降低。此外，我们考虑的情况是，为系统设计人员提供了一组代理，并且对可访问的信息量进行了全局限制。我们的结果表明，最佳设计将座席分成大小相等的集合，并允许座席访问同一集合中所有先前座席的决策。