In this work, we study online submodular maximization, and how the requirement of maintaining a stable solution impacts the approximation. In particular, we seek bounds on the best-possible approximation ratio that is attainable when the algorithm is allowed to make at most a constant number of updates per step. We show a tight information-theoretic bound of $\tfrac{2}{3}$ for general monotone submodular functions, and an improved (also tight) bound of $\tfrac{3}{4}$ for coverage functions. Since both these bounds are attained by non poly-time algorithms, we also give a poly-time randomized algorithm that achieves a $0.51$-approximation. Combined with an information-theoretic hardness of $\tfrac{1}{2}$ for deterministic algorithms from prior work, our work thus shows a separation between deterministic and randomized algorithms, both information theoretically and for poly-time algorithms.

翻译：本文研究在线次模最大化问题，并探讨维持解稳定性对近似性能的影响。具体而言，我们旨在分析当算法在每一步最多允许进行常数次更新时，可达到的最佳近似比上界。对于一般单调次模函数，我们证明其信息论紧界为 $\tfrac{2}{3}$；对于覆盖函数，我们进一步得到改进的（同样紧致的）上界 $\tfrac{3}{4}$。由于这两个界均由非多项式时间算法达成，我们还提出一种多项式时间随机算法，可实现 $0.51$ 近似比。结合先前工作中确定性算法 $\tfrac{1}{2}$ 的信息论下界，我们的研究从信息论和多项式时间算法两个层面，揭示了确定性算法与随机算法之间的分离现象。