The Cost of Consistency: Submodular Maximization with Constant Recourse

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 tfrac23 for general monotone submodular functions, and an improved (also tight) bound of tfrac34 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 tfrac12 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.