Gap between coverage and general submodular functions for deterministic consistent algorithms

Determine whether deterministic consistent algorithms—algorithms that maintain a solution of size at most k for online monotone submodular maximization under a cardinality constraint while making only a constant number of changes per time step—achieve different optimal approximation ratios for coverage functions versus general monotone submodular functions. Specifically, establish whether a strict gap exists in the best attainable approximation ratio between coverage functions and general monotone submodular functions for deterministic consistent algorithms, or prove that no such gap exists.

Background

The paper proves a separation between coverage functions and general submodular functions for randomized consistent algorithms: a tight 3/4 bound for coverage and a tight 2/3 bound for general monotone submodular functions. For deterministic consistent algorithms, prior work gives an information-theoretic hardness of 1/2 (based on coverage functions) and best-known positive results of 0.3818 for general monotone submodular functions.

The authors note they are unaware of other natural submodular optimization problems exhibiting such a gap and explicitly highlight the deterministic setting as unresolved: whether coverage functions admit strictly better deterministic consistent approximations than general monotone submodular functions remains unknown.