Maximize a Monotone Function with a Generic Submodularity Ratio

Approximation guarantees for parallelized maximization of monotone non-submodular function with a cardinality constraint

“…Gong et al (2019) first introduced the generic submodularity ratio γ and derived a continuous greedy algorithm and contention resolution schemes for multilinear optimization problem with a matroid constraint, achieving an approximation ratio (1 − e −γ − o(1)). Later, Nong et al (2019) used the generic submodularity ratio γ in the greedy algorithms for maximizing a strictly monotone and normalized set function, and obtained a (1 − e −γ )-approximation for maximization of a monotone non-submodular function with a cardinality constraint. Recently, for maximizing a monotone submodular function subject to a cardinality constraint, Breuer et al (2020) designed a new algorithm called Fast Adaptive Sequencing Technique (FAST), which is shown deserves an approximation ratio arbitrarily close to 1 − 1/e with O((log(n) log 2 (log k))) adaptivity rounds and the total oracle queries is O(n log log(k)).…”