A polynomial lower bound on adaptive complexity of submodular maximization

In large-data applications, it is desirable to design algorithms with a high degree of parallelization. In the context of submodular optimization, adaptive complexity has become a widely-used measure of an algorithm's "sequentiality". Algorithms in the adaptive model proceed in rounds, and can issue polynomially many queries to a function f in each round. The queries in each round must be independent, produced by a computation that depends only on query results obtained in previous rounds. In this work, we examine two fundamental variants of submodular maximization in the adaptive complexity model: cardinality-constrained monotone maximization, and unconstrained non-monotone maximization. Our main result is that an r -round algorithm for cardinality-constrained monotone maximization cannot achieve an approximation factor better than 1 -1/e -Ω(min 1 r , log 2 n r 3 ), for any r < n c (where c > 0 is some constant). This is the first result showing that the number of rounds must blow up polynomially large as we approach the optimal factor of 1 -1/e. For the unconstrained non-monotone maximization problem, we show a positive result: For every instance, and every δ > 0, either we obtain a (1/2 -δ )-approximation in 1 round, or a (1/2 + Ω(δ 2 ))-approximation in O(1/δ 2 ) rounds. In particular (and in contrast to the cardinalityconstrained case), there cannot be an instance where (i) it is impossible to achieve an approximation factor better than 1/2 regardless of the number of rounds, and (ii) it takes r rounds to achieve a factor of 1/2 -O(1/r ).