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 a factor better than $1 - 1/e - \Omega(\min \{ \frac{1}{r}, \frac{\log^2 n}{r^3} \})$, 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 $\delta>0$, either we obtain a $(1/2-\delta)$-approximation in $1$ round, or a $(1/2+\Omega(\delta^2))$-approximation in $O(1/\delta^2)$ rounds. In particular (in contrast to the cardinality-constrained case), there cannot be an instance where (i) it is impossible to achieve a factor better than $1/2$ regardless of the number of rounds, and (ii) it takes $r$ rounds to achieve a factor of $1/2-O(1/r)$.