# A polynomial lower bound on adaptive complexity of submodular maximization

arXiv: https://arxiv.org/abs/2002.09130 · HTML: https://mlsanity.com/paper/2002.09130/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Feb 2020.

## Built on by (most cited first, 4 of 4)

- [Submodularity In Machine Learning and Artificial Intelligence](https://mlsanity.com/paper/2202.00132/) (2022)
- [Parallel Sampling via Counting](https://mlsanity.com/paper/2408.09442/) (2024)
- [Adaptive Sampling for Fast Constrained Maximization of Submodular Function](https://mlsanity.com/paper/2102.06486/) (2021)
- [Sample-Adaptivity Tradeoff in On-Demand Sampling](https://mlsanity.com/paper/2511.15507/) (2025)

## Abstract (arXiv)

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-mono-tone 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 - Omega(min  frac1r, fraclog2 nr3 ), for any r < nc (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+Omega(δ2))-approximation in O(1/δ2) rounds. In particular (and in contrast to the cardinality-constrained 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).