A polynomial lower bound on adaptive complexity of submodular maximization
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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 - 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 (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).
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