Submodular Maximization subject to a Knapsack Constraint: Combinatorial Algorithms with Near-optimal Adaptive Complexity

• URL: http://arxiv.org/abs/2102.08327v2
• Date: Fri, 20 Oct 2023 08:22:31 GMT
• Title: Submodular Maximization subject to a Knapsack Constraint: Combinatorial Algorithms with Near-optimal Adaptive Complexity
• Authors: Georgios Amanatidis, Federico Fusco, Philip Lazos, Stefano Leonardi, Alberto Marchetti Spaccamela, Rebecca Reiffenh\"auser
• Abstract summary: We obtain the first constant approximation for non-monotone submodular algorithms with near-optimal $O(log n)$ adaptive complexity. Our algorithm asks $tildeO(n2)$ value queries, but can be modified to run with only $tildeO(n)$ instead. This is also the first approach with sublinear adaptive complexity for the problem and yields comparable to the state-of-the-art even for special cases of cardinality constraints or monotone objectives.
• Score: 13.416309759182635
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Submodular maximization is a classic algorithmic problem with multiple applications in data mining and machine learning; there, the growing need to deal with massive instances motivates the design of algorithms balancing the quality of the solution with applicability. For the latter, an important measure is the adaptive complexity, which captures the number of sequential rounds of parallel computation needed by an algorithm to terminate. In this work we obtain the first constant factor approximation algorithm for non-monotone submodular maximization subject to a knapsack constraint with near-optimal $O(\log n)$ adaptive complexity. Low adaptivity by itself, however, is not enough: a crucial feature to account for is represented by the total number of function evaluations (or value queries). Our algorithm asks $\tilde{O}(n^2)$ value queries, but can be modified to run with only $\tilde{O}(n)$ instead, while retaining a low adaptive complexity of $O(\log^2n)$. Besides the above improvement in adaptivity, this is also the first combinatorial approach with sublinear adaptive complexity for the problem and yields algorithms comparable to the state-of-the-art even for the special cases of cardinality constraints or monotone objectives.

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