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

TL;DR

This paper introduces a novel combinatorial algorithm for non-monotone submodular maximization under a knapsack constraint, achieving near-optimal adaptive complexity and improving efficiency in large-scale data applications.

Contribution

It presents the first constant factor approximation algorithm with near-optimal adaptive complexity for this problem, and reduces the number of function evaluations needed.

Findings

Achieves $O(rac{1}{ ext{approximation factor}})$ adaptive complexity.

Reduces function evaluations to $ ilde{O}(n)$ while maintaining low adaptivity.

Provides the first combinatorial approach with sublinear adaptive complexity for the problem.

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.