On Maximization of Weakly Modular Functions: Guarantees of Multi-stage Algorithms, Tractability, and Hardness

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Shinsaku Sakaue ;
Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, PMLR 108:22-33, 2020.

Abstract

Maximization of {\it non-submodular} functions appears in various scenarios, and many previous works studied it based on some measures that quantify the closeness to being submodular. On the other hand, some practical non-submodular functions are actually close to being {\it modular}, which has been utilized in few studies. In this paper, we study cardinality-constrained maximization of {\it weakly modular} functions, whose closeness to being modular is measured by {\it submodularity} and {\it supermodularity ratios}, and reveal what we can and cannot do by using the weak modularity. We first show that guarantees of multi-stage algorithms can be proved with the weak modularity, which generalize and improve some existing results, and experiments confirm their effectiveness. We then show that weakly modular maximization is {\it fixed-parameter tractable} under certain conditions; as a byproduct, we provide a new time–accuracy trade-off for $\ell_0$-constrained minimization. We finally prove that, even if objective functions are weakly modular, no polynomial-time algorithms can improve the existing approximation guarantee achieved by the greedy algorithm in general.

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