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

TL;DR

This paper investigates the maximization of weakly modular functions, providing theoretical guarantees for multi-stage algorithms, establishing fixed-parameter tractability under certain conditions, and proving the limitations of polynomial-time algorithms.

Contribution

It extends analysis of maximization algorithms to weakly modular functions, showing improved guarantees, tractability results, and hardness bounds.

Findings

Multi-stage algorithms have provable guarantees under weak modularity.

Weakly modular maximization is fixed-parameter tractable with certain conditions.

No polynomial-time algorithms can surpass greedy algorithm guarantees for weakly modular functions.

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, many 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 guarantees achieved by the greedy algorithm.