Unified greedy approximability beyond submodular maximization

TL;DR

This paper introduces a new class of objective functions called $oldsymbol{ ext{ extgamma}- ext{ extalpha}}$-augmentable functions, providing tight approximation bounds for greedy algorithms in cardinality constrained maximization problems, unifying and extending existing results.

Contribution

It defines the $ ext{ extgamma}- ext{ extalpha}$-augmentable class, proves tight bounds on greedy approximation ratios, and closes gaps in previous bounds for $ ext{ extalpha}$-augmentable functions and independence systems.

Findings

Introduces $ ext{ extgamma}- ext{ extalpha}$-augmentable functions class.

Provides tight approximation bounds for greedy algorithms.

Unifies bounds for various subclasses like submodular and independence systems.

Abstract

We consider classes of objective functions of cardinality constrained maximization problems for which the greedy algorithm guarantees a constant approximation. We propose the new class of $\gamma$-$\alpha$-augmentable functions and prove that it encompasses several important subclasses, such as functions of bounded submodularity ratio, $\alpha$-augmentable functions, and weighted rank functions of an independence system of bounded rank quotient - as well as additional objective functions for which the greedy algorithm yields an approximation. For this general class of functions, we show a tight bound of $\frac{\alpha}{\gamma}\cdot\frac{\mathrm{e}^\alpha}{\mathrm{e}^\alpha-1}$ on the approximation ratio of the greedy algorithm that tightly interpolates between bounds from the literature for functions of bounded submodularity ratio and for $\alpha$-augmentable functions. In paritcular, as a by-product, we close a gap left in [Math.Prog., 2020] by obtaining a tight lower bound for $\alpha$-augmentable functions for all $\alpha\geq1$. For weighted rank functions of independence systems, our tight bound becomes $\frac{\alpha}{\gamma}$, which recovers the known bound of $1/q$ for independence systems of rank quotient at least $q$.