Unified Greedy Approximability beyond Submodular Maximization

IF 0.9 3区数学 Q2 MATHEMATICS

SIAM Journal on Discrete Mathematics Pub Date : 2024-01-18 DOI:10.1137/22m1526952

Yann Disser, David Weckbecker

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引用次数: 0

Abstract

SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 348-379, March 2024.
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 [math]-[math]-augmentable functions and prove that it encompasses several important subclasses, such as functions of bounded submodularity ratio, [math]-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 [math] on the approximation ratio of the greedy algorithm that tightly interpolates between bounds from the literature for functions of bounded submodularity ratio and for [math]-augmentable functions. In particular, as a by-product, we close a gap in [A. Bernstein et al., Math. Program., 191 (2022), pp. 953–979] by obtaining a tight lower bound for [math]-augmentable functions for all [math]. For weighted rank functions of independence systems, our tight bound becomes [math], which recovers the known bound of [math] for independence systems of rank quotient at least [math].

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超越次模态最大化的统一贪婪逼近性

SIAM 离散数学杂志》，第 38 卷，第 1 期，第 348-379 页，2024 年 3 月。摘要。我们考虑了贪心算法能保证恒定逼近的有数量限制的最大化问题的目标函数类。我们提出了[math]-[math]-可增强函数这一新类，并证明它包含几个重要的子类，例如有界次模性比的函数、[math]-可增强函数、有界秩商独立系统的加权秩函数，以及贪心算法能得到近似值的其他目标函数。对于这一类函数，我们展示了贪心算法近似率的[math]紧约束，它紧密地插值于文献中关于有界次模性比函数和[math]可增强函数的约束之间。特别是，作为副产品，我们弥补了[A. Bernstein 等，Math.Program.，191 (2022)，第 953-979 页]中的空白，获得了所有[math]的[math]可增强函数的严格下界。对于独立系统的加权秩函数，我们的紧约束变成了[math]，它恢复了秩商至少为[math]的独立系统的已知[math]约束。

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来源期刊

SIAM Journal on Discrete Mathematics 数学-应用数学

CiteScore

1.90

自引率

0.00%

发文量

124

审稿时长

4-8 weeks

期刊介绍： SIAM Journal on Discrete Mathematics (SIDMA) publishes research papers of exceptional quality in pure and applied discrete mathematics, broadly interpreted. The journal''s focus is primarily theoretical rather than empirical, but the editors welcome papers that evolve from or have potential application to real-world problems. Submissions must be clearly written and make a significant contribution. Topics include but are not limited to: properties of and extremal problems for discrete structures combinatorial optimization, including approximation algorithms algebraic and enumerative combinatorics coding and information theory additive, analytic combinatorics and number theory combinatorial matrix theory and spectral graph theory design and analysis of algorithms for discrete structures discrete problems in computational complexity discrete and computational geometry discrete methods in computational biology, and bioinformatics probabilistic methods and randomized algorithms.