目标函数可分解的非次模态最大化

IF 0.9 4区 数学 Q4 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS
Cheng Lu, Wenguo Yang
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引用次数: 0

摘要

我们研究的是非次模态最大化问题,其目标函数可以表示为两个集合(DS)函数之间的差值或两个集合(RS)函数之间的比值。对于有卡限和无卡限的 DS 最大化问题,我们提出了几种确定性算法,我们的分析表明,这些算法可以提供可证明的近似保证。作为一种应用,我们设法推导出了 DS 最小化问题在某些条件下比现有结果更好的近似边界。至于 RS 最大化问题,我们证明存在从 RS 最大化近似到 DS 最大化近似的多项式时间还原。基于这一还原,我们推导出了卡方最大化问题的第一个近似边界。我们还为无约束问题设计了算法,并分析了它们的近似保证。通过将我们的结果应用于优化两个超模函数之间比率的问题,我们回答了 Bai 等人提出的问题(载于第 33 届国际机器学习大会(ICML)论文集,2016 年)。此外,我们还举例说明了集合函数是否归一化会对 RS 优化问题的近似性产生影响。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

Non-submodular maximization with a decomposable objective function

Non-submodular maximization with a decomposable objective function

We study the non-submodular maximization problem, whose objective function can be expressed as the Difference between two Set (DS) functions or the Ratio between two Set (RS) functions. For the cardinality-constrained and unconstrained DS maximization problems, we present several deterministic algorithms and our analysis shows that the algorithms can provide provable approximation guarantees. As an application, we manage to derive an improved approximation bound for the DS minimization problem under certain conditions compared with existing results. As for the RS maximization problem, we show that there exists a polynomial-time reduction from the approximation of RS maximization to the approximation of DS maximization. Based on this reduction, we derive the first approximation bound for the cardinality-constrained RS maximization problem. We also devise algorithms for the unconstrained problem and analyze their approximation guarantees. By applying our results to the problem of optimizing the ratio between two supermodular functions, we give an answer to the question posed by Bai et al. (in Proceedings of The 33rd international conference on machine learning (ICML), 2016). Moreover, we give an example to illustrate that whether the set function is normalized has an effect on the approximability of the RS optimization problem.

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来源期刊
Journal of Combinatorial Optimization
Journal of Combinatorial Optimization 数学-计算机:跨学科应用
CiteScore
2.00
自引率
10.00%
发文量
83
审稿时长
6 months
期刊介绍: The objective of Journal of Combinatorial Optimization is to advance and promote the theory and applications of combinatorial optimization, which is an area of research at the intersection of applied mathematics, computer science, and operations research and which overlaps with many other areas such as computation complexity, computational biology, VLSI design, communication networks, and management science. It includes complexity analysis and algorithm design for combinatorial optimization problems, numerical experiments and problem discovery with applications in science and engineering. The Journal of Combinatorial Optimization publishes refereed papers dealing with all theoretical, computational and applied aspects of combinatorial optimization. It also publishes reviews of appropriate books and special issues of journals.
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