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顶点-脱节最短路径的紧密近似和核心化边界.

Matthias Bentert1,2, Fedor V Fomin1, Petr A Golovach1

  • 1University of Bergen, Bergen, Norway.

Theory of computing systems
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PubMed
概括
此摘要是机器生成的。

这项研究表明,近似最大顶点-离合最短路径几乎和确切的问题一样困难. 我们证明在差距-ETH下不存在o(k) 接近,除非P=NP,否则不存在m^(1/2-ε) 接近.

关键词:
对于ETH来说,这是一个很好的选择.固定参数可处理性 固定参数可处理性参数化 (在) 接近性.

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科学领域:

  • 图形理论和算法
  • 计算复杂性 计算复杂性
  • 接近算法近似算法

背景情况:

  • 最大顶点离合最短路径问题旨在使用对对顶点离合最短路径连接最大数量的终端对.
  • 最近的突破已经显示了对固定k的多项式时间可解决性,这意味着ck-近似.
  • 了解近似的极限对于实际应用至关重要.

研究的目的:

  • 为了确定接近最大顶点-离合最短路径问题的硬度.
  • 确定在多项式时间内可实现的最佳近似比.
  • 探索问题的精确复杂性和核心化界限.

主要方法:

  • 利用差距指数时间假设 (gap-ETH) 证明不接近性结果.
  • 使用已知NP-hard问题的减法来证明计算限制.
  • 开发一个简单的sqrt ((l) -近似算法来显示边界的紧密性.
  • 分析精确的算法和核心化复杂性.

主要成果:

  • 假设差距-ETH,对于任何k-依赖函数f,在f{\displaystyle f} k{\displaystyle f} n} 时间内排除一个o{\displaystyle o} k{\displaystyle o} k{\displaystyle o} k} -近似.
  • 在多项式时间中,m^(1/2-ε) 的近似比是不可行的,除非P=NP.
  • 一个紧密的sqrt ((l) -近似算法被介绍,其中l是最佳解决方案中的边缘数.
  • 这个问题可以在2^O{\displaystyle 2^O}{\displaystyle 1^O}{\displaystyle 1^O}{\displaystyle 1^O}{\displaystyle 1^O}{\displaystyle 1^O}{\displaystyle 1^O}{\displaystyle 1^O}{\displaystyle 1^O}{\displaystyle 1^O}{\displaystyle 1^O}}{\displaystyle 1^O}{\displaystyle 1^O}{\displaystyle 1^O}{\displaystyle 1^O}{\displaystyle 1^O}}{\displaystyle 1^O}{\displaystyle 1^O}{\displaystyle 1^O^{\displaystyle 1^O^{\displaystyle 1^O^{{\}}}{\displaystyle 1^O^{\displaystyle 1^O^{\displaystyle 1^O^{\displaystyle 1^O^{\displaystyle 1^O^{\displaystyle 1^O^{\displaystyle 1^O^{\displaystyle 1}
  • 在ETH下,它不能在2^o(l) poly(n) 时间内解决.

结论:

  • 最大顶点-离合最短路径的近似算法接近于最佳,硬度结果与已知的界限相匹配.
  • 该研究提供了对问题的复杂性的全面了解,弥合了理论限制和实际可解决性之间的差距.
  • 硬度结果甚至适用于带有单位权重的非定向图,而正结果则一般用于带有任意非负权重的定向图.