[论文解读] Half-integrality, LP-branching and FPT algorithms

Magnus Wahlström|arXiv (Cornell University)|Jan 5, 2014

Advanced Graph Theory Research参考文献 47被引用 31

一句话总结

本文提出了一种新颖的框架，利用约束满足中的半整数松弛来实现通过线性规划分支的高效FPT算法，显著扩展了此类技术可适用的问题范围。该框架实现了更优的时间复杂度，包括在指数时间假设下最优或近似最优的O*(4^k)-时间算法求解Group Feedback Vertex Set，以及O*(|Σ|^{2k})-时间算法求解Unique Label Cover。

ABSTRACT

A recent trend in parameterized algorithms is the application of polytope tools (specifically, LP-branching) to FPT algorithms (e.g., Cygan et al., 2011; Narayanaswamy et al., 2012). Though the list of work in this direction is short, the results are already interesting, yielding significant speedups for a range of important problems. However, the existing approaches require the underlying polytope to have very restrictive properties, including half-integrality and Nemhauser-Trotter-style persistence properties. To date, these properties are essentially known to hold only for two classes of polytopes, covering the cases of Vertex Cover (Nemhauser and Trotter, 1975) and Node Multiway Cut (Garg et al., 1994).Taking a slightly different approach, we view half-integrality as a discrete relaxation of a problem, e.g., a relaxation of the search space from {0, 1}V to {0, 1/2, 1}V such that the new problem admits a polynomial-time exact solution. Using tools from CSP (in particular Thapper and Zivný, 2012) to study the existence of such relaxations, we are able to provide a much broader class of half-integral polytopes with the required properties.Our results unify and significantly extend the previously known cases. In addition to the new insight into problems with half-integral relaxations, our results yield a range of new and improved FPT algorithms, including an O*(|Σ|2k)-time algorithm for node-deletion Unique Label Cover with label set Σ (improving the previous bound of O*(|Σ|O(k2 log k) due to Chitnis et al., 2012) and an O*(4k)-time algorithm for Group Feedback Vertex Set, including the setting where the group is only given by oracle access (improving on the previous bound of O*(2O(k log k)) due to Cygan et al., 2012). The latter bound is optimal under the Exponential Time Hypothesis. The latter result also implies the first single-exponential time FPT algorithm for Subset Feedback Vertex Set, answering an open question of Cygan et al. (2012).Interestingly, despite the half-integrality, our result do not imply any approximation results (as may be expected, given the Unique Games-hardness of the covered problems).

研究动机与目标

• 将LP分支技术在FPT算法中的适用范围从具有半整数多面体的狭窄问题类别显著扩展。
• 识别出更广泛的问题类别，这些类别具有可在多项式时间内求解的半整数松弛。
• 利用CSP理论工具统一并推广先前关于Vertex Cover和Node Multiway Cut的研究结果。
• 为Group Feedback Vertex Set和Unique Label Cover等基础问题开发更快速的FPT算法。
• 回答开放性问题，包括首次为Subset Feedback Vertex Set提供单指数时间复杂度的FPT算法。

提出的方法

• 利用约束满足（CSP）的工具，特别是Thapper和Zivný（2012）的框架，来刻画半整数松弛存在的条件。
• 将解空间从{0,1}^V离散化为{0,1/2,1}^V，同时保持多项式时间可解性。
• 对这些半整数多面体应用LP分支，设计出时间复杂度更优的FPT算法。
• 在新一类半整数多面体中利用Nemhauser-Trotter风格的持久性性质。
• 证明该松弛在保留分支所需结构的同时，实现了更快的计算效率。
• 在Group Feedback Vertex Set中通过Oracle访问群结构，以维持对参数k的单指数依赖。

实验结果

研究问题

• RQ1 能否将可应用LP分支FPT算法的问题类别显著扩展至超越Vertex Cover和Node Multiway Cut的范围？
• RQ2 问题的何种结构特性可确保存在可在多项式时间内求解的半整数松弛？
• RQ3 能否系统性地使用CSP理论工具刻画半整数松弛，以支持更广泛的应用？
• RQ4 当群仅可通过Oracle访问时，能否为Group Feedback Vertex Set实现单指数时间复杂度的FPT算法？
• RQ5 使用半整数松弛是否能为Unique Label Cover及其相关问题带来更优的参数化算法？

主要发现

• 为具有标签集Σ的节点删除型Unique Label Cover问题实现了O*(|Σ|^{2k})-时间的FPT算法，优于先前的O*(|Σ|^{O(k^2 log k)})复杂度。
• 为Group Feedback Vertex Set问题开发了O*(4^k)-时间算法，该算法在指数时间假设下为最优。
• 该Group Feedback Vertex Set的新算法首次为Subset Feedback Vertex Set提供了单指数时间复杂度的FPT算法，解决了Cygan等人（2012）提出的开放问题。
• 该框架通过使用CSP工具识别出具有所需性质的新类别，将已知的半整数多面体推广至Vertex Cover和Node Multiway Cut之外。
• 尽管使用了半整数性，但结果并不蕴含近似保证，因为这些问题具有Unique Games-hard性。
• 该方法通过聚焦于可 tractable 半整数松弛的存在性，成功地将LP分支扩展到了更广泛的问题类别。