论文标题
亚地铁图中独立集的最终贪婪近似
Ultimate greedy approximation of independent sets in subcubic graphs
论文作者
论文摘要
我们研究有界度图中最大尺寸独立集（MIS）问题的近似性。这是最经典和研究的NP-硬化优化问题之一。我们专注于此问题的众所周知的最低度贪婪算法。该算法迭代地选择了图中的最低度顶点，将其添加到解决方案中并删除其邻居，直到其余图为空为止。该算法的近似比已经进行了广泛的研究，并通过建议将其增强，以告诉贪婪哪种最低度顶点是什么不是唯一的。我们的主要贡献是一种新的数学理论，用于设计这种贪婪算法，并有效地计算建议和分析其近似比。有了这一新理论，我们在图3上获得了最高学位的贪婪的最终近似值为5/4，这完全解决了Halldorsson和Yoshihara（1995）的论文中的开放问题。我们的算法是当前已知的算法最快的算法，在此类图上具有此近似值。我们将新算法应用于图3的最小顶点覆盖问题，以获得比当前已知的算法要快的6/5 approximation算法。我们将积极的上限结果与负面的下限结果补充，这证明为贪婪设计好的建议的问题在计算上很难，甚至很难在各个类别的图表上近似。这些结果显着改善了以前已知的硬度结果。此外，这些结果表明，在贪婪建议的设计和分析上获得上限的结果是不平凡的。
We study the approximability of the maximum size independent set (MIS) problem in bounded degree graphs. This is one of the most classic and widely studied NP-hard optimization problems. We focus on the well known minimum degree greedy algorithm for this problem. This algorithm iteratively chooses a minimum degree vertex in the graph, adds it to the solution and removes its neighbors, until the remaining graph is empty. The approximation ratios of this algorithm have been very widely studied, where it is augmented with an advice that tells the greedy which minimum degree vertex to choose if it is not unique. Our main contribution is a new mathematical theory for the design of such greedy algorithms with efficiently computable advice and for the analysis of their approximation ratios. With this new theory we obtain the ultimate approximation ratio of 5/4 for greedy on graphs with maximum degree 3, which completely solves the open problem from the paper by Halldorsson and Yoshihara (1995). Our algorithm is the fastest currently known algorithm with this approximation ratio on such graphs. We apply our new algorithm to the minimum vertex cover problem on graphs with maximum degree 3 to obtain a substantially faster 6/5-approximation algorithm than the one currently known. We complement our positive, upper bound results with negative, lower bound results which prove that the problem of designing good advice for greedy is computationally hard and even hard to approximate on various classes of graphs. These results significantly improve on such previously known hardness results. Moreover, these results suggest that obtaining the upper bound results on the design and analysis of greedy advice is non-trivial.
