使用邻域多样性解决困难问题
Using Neighborhood Diversity to Solve Hard Problems
摘要：
参数化算法是处理图上NP难问题的一个非常有用的工具。然而，为了正确地利用参数化算法，有必要根据问题的类型和目标图类的属性选择正确的参数。树宽是一个非常成功的图形参数的例子，但是它不能用于稠密的图形类，也存在一些问题，甚至在有界树宽的图形上也很难。此类问题可以通过使用顶点覆盖作为参数来解决，但这对可接受的图类施加了严格的限制。Michael Lampis最近引入了邻域多样性，这是一个新的图参数，它将顶点覆盖推广到稠密图。在其他结果中，他已经表明，简单的参数化算法存在的几个问题的图有界邻域多样性。我们的文章进一步研究了这一领域，并提供了新的算法参数的邻域多样性的p-顶点不相交的路径，图形图案和预着色扩展问题-后两个是硬的，甚至在图的有界树宽度。
英文摘要：
Parameterized algorithms are a very useful tool for dealing with NP-hard problems on graphs. Yet, to properly utilize parameterized algorithms it is necessary to choose the right parameter based on the type of problem and properties of the target graph class. Tree-width is an example of a very successful graph parameter, however it cannot be used on dense graph classes and there also exist problems which are hard even on graphs of bounded tree-width. Such problems can be tackled by using vertex cover as a parameter, however this places severe restrictions on admissible graph classes. Michael Lampis has recently introduced neighborhood diversity, a new graph parameter which generalizes vertex cover to dense graphs. Among other results, he has shown that simple parameterized algorithms exist for a few problems on graphs of bounded neighborhood diversity. Our article further studies this area and provides new algorithms parameterized by neighborhood diversity for the p-Vertex-Disjoint Paths, Graph Motif and Precoloring Extension problems -- the latter two being hard even on graphs of bounded tree-width.
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