Vertex Cover parameterized by the solution size k is the quintessential fixed-parameter tractable problem. FPT algorithms are most interesting when the parameter is small. Several lower bounds on k are well-known, such as the maximum size of a matching. This has led to a line of research on parameterizations of Vertex Cover by the difference of the solution size k and a lower bound. The most prominent cases for such lower bounds for which the problem is FPT are the matching number or the optimal fractional LP solution. We investigate parameterizations by the difference between k and other graph parameters including the feedback vertex number, the degeneracy, cluster deletion number, and treewidth with the goal of finding the border of fixed-parameter tractability for said difference parameterizations. We also consider similar parameterizations of the Feedback Vertex Set problem.
翻译:由溶液大小 K 所设定的垂直覆盖参数参数是典型的固定参数可定位问题。 FPT 算法在参数小时最有趣。 k 上有几个下限是众所周知的, 例如匹配的最大大小。 这导致对Vetex 覆盖参数化的研究线, 其间溶液大小和下限的差别。 问题在于 FPT 的下限最突出的例子是匹配数或最佳的分数LP 解决方案。 我们根据 k 与其他图形参数的差异来调查参数化参数化, 包括反馈的顶点数、 退化性、 集组删除数字和树宽, 目的是找到上述差异参数化的固定参数可定位的边框。 我们还考虑反馈点设置问题的类似参数化。