We study the parameterized complexity of various classic vertex-deletion problems such as Odd cycle transversal, Vertex planarization, and Chordal vertex deletion under hybrid parameterizations. Existing FPT algorithms for these problems either focus on the parameterization by solution size, detecting solutions of size $k$ in time $f(k) \cdot n^{O(1)}$, or width parameterizations, finding arbitrarily large optimal solutions in time $f(w) \cdot n^{O(1)}$ for some width measure $w$ like treewidth. We unify these lines of research by presenting FPT algorithms for parameterizations that can simultaneously be arbitrarily much smaller than the solution size and the treewidth. We consider two classes of parameterizations which are relaxations of either treedepth of treewidth. They are related to graph decompositions in which subgraphs that belong to a target class H (e.g., bipartite or planar) are considered simple. First, we present a framework for computing approximately optimal decompositions for miscellaneous classes H. Namely, if the cost of an optimal decomposition is $k$, we show how to find a decomposition of cost $k^{O(1)}$ in time $f(k) \cdot n^{O(1)}$. This is applicable to any graph class H for which the corresponding vertex-deletion problem admits a constant-factor approximation algorithm or an FPT algorithm paramaterized by the solution size. Secondly, we exploit the constructed decompositions for solving vertex-deletion problems by extending ideas from algorithms using iterative compression and the finite state property. For the three mentioned vertex-deletion problems, and all problems which can be formulated as hitting a finite set of connected forbidden (a) minors or (b) (induced) subgraphs, we obtain FPT algorithms with respect to both studied parameterizations.
翻译:我们研究各种典型的顶端递解问题的参数复杂性, 如奥德周期转折、 Vertex 平面图和Chordal 顶点在混合参数化下删除。 这些问题的现有 FPT 算法要么关注以溶解大小计算的参数化参数化, 发现美元大小的溶液, 或宽度参数化, 在时间( f( k)\ cdotn ⁇ O(1)} 或宽度参数化中找到任意的大型最佳解决方案, 诸如树宽度等宽度度测量$w美元。 我们将这些研究线统一起来, 我们为参数化提供FPT 算法, 这些算法可以任意地大大小于溶解析的溶液大小和树宽度。 我们考虑两种参数化的解析方法, 这些方法可以松动树枝叶色深度的树枝。 它们与图形解析分解的分解状态( 例如, bipartite, 或plantarnalalations) 可以简单化为目标级( 例如, lideal- dal- devoild) listal devodition laveal) ex) orations a modemotion lademodemodemotion, lax the demotion a motional demotional demotional demotion.