We study the following two fixed-cardinality optimization problems (a maximization and a minimization variant). For a fixed $\alpha$ between zero and one we are given a graph and two numbers $k \in \mathbb{N}$ and $t \in \mathbb{Q}$. The task is to find a vertex subset $S$ of exactly $k$ vertices that has value at least (resp. at most for minimization) $t$. Here, the value of a vertex set computes as $\alpha$ times the number of edges with exactly one endpoint in $S$ plus $1-\alpha$ times the number of edges with both endpoints in $S$. These two problems generalize many prominent graph problems, such as Densest $k$-Subgraph, Sparsest $k$-Subgraph, Partial Vertex Cover, and Max ($k$,$n-k$)-Cut. In this work, we complete the picture of their parameterized complexity on several types of sparse graphs that are described by structural parameters. In particular, we provide kernelization algorithms and kernel lower bounds for these problems. A somewhat surprising consequence of our kernelizations is that Partial Vertex Cover and Max $(k,n-k)$-Cut not only behave in the same way but that the kernels for both problems can be obtained by the same algorithms.
翻译:我们研究了以下两个固定心率优化问题( 最大化和最小化变体 ) 。 对于一个在零到一之间固定的 alpha$, 我们给出了一个图表和两个数字 $k $ $\ $\ mathb{N} $ 和 $ t $ $ $ 美元。 任务在于找到一个顶点子子子 $S, 确切的为 $k美元, 其价值至少( 最多可以最小化) $t 。 这里, 顶点的值是 $ + $ 和 1 - alpha$ 的两倍 。 对于结构参数所描述的“ 顶端点 ”, 顶点是许多突出的图形问题, 例如 Densest $k suphrat, sprassarst $k $k$- Subgraph, 部分 Vertex Cover, 和 Max $k, $n- k)- Cut。 在此工作中, 我们只能通过 缩略图解解解解的“ ” 的“ 底端点” 问题, 我们只能用“ ” 的“ ” 的“ 的“ 底端点” 问题” 的“ 。