给定一个有向图$D$，最小叶子分支问题（MinLOB）是在$D$中找到具有最小可能叶子数量的分支的问题，即，出度为0的顶点。Gutin，Razgon and Kim（2008）证明了MinLOB对于无圈有向图是多项式时间可解的，无圈有向图恰好是有向路宽（DAG-宽度，有向树宽）为0的有向图。我们调查有多少人可以延长这个多项式的结果。我们已经证明，有向路径宽度（有向树宽度，DAG宽度，分别）1，MinLOB是NP-困难的有向图。另一方面，我们表明，限制有向树宽度（有向路径宽度，DAG宽度，分别）和一个固定的整数$k$的有向图，检查是否有一个最多$k$叶的外分支的问题是多项式时间可解的。
英文摘要：
Given a digraph $D$, the Minimum Leaf Out-Branching problem (MinLOB) is the problem of finding in $D$ an out-branching with the minimum possible number of leaves, i.e., vertices of out-degree 0. Gutin, Razgon and Kim (2008) proved that MinLOB is polynomial time solvable for acyclic digraphs which are exactly the digraphs of directed path-width (DAG-width, directed tree-width, respectively) 0. We investigate how much one can extend this polynomiality result. We prove that already for digraphs of directed path-width (directed tree-width, DAG-width, respectively) 1, MinLOB is NP-hard. On the other hand, we show that for digraphs of restricted directed tree-width (directed path-width, DAG-width, respectively) and a fixed integer $k$, the problem of checking whether there is an out-branching with at most $k$ leaves is polynomial time solvable.
Kernel(s) for Problems With no Kernel: On Out-Trees With Many Leaves
The { \sc $k$ -Leaf Out-Branching} problem is to find an out-branching (i.e. a rooted oriented spanning tree) with at least $k$ leaves in a given digraph. The problem has recently received much attention from the viewpoint of parameterized algorithms {alonLNCS4596,AlonFGKS07fsttcs,BoDo2,KnLaRo}. In this paper we step aside and take a kernelization based approach to the { \sc $k$ -Leaf-Out-Branching} problem. We give the first polynomial kernel for { \sc Rooted $k$ -Leaf-Out-Branching}, a variant of { \sc $k$ -Leaf-Out-Branching} where the root of the tree searched for is also a part of the inp…
Better Algorithms and Bounds for Directed Maximum Leaf Problems
{ \sc有向最大叶子出分支}问题是在给定的有向图中找到具有最大叶子数的出分支（即有根有向生成树）。在本文中，我们改进了已知的参数化算法和组合界的叶数在出分支。我们表明，\begin{itemize} \item every strongly connected digraph $D$ of order $n$ with minimum in-degree at least 3 has an out-branching with at least $(n/4)^{1/3}-1$ leaves; \item if a strongly connected digraph $D$ does not contain an out-branching with $k$ leaves, then the pathwidth of its underlying graph is $O(k\log k)$; \item it can be decided in time $2^{O(k\log^2 k)}\cdot n^{O(1)}$ whether a strongly connected digraph on $n$ vertices has an out-branching with at least $k$ leaves. \end{itemize}所有的改进使用的极值结构后，应用本地搜…
Perfect phylogenies via branchings in acyclic digraphs and a generalization of Dilworth's theorem
受癌症基因组学应用的启发，并遵循Hajirasouliha和Raphael的工作（WABI 2014），Hujdurović et al.（IEEE TCBB，即将出现）介绍了最小无冲突行分割（MCRS）问题：将给定二进制矩阵的每一行拆分为一组行的按位或，使得所得矩阵对应于完美的同构，并且在所有矩阵中具有最小可能的行数与此属性。Hajirasouliha和Raphael也提出了类似问题的研究，其中的任务是最小化所得矩阵的不同行的数量。Hujdurović等人证明了这两个问题都是NP-难的，给出了传递可定向图的相关刻画，并提出了一个基于着色协可比图的MCRS问题的多项式时间启发式算法。我们给出了新的，更透明的配方的两个问题，表明这些问题是等价于两个优化问题的分支在派生的有向无环图。在这些公式的基础上，我们得到了关于这两个问题的新结果，包括：（i）Hujdurović等人通过推广Dilworth定理的有向图中的新的最小-最大结果来加强启发式，这可能是独立的兴趣，（ii）两个问题的APX-硬度结果，（iii）近似算法，和（iv）指数时间算法解决这两个问题的最优性比天真的蛮力方法更快。我们的工作涉及到几个很好的研究概念在组合优化：链分区在偏序集，层状超图，（经典和加权）着色的图形。
