有向区间图的着色与识别
Coloring and Recognizing Directed Interval Graphs
AI总结：
本文研究混合区间图着色，给出一般情形 min{ω,λ+1} 近似，证明双向区间图最优着色 NP 难，并提出可多项式识别、着色具 2 近似的包含区间图。
摘要：
混合区间图是这样一种区间图：对于每一对相交的区间，它们之间或者有一条弧（方向任意），或者有一条（无向）边。我们尤其关注边和弧由区间几何关系定义的情形。在混合区间图 G 的正常着色中，如果 G 包含弧 (u,v)，则区间 u 获得比区间 v 更低的颜色；如果包含边 {u,v}，则二者获得不同的颜色。混合图着色在例如带优先约束的调度中有应用；参见 Sotskov 的综述[Mathematics, 2020]。对于一般混合区间图的着色，我们提出一种 min{ω(G), λ(G)+1}-近似算法，其中 ω(G) 是最大团的大小，λ(G) 是 G 中最长有向路径的长度。对于双向区间图（最近因图绘制应用而引入）这一子类，我们证明最优着色是 NP 难的；这一点此前对一般混合区间图已知。我们引入一种新的自然混合区间图类别，称之为包含区间图。在此类图中，如果区间 u 包含区间 v，则存在弧 (u,v)；如果 u 与 v 重叠，则存在边 {u,v}。我们证明这些图可在多项式时间内识别，用最少颜色为其着色是 NP 难的，并且存在一种着色的 2-近似算法。
英文摘要：
A \emph{mixed interval graph} is an interval graph that has, for every pair of intersecting intervals, either an arc (directed arbitrarily) or an (undirected) edge. We are particularly interested in scenarios where edges and arcs are defined by the geometry of intervals. In a proper coloring of a mixed interval graph $G$, an interval $u$ receives a lower (different) color than an interval $v$ if $G$ contains arc $(u,v)$ (edge $\{u,v\}$). Coloring of mixed graphs has applications, for example, in scheduling with precedence constraints; see a survey by Sotskov [Mathematics, 2020]. For coloring general mixed interval graphs, we present a $\min \{ω(G), λ(G)+1 \}$-approximation algorithm, where $ω(G)$ is the size of a largest clique and $λ(G)$ is the length of a longest directed path in $G$. For the subclass of \emph{bidirectional interval graphs} (introduced recently for an application in graph drawing), we show that optimal coloring is NP-hard. This was known for general mixed interval graphs. We introduce a new natural class of mixed interval graphs, which we call \emph{containment interval graphs}. In such a graph, there is an arc $(u,v)$ if interval $u$ contains interval $v$, and there is an edge $\{u,v\}$ if $u$ and $v$ overlap. We show that these graphs can be recognized in polynomial time, that coloring them with the minimum number of colors is NP-hard, and that there is a 2-approximation algorithm for coloring.
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