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Coloring and Recognizing Directed Interval Graphs

Grzegorz Gutowski, Konstanty Junosza-Szaniawski, Felix Klesen, Paweł Rzążewski, Alexander Wolff, Johannes Zink

TL;DR

The paper studies coloring mixed interval graphs where containment yields oriented arcs and overlaps yield undirected edges, introducing containment interval graphs as a natural geometric subclass. It proves recognition in polynomial time $O(nm)$ via a PQ-tree-based procedure and rotation to obtain a containment representation, while showing that optimal coloring is NP-hard for containment and bidirectional interval graphs. A 2-approximation algorithm for coloring containment interval graphs is given, with a tight bound of $\chi \le 2\omega-1$ and an $O(n\log n)$ running time. For general mixed interval graphs without directed cycles, it shows $\chi(G) \le \omega(G) \cdot (\lambda(G) + 1)$ and provides a $\min\{\omega, \lambda+1\}$-approximation, highlighting both the algorithmic prospects and inherent hardness in these geometric graph classes.

Abstract

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.

Coloring and Recognizing Directed Interval Graphs

TL;DR

The paper studies coloring mixed interval graphs where containment yields oriented arcs and overlaps yield undirected edges, introducing containment interval graphs as a natural geometric subclass. It proves recognition in polynomial time via a PQ-tree-based procedure and rotation to obtain a containment representation, while showing that optimal coloring is NP-hard for containment and bidirectional interval graphs. A 2-approximation algorithm for coloring containment interval graphs is given, with a tight bound of and an running time. For general mixed interval graphs without directed cycles, it shows and provides a -approximation, highlighting both the algorithmic prospects and inherent hardness in these geometric graph classes.

Abstract

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 , an interval receives a lower (different) color than an interval if contains arc (edge ). 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 -approximation algorithm, where is the size of a largest clique and is the length of a longest directed path in . 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 if interval contains interval , and there is an edge if and 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.
Paper Structure (2 sections, 3 theorems, 1 figure, 1 table)

This paper contains 2 sections, 3 theorems, 1 figure, 1 table.

Table of Contents

  1. Recognition of Containment Interval Graphs
  2. A 2-Approximation Algorithm for Coloring Containment Interval Graphs

Key Result

Theorem 1

There is an algorithm that, given a mixed graph $G$, decides whether $G$ is a containment interval graph. The algorithm runs in $O(nm)$ time, where $n$ is the number of vertices of $G$ and $m$ is the total number of edges and arcs of $G$, and produces a containment representation of $G$ if $G$ admit

Figures (1)

  • Figure 1: Let $\mathcal{D}$, $\mathcal{B}$, and $\mathcal{C}$ be the classes of directional, bidirectional, and containment interval graphs. Clearly, $\mathcal{D} \subseteq \mathcal{B}$. The above sets of intervals and the corresponding directed graphs show that the classes $\mathcal{D}$ and $\mathcal{B}$ are incomparable with the class $\mathcal{C}$ and that $\mathcal{D}$ is properly contained in $\mathcal{B}$.

Theorems & Definitions (3)

  • Theorem 1: \ref{['thm:recognition*']}
  • Theorem 2
  • Corollary 5