[Paper Review] Acyclic coloring of special digraphs

Frank Gurski, Dominique Komander+1 more|arXiv (Cornell University)|Jun 24, 2020

Advanced Graph Theory Research52 references4 citations

TL;DR

This paper presents the first polynomial-time algorithm for computing the dichromatic number of directed graphs with constant directed clique-width, using bottom-up dynamic programming on a directed clique-width expression tree. The key contribution is an XP algorithm parameterized by directed clique-width and an FPT algorithm when parameterized by both directed clique-width and the number of colors, with linear-time solutions for directed co-graphs and their generalizations.

ABSTRACT

An acyclic r-coloring of a directed graph G=(V,E) is a partition of the vertex set V into r acyclic sets. The dichromatic number of a directed graph G is the smallest r such that G allows an acyclic r-coloring. For symmetric digraphs the dichromatic number equals the well-known chromatic number of the underlying undirected graph. This allows us to carry over the W[1]-hardness and lower bounds for running times of the chromatic number problem parameterized by clique-width to the dichromatic number problem parameterized by directed clique-width. We introduce the first polynomial-time algorithm for the acyclic coloring problem on digraphs of constant directed clique-width. From a parameterized point of view our algorithm shows that the Dichromatic Number problem is in XP when parameterized by directed clique-width and extends the only known structural parameterization by directed modular width for this problem. Furthermore, we apply defineability within monadic second order logic in order to show that Dichromatic Number problem is in FPT when parameterized by the directed clique-width and r. For directed co-graphs, which is a class of digraphs of directed clique-width 2, and several generalizations we even show linear time solutions for computing the dichromatic number. Furthermore, we conclude that directed co-graphs and the considered generalizations lead to subclasses of perfect digraphs. For directed cactus forests, which is a set of digraphs of directed tree-width 1, we conclude an upper bound of 2 for the dichromatic number and we show that an optimal acyclic coloring can be computed in linear time.

Motivation & Objective

• To develop efficient algorithms for the dichromatic number problem on special classes of directed graphs.
• To address the computational complexity of acyclic coloring in directed graphs, particularly for classes with bounded structural parameters.
• To extend parameterized complexity results from undirected graph coloring to directed graphs using directed clique-width.
• To show that the dichromatic number problem is in FPT when parameterized by directed clique-width and the number of colors.
• To identify subclasses of perfect digraphs, such as directed co-graphs and directed cactus forests, where the dichromatic number can be computed efficiently.

Proposed method

• Using bottom-up dynamic programming on a rooted expression tree derived from a directed clique-width expression of the input digraph.
• For each subtree, maintaining label-based reachability information via multi-sets of reachability pairs for each acyclic color class.
• Computing the minimal number of colors required such that each color class induces an acyclic subdigraph.
• Applying monadic second-order logic (MSO) definability to show FPT membership when parameterized by directed clique-width and the number of colors.
• Leveraging known structural properties of directed co-graphs (clique-width 2) and directed cactus forests (tree-width 1) to derive linear-time algorithms.
• Transforming the acyclic coloring problem into an integer program with O(n²) variables to show FPT membership when parameterized by the number of vertices.

Experimental results

Research questions

• RQ1 Can the dichromatic number problem be solved in polynomial time for digraphs of constant directed clique-width?
• RQ2 Is the dichromatic number problem fixed-parameter tractable when parameterized by directed clique-width and the number of colors?
• RQ3 Do directed co-graphs and their generalizations admit linear-time algorithms for computing the dichromatic number?
• RQ4 Can the dichromatic number problem be solved in XP time when parameterized by directed clique-width?
• RQ5 Is the dichromatic number problem FPT when parameterized by the clique-width of the underlying undirected graph?

Key findings

• The paper presents the first polynomial-time algorithm for the dichromatic number problem on digraphs of constant directed clique-width, with a running time of n^{2^{O(k²)}} where k is the directed clique-width.
• The dichromatic number problem is shown to be in FPT when parameterized by both directed clique-width and the number of colors, using MSO definability.
• For directed co-graphs and their generalizations, which have directed clique-width 2, the dichromatic number can be computed in linear time.
• Directed cactus forests, which have directed tree-width 1, have a dichromatic number of at most 2, and an optimal acyclic coloring can be computed in linear time.
• The problem is not in FPT when parameterized by the clique-width of the underlying undirected graph, unless P = NP.
• The problem is not in XP when parameterized by the number of colors r, since DCN₂ is NP-complete.