Are there any good digraph width measures?
📝 Abstract
Several different measures for digraph width have appeared in the last few years. However, none of them shares all the “nice” properties of treewidth: First, being \emph{algorithmically useful} i.e. admitting polynomial-time algorithms for all $\MS1 $-definable problems on digraphs of bounded width. And, second, having nice \emph{structural properties} i.e. being monotone under taking subdigraphs and some form of arc contractions. As for the former, (undirected) $\MS1$ seems to be the least common denominator of all reasonably expressive logical languages on digraphs that can speak about the edge/arc relation on the vertex set.The latter property is a necessary condition for a width measure to be characterizable by some version of the cops-and-robber game characterizing the ordinary treewidth. Our main result is that \emph{any reasonable} algorithmically useful and structurally nice digraph measure cannot be substantially different from the treewidth of the underlying undirected graph. Moreover, we introduce \emph{directed topological minors} and argue that they are the weakest useful notion of minors for digraphs.
💡 Analysis
Several different measures for digraph width have appeared in the last few years. However, none of them shares all the “nice” properties of treewidth: First, being \emph{algorithmically useful} i.e. admitting polynomial-time algorithms for all $\MS1 $-definable problems on digraphs of bounded width. And, second, having nice \emph{structural properties} i.e. being monotone under taking subdigraphs and some form of arc contractions. As for the former, (undirected) $\MS1$ seems to be the least common denominator of all reasonably expressive logical languages on digraphs that can speak about the edge/arc relation on the vertex set.The latter property is a necessary condition for a width measure to be characterizable by some version of the cops-and-robber game characterizing the ordinary treewidth. Our main result is that \emph{any reasonable} algorithmically useful and structurally nice digraph measure cannot be substantially different from the treewidth of the underlying undirected graph. Moreover, we introduce \emph{directed topological minors} and argue that they are the weakest useful notion of minors for digraphs.
📄 Content
An intensely investigated field in algorithmic graph theory is the design of graph width parameters that satisfy two seemingly contradictory requirements: (1) graphs of bounded width should have a reasonably rich structure; and, (2) a large class of problems must be efficiently solvable on graphs of bounded width. For undirected graphs, research into width parameters has been extremely successful with a number of algorithmically useful measures being proposed over the years, chief among them being treewidth [16], clique-width [6], branchwidth [18] and related measures (see also [3]). Many problems that are hard on general graphs turned out to be tractable on graphs of bounded treewidth. These results were combined and generalized by Courcelle’s celebrated theorem which states that a very large class of problems (MSO 2 ) is tractable on graphs of bounded treewidth [4].
However, there still do not exist directed graph width measures that are as successful as treewidth. This is because, despite many achievements and interesting results, most known digraph width measures do not allow for efficient algorithms for many problems. During the last decade, many digraph width measures were introduced, the prominent ones being directed treewidth [12], DAG-width [2,14], and Kelly-width [11]. These width measures proved useful for some problems. For instance, one can obtain polynomial-time (XP to be more precise) algorithms for Hamiltonian Path on digraphs of bounded directed treewidth [12] and for Parity Games on digraphs of bounded DAG-width [2] and Kelly-width [11]. But there is the negative side, too. Hamiltonian Path, for instance, likely cannot be solved on digraphs of directed treewidth, DAG-width, or Kelly-width at most k in time O(f (k) • n c ), where c is a constant independent of k. Note that Hamiltonian Path can be solved in such a running time for undirected graphs of treewidth at most k [4].
Additionally, for the newly introduced [9] DAG-depth and Kenny-width 3 -digraph width measures that are much more restrictive than DAG-width -problems such as Directed Dominating Set, Directed Cut and k-Path remain NP-complete on digraphs of constant width [9]. In contrast, another recent digraph measure bi-rank-width [13] looks more promising. A Courcelle-like MSO 1 theorem exists for digraphs of bounded bi-rank-width, and many other interesting problems can be solved in polynomial (XP) time on these [13,10]. For a recent survey on complexity results for DAG-width, Kelly-width, bi-rank-width, and other digraph measures, see [9].
In this paper, we boldly ask whether there exist digraph width measures that are algorithmically useful, and if so what properties can they be expected to satisfy. We first address the question of what it means for a width measure to be algorithmically useful. While there is no formal definition of this notion, we appeal to what is known about width measures for undirected graphs, in particular, about treewidth. As mentioned earlier, Courcelle’s Theorem states that all problems expressible in MSO 2 logic are (fixed-parameter, or FPT) tractable on graphs of bounded treewidth. It would be nice to have such a strong result for a digraph width measure, but to this day there exists no widely accepted logical language specifically aimed at digraphs at all. This fact then prompts us to consider the least common denominator of all possible descriptive languages over digraphs that; a) have sufficient expressive power (meaning they can quantify over sets, not only over singletons), and b) can identify the arc/edge relation over the vertex set. Clearly, this least common denominator includes at least the ordinary MSO 1 logic (see Section 2) of the underlying undirected graph.
We thus define algorithmic usefulness as the property of admitting polynomial-time (XP to be precise) algorithms for all MSO 1 -definable problems on digraphs of bounded width as the parameter. Note that we even relax the required time bound from FPT-time to XP-time (see in Section 2). It is easy to see that algorithmically useful digraph width measures do indeed exist. Besides some simplistic examples, such as the measure that counts the number of vertices in the input graph, there is the treewidth of the underlying undirected graph. In the latter case we can apply the rich theory of (undirected) graphs of bounded treewidth, but we would not get anything substantially new for digraphs. As such, we are interested in digraph width measures that are incomparable to undirected treewidth.
Our second question is what properties can an algorithmically useful digraph width measure be expected to satisfy. In particular, can we expect any such properties typical for undirected width measures also in the directed case? An important feature of treewidth is that it allows a copsand-robber game characterization. In fact, several digraph width measures such as DAG-width [2,14], Kelly-width [11], and DAG-depth [9] admit some variants of a cops
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