96 research outputs found
Are there any good digraph width measures?
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
Clique-width: When Hard Does Not Mean Impossible
In recent years, the parameterized complexity approach has lead to the introduction of many new algorithms and frameworks on graphs and digraphs of bounded clique-width and, equivalently, rank-width. However, despite intensive work on the subject, there still exist well-established hard problems where neither a parameterized algorithm nor a theoretical obstacle to its existence are known. Our article is interested mainly in the digraph case, targeting the well-known Minimum Leaf Out-Branching (cf. also Minimum Leaf Spanning Tree) and Edge Disjoint Paths problems on digraphs of bounded clique-width with non-standard new approaches.
The first part of the article deals with the Minimum Leaf Out-Branching problem and introduces a novel XP-time algorithm wrt. clique-width. We remark that this problem is known to be W[2]-hard, and that our algorithm does not resemble any of the previously published attempts solving special cases of it such as the Hamiltonian Path. The second part then looks at the Edge Disjoint Paths problem (both on graphs and digraphs) from a different perspective -- rather surprisingly showing that this problem has a definition in the MSO_1 logic of graphs. The linear-time FPT algorithm wrt. clique-width then follows as a direct consequence
07281 Abstracts Collection -- Structure Theory and FPT Algorithmics for Graphs, Digraphs and Hypergraphs
From 8th to 13th July 2007, the Dagstuhl Seminar ``Structure Theory and FPT Algorithmics for Graphs, Digraphs and Hypergraphs\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available
Digraph Complexity Measures and Applications in Formal Language Theory
We investigate structural complexity measures on digraphs, in particular the
cycle rank. This concept is intimately related to a classical topic in formal
language theory, namely the star height of regular languages. We explore this
connection, and obtain several new algorithmic insights regarding both cycle
rank and star height. Among other results, we show that computing the cycle
rank is NP-complete, even for sparse digraphs of maximum outdegree 2.
Notwithstanding, we provide both a polynomial-time approximation algorithm and
an exponential-time exact algorithm for this problem. The former algorithm
yields an O((log n)^(3/2))- approximation in polynomial time, whereas the
latter yields the optimum solution, and runs in time and space O*(1.9129^n) on
digraphs of maximum outdegree at most two. Regarding the star height problem,
we identify a subclass of the regular languages for which we can precisely
determine the computational complexity of the star height problem. Namely, the
star height problem for bideterministic languages is NP-complete, and this
holds already for binary alphabets. Then we translate the algorithmic results
concerning cycle rank to the bideterministic star height problem, thus giving a
polynomial-time approximation as well as a reasonably fast exact exponential
algorithm for bideterministic star height.Comment: 19 pages, 1 figur
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