16 research outputs found
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
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
Are there any good digraph width measures?
Many width measures for directed graphs have been proposed in the last few years in pursuit of generalizing (the notion of) treewidth to directed graphs. However, none of these measures possesses, at the same time, the major properties of treewidth, namely, 1. being algorithmically useful , that is, admitting polynomial-time algorithms for a large class of problems on digraphs of bounded width (e.g. the problems definable in MSO1MSO1); 2. having nice structural properties such as being (at least nearly) monotone under taking subdigraphs and some form of arc contractions (property closely related to characterizability by particular cops-and-robber games). We investigate the question whether the search for directed treewidth counterparts has been unsuccessful by accident, or whether it has been doomed to fail from the beginning. Our main result states that any reasonable width measure for directed graphs which satisfies the two properties above must necessarily be similar to treewidth of the underlying undirected graph
An Algorithmic Metatheorem for Directed Treewidth
The notion of directed treewidth was introduced by Johnson, Robertson,
Seymour and Thomas [Journal of Combinatorial Theory, Series B, Vol 82, 2001] as
a first step towards an algorithmic metatheory for digraphs. They showed that
some NP-complete properties such as Hamiltonicity can be decided in polynomial
time on digraphs of constant directed treewidth. Nevertheless, despite more
than one decade of intensive research, the list of hard combinatorial problems
that are known to be solvable in polynomial time when restricted to digraphs of
constant directed treewidth has remained scarce. In this work we enrich this
list by providing for the first time an algorithmic metatheorem connecting the
monadic second order logic of graphs to directed treewidth. We show that most
of the known positive algorithmic results for digraphs of constant directed
treewidth can be reformulated in terms of our metatheorem. Additionally, we
show how to use our metatheorem to provide polynomial time algorithms for two
classes of combinatorial problems that have not yet been studied in the context
of directed width measures. More precisely, for each fixed , we show how to count in polynomial time on digraphs of directed
treewidth , the number of minimum spanning strong subgraphs that are the
union of directed paths, and the number of maximal subgraphs that are the
union of directed paths and satisfy a given minor closed property. To prove
our metatheorem we devise two technical tools which we believe to be of
independent interest. First, we introduce the notion of tree-zig-zag number of
a digraph, a new directed width measure that is at most a constant times
directed treewidth. Second, we introduce the notion of -saturated tree slice
language, a new formalism for the specification and manipulation of infinite
sets of digraphs.Comment: 41 pages, 6 figures, Accepted to Discrete Applied Mathematic
Computations by fly-automata beyond monadic second-order logic
We present logically based methods for constructing XP and FPT graph
algorithms, parametrized by tree-width or clique-width. We will use
fly-automata introduced in a previous article. They make possible to check
properties that are not monadic second-order expressible because their states
may include counters, so that their sets of states may be infinite. We equip
these automata with output functions, so that they can compute values
associated with terms or graphs. Rather than new algorithmic results we present
tools for constructing easily certain dynamic programming algorithms by
combining predefined automata for basic functions and properties.Comment: Accepted for publication in Theoretical Computer Scienc
Directed branch-width: A directed analogue of tree-width
We introduce a new digraph width measure called directed branch-width. To do
this, we generalize a characterization of graph classes of bounded tree-width
in terms of their line graphs to digraphs.
Under parameterizations by directed branch-width we obtain linear time
algorithms for many problems, such as directed Hamilton path and Max-Cut, which
are hard when parameterized by other known directed width measures. More
generally, we obtain an algorithmic meta-theorem for the model-checking problem
for a restricted variant of MSO_2-logic on classes of bounded directed
branch-width