1,280 research outputs found
Generating Functions For Kernels of Digraphs (Enumeration & Asymptotics for Nim Games)
In this article, we study directed graphs (digraphs) with a coloring
constraint due to Von Neumann and related to Nim-type games. This is equivalent
to the notion of kernels of digraphs, which appears in numerous fields of
research such as game theory, complexity theory, artificial intelligence
(default logic, argumentation in multi-agent systems), 0-1 laws in monadic
second order logic, combinatorics (perfect graphs)... Kernels of digraphs lead
to numerous difficult questions (in the sense of NP-completeness,
#P-completeness). However, we show here that it is possible to use a generating
function approach to get new informations: we use technique of symbolic and
analytic combinatorics (generating functions and their singularities) in order
to get exact and asymptotic results, e.g. for the existence of a kernel in a
circuit or in a unicircuit digraph. This is a first step toward a
generatingfunctionology treatment of kernels, while using, e.g., an approach "a
la Wright". Our method could be applied to more general "local coloring
constraints" in decomposable combinatorial structures.Comment: Presented (as a poster) to the conference Formal Power Series and
Algebraic Combinatorics (Vancouver, 2004), electronic proceeding
Out-degree reducing partitions of digraphs
Let be a fixed integer. We determine the complexity of finding a
-partition of the vertex set of a given digraph such
that the maximum out-degree of each of the digraphs induced by , () is at least smaller than the maximum out-degree of . We show
that this problem is polynomial-time solvable when and -complete otherwise. The result for and answers a question
posed in \cite{bangTCS636}. We also determine, for all fixed non-negative
integers , the complexity of deciding whether a given digraph of
maximum out-degree has a -partition such that the digraph
induced by has maximum out-degree at most for . It
follows from this characterization that the problem of deciding whether a
digraph has a 2-partition such that each vertex has at
least as many neighbours in the set as in , for is
-complete. This solves a problem from \cite{kreutzerEJC24} on
majority colourings.Comment: 11 pages, 1 figur
Beyond Bidimensionality: Parameterized Subexponential Algorithms on Directed Graphs
We develop two different methods to achieve subexponential time parameterized
algorithms for problems on sparse directed graphs. We exemplify our approaches
with two well studied problems.
For the first problem, {\sc -Leaf Out-Branching}, which is to find an
oriented spanning tree with at least leaves, we obtain an algorithm solving
the problem in time on directed graphs
whose underlying undirected graph excludes some fixed graph as a minor. For
the special case when the input directed graph is planar, the running time can
be improved to . The second example is a
generalization of the {\sc Directed Hamiltonian Path} problem, namely {\sc
-Internal Out-Branching}, which is to find an oriented spanning tree with at
least internal vertices. We obtain an algorithm solving the problem in time
on directed graphs whose underlying
undirected graph excludes some fixed apex graph as a minor. Finally, we
observe that for any , the {\sc -Directed Path} problem is
solvable in time , where is some
function of \ve.
Our methods are based on non-trivial combinations of obstruction theorems for
undirected graphs, kernelization, problem specific combinatorial structures and
a layering technique similar to the one employed by Baker to obtain PTAS for
planar graphs
Nondeterministic graph property testing
A property of finite graphs is called nondeterministically testable if it has
a "certificate" such that once the certificate is specified, its correctness
can be verified by random local testing. In this paper we study certificates
that consist of one or more unary and/or binary relations on the nodes, in the
case of dense graphs. Using the theory of graph limits, we prove that
nondeterministically testable properties are also deterministically testable.Comment: Version 2: 11 pages; we allow orientation in the certificate,
describe new application
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