4,287 research outputs found
A Faster Exact Algorithm for the Directed Maximum Leaf Spanning Tree Problem
Given a directed graph , the Directed Maximum Leaf Spanning Tree
problem asks to compute a directed spanning tree (i.e., an out-branching) with
as many leaves as possible. By designing a Branch-and-Reduced algorithm
combined with the Measure & Conquer technique for running time analysis, we
show that the problem can be solved in time \Oh^*(1.9043^n) using polynomial
space. Hitherto, there have been only few examples. Provided exponential space
this run time upper bound can be lowered to \Oh^*(1.8139^n)
Directed Hamiltonicity and Out-Branchings via Generalized Laplacians
We are motivated by a tantalizing open question in exact algorithms: can we
detect whether an -vertex directed graph has a Hamiltonian cycle in time
significantly less than ? We present new randomized algorithms that
improve upon several previous works:
1. We show that for any constant and prime we can count the
Hamiltonian cycles modulo in
expected time less than for a constant that depends only on and
. Such an algorithm was previously known only for the case of counting
modulo two [Bj\"orklund and Husfeldt, FOCS 2013].
2. We show that we can detect a Hamiltonian cycle in
time and polynomial space, where is the size of the maximum
independent set in . In particular, this yields an time
algorithm for bipartite directed graphs, which is faster than the
exponential-space algorithm in [Cygan et al., STOC 2013].
Our algorithms are based on the algebraic combinatorics of "incidence
assignments" that we can capture through evaluation of determinants of
Laplacian-like matrices, inspired by the Matrix--Tree Theorem for directed
graphs. In addition to the novel algorithms for directed Hamiltonicity, we use
the Matrix--Tree Theorem to derive simple algebraic algorithms for detecting
out-branchings. Specifically, we give an -time randomized algorithm
for detecting out-branchings with at least internal vertices, improving
upon the algorithms of [Zehavi, ESA 2015] and [Bj\"orklund et al., ICALP 2015].
We also present an algebraic algorithm for the directed -Leaf problem, based
on a non-standard monomial detection problem
Spotting Trees with Few Leaves
We show two results related to the Hamiltonicity and -Path algorithms in
undirected graphs by Bj\"orklund [FOCS'10], and Bj\"orklund et al., [arXiv'10].
First, we demonstrate that the technique used can be generalized to finding
some -vertex tree with leaves in an -vertex undirected graph in
time. It can be applied as a subroutine to solve the
-Internal Spanning Tree (-IST) problem in
time using polynomial space, improving upon previous algorithms for this
problem. In particular, for the first time we break the natural barrier of
. Second, we show that the iterated random bipartition employed by
the algorithm can be improved whenever the host graph admits a vertex coloring
with few colors; it can be an ordinary proper vertex coloring, a fractional
vertex coloring, or a vector coloring. In effect, we show improved bounds for
-Path and Hamiltonicity in any graph of maximum degree
or with vector chromatic number at most 8
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
08431 Abstracts Collection -- Moderately Exponential Time Algorithms
From to , the Dagstuhl Seminar 08431 ``Moderately Exponential Time Algorithms \u27\u27 was held in Schloss Dagstuhl~--~Leibniz Center for Informatics.
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
- …