20,899 research outputs found
Bounded Search Tree Algorithms for Parameterized Cograph Deletion: Efficient Branching Rules by Exploiting Structures of Special Graph Classes
Many fixed-parameter tractable algorithms using a bounded search tree have
been repeatedly improved, often by describing a larger number of branching
rules involving an increasingly complex case analysis. We introduce a novel and
general search strategy that branches on the forbidden subgraphs of a graph
class relaxation. By using the class of -sparse graphs as the relaxed
graph class, we obtain efficient bounded search tree algorithms for several
parameterized deletion problems. We give the first non-trivial bounded search
tree algorithms for the cograph edge-deletion problem and the trivially perfect
edge-deletion problems. For the cograph vertex deletion problem, a refined
analysis of the runtime of our simple bounded search algorithm gives a faster
exponential factor than those algorithms designed with the help of complicated
case distinctions and non-trivial running time analysis [21] and computer-aided
branching rules [11].Comment: 23 pages. Accepted in Discrete Mathematics, Algorithms and
Applications (DMAA
On the complexity of finding and counting solution-free sets of integers
Given a linear equation , a set of integers is
-free if does not contain any `non-trivial' solutions to
. This notion incorporates many central topics in combinatorial
number theory such as sum-free and progression-free sets. In this paper we
initiate the study of (parameterised) complexity questions involving
-free sets of integers. The main questions we consider involve
deciding whether a finite set of integers has an -free subset
of a given size, and counting all such -free subsets. We also
raise a number of open problems.Comment: 27 page
Hitting time for Bessel processes - walk on moving spheres algorithm (WoMS)
In this article we investigate the hitting time of some given boundaries for
Bessel processes. The main motivation comes from mathematical finance when
dealing with volatility models, but the results can also be used in optimal
control problems. The aim here is to construct a new and efficient algorithm in
order to approach this hitting time. As an application we will consider the
hitting time of a given level for the Cox-Ingersoll-Ross process. The main
tools we use are on one side, an adaptation of the method of images to this
particular situation and on the other side, the connection that exists between
Cox-Ingersoll-Ross processes and Bessel processes.Comment: Published in at http://dx.doi.org/10.1214/12-AAP900 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Lossy Kernelization
In this paper we propose a new framework for analyzing the performance of
preprocessing algorithms. Our framework builds on the notion of kernelization
from parameterized complexity. However, as opposed to the original notion of
kernelization, our definitions combine well with approximation algorithms and
heuristics. The key new definition is that of a polynomial size
-approximate kernel. Loosely speaking, a polynomial size
-approximate kernel is a polynomial time pre-processing algorithm that
takes as input an instance to a parameterized problem, and outputs
another instance to the same problem, such that . Additionally, for every , a -approximate solution
to the pre-processed instance can be turned in polynomial time into a
-approximate solution to the original instance .
Our main technical contribution are -approximate kernels of
polynomial size for three problems, namely Connected Vertex Cover, Disjoint
Cycle Packing and Disjoint Factors. These problems are known not to admit any
polynomial size kernels unless . Our approximate
kernels simultaneously beat both the lower bounds on the (normal) kernel size,
and the hardness of approximation lower bounds for all three problems. On the
negative side we prove that Longest Path parameterized by the length of the
path and Set Cover parameterized by the universe size do not admit even an
-approximate kernel of polynomial size, for any , unless
. In order to prove this lower bound we need to combine
in a non-trivial way the techniques used for showing kernelization lower bounds
with the methods for showing hardness of approximationComment: 58 pages. Version 2 contain new results: PSAKS for Cycle Packing and
approximate kernel lower bounds for Set Cover and Hitting Set parameterized
by universe siz
Kernels for Below-Upper-Bound Parameterizations of the Hitting Set and Directed Dominating Set Problems
In the {\sc Hitting Set} problem, we are given a collection of
subsets of a ground set and an integer , and asked whether has a
-element subset that intersects each set in . We consider two
parameterizations of {\sc Hitting Set} below tight upper bounds: and
. In both cases is the parameter. We prove that the first
parameterization is fixed-parameter tractable, but has no polynomial kernel
unless coNPNP/poly. The second parameterization is W[1]-complete,
but the introduction of an additional parameter, the degeneracy of the
hypergraph , makes the problem not only fixed-parameter
tractable, but also one with a linear kernel. Here the degeneracy of
is the minimum integer such that for each the
hypergraph with vertex set and edge set containing all edges of
without vertices in , has a vertex of degree at most
In {\sc Nonblocker} ({\sc Directed Nonblocker}), we are given an undirected
graph (a directed graph) on vertices and an integer , and asked
whether has a set of vertices such that for each vertex there is an edge (arc) from a vertex in to . {\sc Nonblocker} can be
viewed as a special case of {\sc Directed Nonblocker} (replace an undirected
graph by a symmetric digraph). Dehne et al. (Proc. SOFSEM 2006) proved that
{\sc Nonblocker} has a linear-order kernel. We obtain a linear-order kernel for
{\sc Directed Nonblocker}
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