103 research outputs found
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}
Hierarchies of Inefficient Kernelizability
The framework of Bodlaender et al. (ICALP 2008) and Fortnow and Santhanam
(STOC 2008) allows us to exclude the existence of polynomial kernels for a
range of problems under reasonable complexity-theoretical assumptions. However,
there are also some issues that are not addressed by this framework, including
the existence of Turing kernels such as the "kernelization" of Leaf Out
Branching(k) into a disjunction over n instances of size poly(k). Observing
that Turing kernels are preserved by polynomial parametric transformations, we
define a kernelization hardness hierarchy, akin to the M- and W-hierarchy of
ordinary parameterized complexity, by the PPT-closure of problems that seem
likely to be fundamentally hard for efficient Turing kernelization. We find
that several previously considered problems are complete for our fundamental
hardness class, including Min Ones d-SAT(k), Binary NDTM Halting(k), Connected
Vertex Cover(k), and Clique(k log n), the clique problem parameterized by k log
n
Alternative parameterizations of Metric Dimension
A set of vertices in a graph is called resolving if for any two
distinct , there is such that , where denotes the length of a shortest path
between and in the graph . The metric dimension of
is the minimum cardinality of a resolving set. The Metric Dimension problem,
i.e. deciding whether , is NP-complete even for interval
graphs (Foucaud et al., 2017). We study Metric Dimension (for arbitrary graphs)
from the lens of parameterized complexity. The problem parameterized by was
proved to be -hard by Hartung and Nichterlein (2013) and we study the
dual parameterization, i.e., the problem of whether
where is the order of . We prove that the dual parameterization admits
(a) a kernel with at most vertices and (b) an algorithm of runtime
Hartung and Nichterlein (2013) also observed that Metric
Dimension is fixed-parameter tractable when parameterized by the vertex cover
number of the input graph. We complement this observation by showing
that it does not admit a polynomial kernel even when parameterized by . Our reduction also gives evidence for non-existence of polynomial Turing
kernels
Parameterized Study of the Test Cover Problem
We carry out a systematic study of a natural covering problem, used for
identification across several areas, in the realm of parameterized complexity.
In the {\sc Test Cover} problem we are given a set of items
together with a collection, , of distinct subsets of these items called
tests. We assume that is a test cover, i.e., for each pair of items
there is a test in containing exactly one of these items. The
objective is to find a minimum size subcollection of , which is still a
test cover. The generic parameterized version of {\sc Test Cover} is denoted by
-{\sc Test Cover}. Here, we are given and a
positive integer parameter as input and the objective is to decide whether
there is a test cover of size at most . We study four
parameterizations for {\sc Test Cover} and obtain the following:
(a) -{\sc Test Cover}, and -{\sc Test Cover} are fixed-parameter
tractable (FPT).
(b) -{\sc Test Cover} and -{\sc Test Cover} are
W[1]-hard. Thus, it is unlikely that these problems are FPT
The Directed Dominating Set Problem: Generalized Leaf Removal and Belief Propagation
A minimum dominating set for a digraph (directed graph) is a smallest set of
vertices such that each vertex either belongs to this set or has at least one
parent vertex in this set. We solve this hard combinatorial optimization
problem approximately by a local algorithm of generalized leaf removal and by a
message-passing algorithm of belief propagation. These algorithms can construct
near-optimal dominating sets or even exact minimum dominating sets for random
digraphs and also for real-world digraph instances. We further develop a core
percolation theory and a replica-symmetric spin glass theory for this problem.
Our algorithmic and theoretical results may facilitate applications of
dominating sets to various network problems involving directed interactions.Comment: 11 pages, 3 figures in EPS forma
A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms
Parameterization and approximation are two popular ways of coping with
NP-hard problems. More recently, the two have also been combined to derive many
interesting results. We survey developments in the area both from the
algorithmic and hardness perspectives, with emphasis on new techniques and
potential future research directions
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
- …