94 research outputs found
Towards Work-Efficient Parallel Parameterized Algorithms
Parallel parameterized complexity theory studies how fixed-parameter
tractable (fpt) problems can be solved in parallel. Previous theoretical work
focused on parallel algorithms that are very fast in principle, but did not
take into account that when we only have a small number of processors (between
2 and, say, 1024), it is more important that the parallel algorithms are
work-efficient. In the present paper we investigate how work-efficient fpt
algorithms can be designed. We review standard methods from fpt theory, like
kernelization, search trees, and interleaving, and prove trade-offs for them
between work efficiency and runtime improvements. This results in a toolbox for
developing work-efficient parallel fpt algorithms.Comment: Prior full version of the paper that will appear in Proceedings of
the 13th International Conference and Workshops on Algorithms and Computation
(WALCOM 2019), February 27 - March 02, 2019, Guwahati, India. The final
authenticated version is available online at
https://doi.org/10.1007/978-3-030-10564-8_2
Point Line Cover: The Easy Kernel is Essentially Tight
The input to the NP-hard Point Line Cover problem (PLC) consists of a set
of points on the plane and a positive integer , and the question is
whether there exists a set of at most lines which pass through all points
in . A simple polynomial-time reduction reduces any input to one with at
most points. We show that this is essentially tight under standard
assumptions. More precisely, unless the polynomial hierarchy collapses to its
third level, there is no polynomial-time algorithm that reduces every instance
of PLC to an equivalent instance with points, for
any . This answers, in the negative, an open problem posed by
Lokshtanov (PhD Thesis, 2009).
Our proof uses the machinery for deriving lower bounds on the size of kernels
developed by Dell and van Melkebeek (STOC 2010). It has two main ingredients:
We first show, by reduction from Vertex Cover, that PLC---conditionally---has
no kernel of total size bits. This does not directly imply
the claimed lower bound on the number of points, since the best known
polynomial-time encoding of a PLC instance with points requires
bits. To get around this we build on work of Goodman et al.
(STOC 1989) and devise an oracle communication protocol of cost
for PLC; its main building block is a bound of for the order
types of points that are not necessarily in general position, and an
explicit algorithm that enumerates all possible order types of n points. This
protocol and the lower bound on total size together yield the stated lower
bound on the number of points.
While a number of essentially tight polynomial lower bounds on total sizes of
kernels are known, our result is---to the best of our knowledge---the first to
show a nontrivial lower bound for structural/secondary parameters
Paradigms for Parameterized Enumeration
The aim of the paper is to examine the computational complexity and
algorithmics of enumeration, the task to output all solutions of a given
problem, from the point of view of parameterized complexity. First we define
formally different notions of efficient enumeration in the context of
parameterized complexity. Second we show how different algorithmic paradigms
can be used in order to get parameter-efficient enumeration algorithms in a
number of examples. These paradigms use well-known principles from the design
of parameterized decision as well as enumeration techniques, like for instance
kernelization and self-reducibility. The concept of kernelization, in
particular, leads to a characterization of fixed-parameter tractable
enumeration problems.Comment: Accepted for MFCS 2013; long version of the pape
Vertex Cover Kernelization Revisited: Upper and Lower Bounds for a Refined Parameter
An important result in the study of polynomial-time preprocessing shows that
there is an algorithm which given an instance (G,k) of Vertex Cover outputs an
equivalent instance (G',k') in polynomial time with the guarantee that G' has
at most 2k' vertices (and thus O((k')^2) edges) with k' <= k. Using the
terminology of parameterized complexity we say that k-Vertex Cover has a kernel
with 2k vertices. There is complexity-theoretic evidence that both 2k vertices
and Theta(k^2) edges are optimal for the kernel size. In this paper we consider
the Vertex Cover problem with a different parameter, the size fvs(G) of a
minimum feedback vertex set for G. This refined parameter is structurally
smaller than the parameter k associated to the vertex covering number vc(G)
since fvs(G) <= vc(G) and the difference can be arbitrarily large. We give a
kernel for Vertex Cover with a number of vertices that is cubic in fvs(G): an
instance (G,X,k) of Vertex Cover, where X is a feedback vertex set for G, can
be transformed in polynomial time into an equivalent instance (G',X',k') such
that |V(G')| <= 2k and |V(G')| <= O(|X'|^3). A similar result holds when the
feedback vertex set X is not given along with the input. In sharp contrast we
show that the Weighted Vertex Cover problem does not have a polynomial kernel
when parameterized by the cardinality of a given vertex cover of the graph
unless NP is in coNP/poly and the polynomial hierarchy collapses to the third
level.Comment: Published in "Theory of Computing Systems" as an Open Access
publicatio
Tight Kernel Bounds for Problems on Graphs with Small Degeneracy
In this paper we consider kernelization for problems on d-degenerate graphs,
i.e. graphs such that any subgraph contains a vertex of degree at most .
This graph class generalizes many classes of graphs for which effective
kernelization is known to exist, e.g. planar graphs, H-minor free graphs, and
H-topological-minor free graphs. We show that for several natural problems on
d-degenerate graphs the best known kernelization upper bounds are essentially
tight.Comment: Full version of ESA 201
On Structural Parameterizations of Hitting Set: Hitting Paths in Graphs Using 2-SAT
Hitting Set is a classic problem in combinatorial optimization. Its input
consists of a set system F over a finite universe U and an integer t; the
question is whether there is a set of t elements that intersects every set in
F. The Hitting Set problem parameterized by the size of the solution is a
well-known W[2]-complete problem in parameterized complexity theory. In this
paper we investigate the complexity of Hitting Set under various structural
parameterizations of the input. Our starting point is the folklore result that
Hitting Set is polynomial-time solvable if there is a tree T on vertex set U
such that the sets in F induce connected subtrees of T. We consider the case
that there is a treelike graph with vertex set U such that the sets in F induce
connected subgraphs; the parameter of the problem is a measure of how treelike
the graph is. Our main positive result is an algorithm that, given a graph G
with cyclomatic number k, a collection P of simple paths in G, and an integer
t, determines in time 2^{5k} (|G| +|P|)^O(1) whether there is a vertex set of
size t that hits all paths in P. It is based on a connection to the 2-SAT
problem in multiple valued logic. For other parameterizations we derive
W[1]-hardness and para-NP-completeness results.Comment: Presented at the 41st International Workshop on Graph-Theoretic
Concepts in Computer Science, WG 2015. (The statement of Lemma 4 was
corrected in this update.
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