1,804 research outputs found
On the (non-)existence of polynomial kernels for Pl-free edge modification problems
Given a graph G = (V,E) and an integer k, an edge modification problem for a
graph property P consists in deciding whether there exists a set of edges F of
size at most k such that the graph H = (V,E \vartriangle F) satisfies the
property P. In the P edge-completion problem, the set F of edges is constrained
to be disjoint from E; in the P edge-deletion problem, F is a subset of E; no
constraint is imposed on F in the P edge-edition problem. A number of
optimization problems can be expressed in terms of graph modification problems
which have been extensively studied in the context of parameterized complexity.
When parameterized by the size k of the edge set F, it has been proved that if
P is an hereditary property characterized by a finite set of forbidden induced
subgraphs, then the three P edge-modification problems are FPT. It was then
natural to ask whether these problems also admit a polynomial size kernel.
Using recent lower bound techniques, Kratsch and Wahlstrom answered this
question negatively. However, the problem remains open on many natural graph
classes characterized by forbidden induced subgraphs. Kratsch and Wahlstrom
asked whether the result holds when the forbidden subgraphs are paths or cycles
and pointed out that the problem is already open in the case of P4-free graphs
(i.e. cographs). This paper provides positive and negative results in that line
of research. We prove that parameterized cograph edge modification problems
have cubic vertex kernels whereas polynomial kernels are unlikely to exist for
the Pl-free and Cl-free edge-deletion problems for large enough l
Compression via Matroids: A Randomized Polynomial Kernel for Odd Cycle Transversal
The Odd Cycle Transversal problem (OCT) asks whether a given graph can be
made bipartite by deleting at most of its vertices. In a breakthrough
result Reed, Smith, and Vetta (Operations Research Letters, 2004) gave a
\BigOh(4^kkmn) time algorithm for it, the first algorithm with polynomial
runtime of uniform degree for every fixed . It is known that this implies a
polynomial-time compression algorithm that turns OCT instances into equivalent
instances of size at most \BigOh(4^k), a so-called kernelization. Since then
the existence of a polynomial kernel for OCT, i.e., a kernelization with size
bounded polynomially in , has turned into one of the main open questions in
the study of kernelization.
This work provides the first (randomized) polynomial kernelization for OCT.
We introduce a novel kernelization approach based on matroid theory, where we
encode all relevant information about a problem instance into a matroid with a
representation of size polynomial in . For OCT, the matroid is built to
allow us to simulate the computation of the iterative compression step of the
algorithm of Reed, Smith, and Vetta, applied (for only one round) to an
approximate odd cycle transversal which it is aiming to shrink to size . The
process is randomized with one-sided error exponentially small in , where
the result can contain false positives but no false negatives, and the size
guarantee is cubic in the size of the approximate solution. Combined with an
\BigOh(\sqrt{\log n})-approximation (Agarwal et al., STOC 2005), we get a
reduction of the instance to size \BigOh(k^{4.5}), implying a randomized
polynomial kernelization.Comment: Minor changes to agree with SODA 2012 version of the pape
Polynomial kernels for 3-leaf power graph modification problems
A graph G=(V,E) is a 3-leaf power iff there exists a tree T whose leaves are
V and such that (u,v) is an edge iff u and v are at distance at most 3 in T.
The 3-leaf power graph edge modification problems, i.e. edition (also known as
the closest 3-leaf power), completion and edge-deletion, are FTP when
parameterized by the size of the edge set modification. However polynomial
kernel was known for none of these three problems. For each of them, we provide
cubic kernels that can be computed in linear time for each of these problems.
We thereby answer an open problem first mentioned by Dom, Guo, Huffner and
Niedermeier (2005).Comment: Submitte
Polynomial kernelization for removing induced claws and diamonds
A graph is called (claw,diamond)-free if it contains neither a claw (a
) nor a diamond (a with an edge removed) as an induced subgraph.
Equivalently, (claw,diamond)-free graphs can be characterized as line graphs of
triangle-free graphs, or as linear dominoes, i.e., graphs in which every vertex
is in at most two maximal cliques and every edge is in exactly one maximal
clique.
In this paper we consider the parameterized complexity of the
(claw,diamond)-free Edge Deletion problem, where given a graph and a
parameter , the question is whether one can remove at most edges from
to obtain a (claw,diamond)-free graph. Our main result is that this problem
admits a polynomial kernel. We complement this finding by proving that, even on
instances with maximum degree , the problem is NP-complete and cannot be
solved in time unless the Exponential Time
Hypothesis fai
Network Sparsification for Steiner Problems on Planar and Bounded-Genus Graphs
We propose polynomial-time algorithms that sparsify planar and bounded-genus
graphs while preserving optimal or near-optimal solutions to Steiner problems.
Our main contribution is a polynomial-time algorithm that, given an unweighted
graph embedded on a surface of genus and a designated face bounded
by a simple cycle of length , uncovers a set of size
polynomial in and that contains an optimal Steiner tree for any set of
terminals that is a subset of the vertices of .
We apply this general theorem to prove that: * given an unweighted graph
embedded on a surface of genus and a terminal set , one
can in polynomial time find a set that contains an optimal
Steiner tree for and that has size polynomial in and ; * an
analogous result holds for an optimal Steiner forest for a set of terminal
pairs; * given an unweighted planar graph and a terminal set , one can in polynomial time find a set that contains
an optimal (edge) multiway cut separating and that has size polynomial
in .
In the language of parameterized complexity, these results imply the first
polynomial kernels for Steiner Tree and Steiner Forest on planar and
bounded-genus graphs (parameterized by the size of the tree and forest,
respectively) and for (Edge) Multiway Cut on planar graphs (parameterized by
the size of the cutset). Additionally, we obtain a weighted variant of our main
contribution
On Polynomial Kernelization of H-free Edge Deletion
For a set H of graphs, the H-free Edge Deletion problem is to decide whether there exist at most k edges in the input graph, for some k∈N, whose deletion results in a graph without an induced copy of any of the graphs in H . The problem is known to be fixed-parameter tractable if H is of finite cardinality. In this paper, we present a polynomial kernel for this problem for any fixed finite set H of connected graphs for the case where the input graphs are of bounded degree. We use a single kernelization rule which deletes vertices ‘far away’ from the induced copies of every H∈H in the input graph. With a slightly modified kernelization rule, we obtain polynomial kernels for H-free Edge Deletion under the following three settings
On Polynomial Kernelization of -free Edge Deletion
For a set of graphs , the \textsc{-free Edge
Deletion} problem asks to find whether there exist at most edges in the
input graph whose deletion results in a graph without any induced copy of
. In \cite{cai1996fixed}, it is shown that the problem is
fixed-parameter tractable if is of finite cardinality. However,
it is proved in \cite{cai2013incompressibility} that if is a
singleton set containing , for a large class of , there exists no
polynomial kernel unless . In this paper, we present a
polynomial kernel for this problem for any fixed finite set of
connected graphs and when the input graphs are of bounded degree. We note that
there are \textsc{-free Edge Deletion} problems which remain
NP-complete even for the bounded degree input graphs, for example
\textsc{Triangle-free Edge Deletion}\cite{brugmann2009generating} and
\textsc{Custer Edge Deletion(-free Edge
Deletion)}\cite{komusiewicz2011alternative}. When contains
, we obtain a stronger result - a polynomial kernel for -free
input graphs (for any fixed ). We note that for , there is an
incompressibility result for \textsc{-free Edge Deletion} for general
graphs \cite{cai2012polynomial}. Our result provides first polynomial kernels
for \textsc{Claw-free Edge Deletion} and \textsc{Line Edge Deletion} for
-free input graphs which are NP-complete even for -free
graphs\cite{yannakakis1981edge} and were raised as open problems in
\cite{cai2013incompressibility,open2013worker}.Comment: 12 pages. IPEC 2014 accepted pape
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