186 research outputs found
Hitting forbidden subgraphs in graphs of bounded treewidth
We study the complexity of a generic hitting problem H-Subgraph Hitting,
where given a fixed pattern graph and an input graph , the task is to
find a set of minimum size that hits all subgraphs of
isomorphic to . In the colorful variant of the problem, each vertex of
is precolored with some color from and we require to hit only
-subgraphs with matching colors. Standard techniques shows that for every
fixed , the problem is fixed-parameter tractable parameterized by the
treewidth of ; however, it is not clear how exactly the running time should
depend on treewidth. For the colorful variant, we demonstrate matching upper
and lower bounds showing that the dependence of the running time on treewidth
of is tightly governed by , the maximum size of a minimal vertex
separator in . That is, we show for every fixed that, on a graph of
treewidth , the colorful problem can be solved in time
, but cannot be solved in time
, assuming the Exponential Time
Hypothesis (ETH). Furthermore, we give some preliminary results showing that,
in the absence of colors, the parameterized complexity landscape of H-Subgraph
Hitting is much richer.Comment: A full version of a paper presented at MFCS 201
Hitting forbidden subgraphs in graphs of bounded treewidth
We study the complexity of a generic hitting problem H -Subgraph Hitting , where given a fixed pattern graph H and an input graph G, we seek for the minimum size of a set X ⊆ V(G) that hits all subgraphs of G isomorphic to H. In the colorful variant of the problem, each vertex of G is precolored with some color from V(H) and we require to hit only H-subgraphs with matching colors. Standard techniques (e.g., Courcelle’s theorem) show that, for every fixed H and the problem is fixed-parameter tractable parameterized by the treewidth of G; however, it is not clear how exactly the running time should depend on treewidth. For the colorful variant, we demonstrate matching upper and lower bounds showing that the dependence of the running time on treewidth of G is tightly governed by μ(H), the maximum size of a minimal vertex separator in H. That is, we show for every fixed H that, on a graph of treewidth t, the colorful problem can be solved in time 2O(tμ(H))⋅|V(G)|, but cannot be solved in time 2o(tμ(H))⋅|V(G)|O(1), assuming the Exponential Time Hypothesis (ETH). Furthermore, we give some preliminary results showing that, in the absence of colors, the parameterized complexity landscape of H -Subgraph Hitting is much richer
Hitting forbidden minors: Approximation and Kernelization
We study a general class of problems called F-deletion problems. In an
F-deletion problem, we are asked whether a subset of at most vertices can
be deleted from a graph such that the resulting graph does not contain as a
minor any graph from the family F of forbidden minors.
We obtain a number of algorithmic results on the F-deletion problem when F
contains a planar graph. We give (1) a linear vertex kernel on graphs excluding
-claw , the star with leves, as an induced subgraph, where
is a fixed integer. (2) an approximation algorithm achieving an approximation
ratio of , where is the size of an optimal solution on
general undirected graphs. Finally, we obtain polynomial kernels for the case
when F contains graph as a minor for a fixed integer . The graph
consists of two vertices connected by parallel edges. Even
though this may appear to be a very restricted class of problems it already
encompasses well-studied problems such as {\sc Vertex Cover}, {\sc Feedback
Vertex Set} and Diamond Hitting Set. The generic kernelization algorithm is
based on a non-trivial application of protrusion techniques, previously used
only for problems on topological graph classes
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
Meta-Kernelization using Well-Structured Modulators
Kernelization investigates exact preprocessing algorithms with performance
guarantees. The most prevalent type of parameters used in kernelization is the
solution size for optimization problems; however, also structural parameters
have been successfully used to obtain polynomial kernels for a wide range of
problems. Many of these parameters can be defined as the size of a smallest
modulator of the given graph into a fixed graph class (i.e., a set of vertices
whose deletion puts the graph into the graph class). Such parameters admit the
construction of polynomial kernels even when the solution size is large or not
applicable. This work follows up on the research on meta-kernelization
frameworks in terms of structural parameters.
We develop a class of parameters which are based on a more general view on
modulators: instead of size, the parameters employ a combination of rank-width
and split decompositions to measure structure inside the modulator. This allows
us to lift kernelization results from modulator-size to more general
parameters, hence providing smaller kernels. We show (i) how such large but
well-structured modulators can be efficiently approximated, (ii) how they can
be used to obtain polynomial kernels for any graph problem expressible in
Monadic Second Order logic, and (iii) how they allow the extension of previous
results in the area of structural meta-kernelization
Hitting forbidden induced subgraphs on bounded treewidth graphs
For a fixed graph , the -IS-Deletion problem asks, given a graph ,
for the minimum size of a set such that does
not contain as an induced subgraph. Motivated by previous work about
hitting (topological) minors and subgraphs on bounded treewidth graphs, we are
interested in determining, for a fixed graph , the smallest function
such that -IS-Deletion can be solved in time assuming the Exponential Time Hypothesis (ETH), where and
denote the treewidth and the number of vertices of the input graph,
respectively.
We show that for every graph on
vertices, and that if is a clique or an independent
set. We present a number of lower bounds by generalizing a reduction of Cygan
et al. [MFCS 2014] for the subgraph version. In particular, we show that when
deviates slightly from a clique, the function suffers a sharp
jump: if is obtained from a clique of size by removing one edge, then
. We also show that
when , and this reduction answers an open question of Mi. Pilipczuk
[MFCS 2011] about the function for the subgraph version.
Motivated by Cygan et al. [MFCS 2014], we also consider the colorful variant
of the problem, where each vertex of is colored with some color from
and we require to hit only induced copies of with matching colors. In this
case, we determine, under the ETH, the function for every connected
graph on vertices: if the problem can be solved in polynomial
time; if , if is a clique, and otherwise.Comment: 24 pages, 3 figure
Hitting Forbidden Induced Subgraphs on Bounded Treewidth Graphs
For a fixed graph H, the H-IS-Deletion problem asks, given a graph G, for the minimum size of a set S ? V(G) such that G? S does not contain H as an induced subgraph. Motivated by previous work about hitting (topological) minors and subgraphs on bounded treewidth graphs, we are interested in determining, for a fixed graph H, the smallest function f_H(t) such that H-IS-Deletion can be solved in time f_H(t) ? n^{?(1)} assuming the Exponential Time Hypothesis (ETH), where t and n denote the treewidth and the number of vertices of the input graph, respectively.
We show that f_H(t) = 2^{?(t^{h-2})} for every graph H on h ? 3 vertices, and that f_H(t) = 2^{?(t)} if H is a clique or an independent set. We present a number of lower bounds by generalizing a reduction of Cygan et al. [MFCS 2014] for the subgraph version. In particular, we show that when H deviates slightly from a clique, the function f_H(t) suffers a sharp jump: if H is obtained from a clique of size h by removing one edge, then f_H(t) = 2^{?(t^{h-2})}. We also show that f_H(t) = 2^{?(t^{h})} when H = K_{h,h}, and this reduction answers an open question of Mi. Pilipczuk [MFCS 2011] about the function f_{C?}(t) for the subgraph version.
Motivated by Cygan et al. [MFCS 2014], we also consider the colorful variant of the problem, where each vertex of G is colored with some color from V(H) and we require to hit only induced copies of H with matching colors. In this case, we determine, under the ETH, the function f_H(t) for every connected graph H on h vertices: if h ? 2 the problem can be solved in polynomial time; if h ? 3, f_H(t) = 2^{?(t)} if H is a clique, and f_H(t) = 2^{?(t^{h-2})} otherwise
Hitting Subgraphs in Sparse Graphs and Geometric Intersection Graphs
We investigate a fundamental vertex-deletion problem called (Induced)
Subgraph Hitting: given a graph and a set of forbidden
graphs, the goal is to compute a minimum-sized set of vertices of such
that does not contain any graph in as an (induced)
subgraph. This is a generic problem that encompasses many well-known problems
that were extensively studied on their own, particularly (but not only) from
the perspectives of both approximation and parameterization. We focus on the
design of efficient approximation schemes, i.e., with running time
, which are also of significant
interest to both communities. Technically, our main contribution is a
linear-time approximation-preserving reduction from (Induced) Subgraph Hitting
on any graph class of bounded expansion to the same problem on
bounded degree graphs within . This yields a novel algorithmic
technique to design (efficient) approximation schemes for the problem on very
broad graph classes, well beyond the state-of-the-art. Specifically, applying
this reduction, we derive approximation schemes with (almost) linear running
time for the problem on any graph classes that have strongly sublinear
separators and many important classes of geometric intersection graphs (such as
fat-object graphs, pseudo-disk graphs, etc.). Our proofs introduce novel
concepts and combinatorial observations that may be of independent interest
(and, which we believe, will find other uses) for studies of approximation
algorithms, parameterized complexity, sparse graph classes, and geometric
intersection graphs. As a byproduct, we also obtain the first robust algorithm
for -Subgraph Isomorphism on intersection graphs of fat objects and
pseudo-disks, with running time .Comment: 60 pages, abstract shortened to fulfill the length limi
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