624 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
Fast branching algorithm for Cluster Vertex Deletion
In the family of clustering problems, we are given a set of objects (vertices
of the graph), together with some observed pairwise similarities (edges). The
goal is to identify clusters of similar objects by slightly modifying the graph
to obtain a cluster graph (disjoint union of cliques). Hueffner et al. [Theory
Comput. Syst. 2010] initiated the parameterized study of Cluster Vertex
Deletion, where the allowed modification is vertex deletion, and presented an
elegant O(2^k * k^9 + n * m)-time fixed-parameter algorithm, parameterized by
the solution size. In our work, we pick up this line of research and present an
O(1.9102^k * (n + m))-time branching algorithm
Structural solutions to maximum independent set and related problems
In this thesis, we study some fundamental problems in algorithmic graph theory. Most
natural problems in this area are hard from a computational point of view. However,
many applications demand that we do solve such problems, even if they are intractable.
There are a number of methods in which we can try to do this:
1) We may use an approximation algorithm if we do not necessarily require the best
possible solution to a problem.
2) Heuristics can be applied and work well enough to be useful for many applications.
3) We can construct randomised algorithms for which the probability of failure is very
small.
4) We may parameterize the problem in some way which limits its complexity.
In other cases, we may also have some information about the structure of the
instances of the problem we are trying to solve. If we are lucky, we may and that we
can exploit this extra structure to find efficient ways to solve our problem. The question
which arises is - How far must we restrict the structure of our graph to be able to solve
our problem efficiently?
In this thesis we study a number of problems, such as Maximum Indepen-
dent Set, Maximum Induced Matching, Stable-II, Efficient Edge Domina-
tion, Vertex Colouring and Dynamic Edge-Choosability. We try to solve problems
on various hereditary classes of graphs and analyse the complexity of the resulting
problem, both from a classical and parameterized point of view
Search-Space Reduction via Essential Vertices
We investigate preprocessing for vertex-subset problems on graphs. While the notion of kernelization, originating in parameterized complexity theory, is a formalization of provably effective preprocessing aimed at reducing the total instance size, our focus is on finding a non-empty vertex set that belongs to an optimal solution. This decreases the size of the remaining part of the solution which still has to be found, and therefore shrinks the search space of fixed-parameter tractable algorithms for parameterizations based on the solution size. We introduce the notion of a c-essential vertex as one that is contained in all c-approximate solutions. For several classic combinatorial problems such as Odd Cycle Transversal and Directed Feedback Vertex Set, we show that under mild conditions a polynomial-time preprocessing algorithm can find a subset of an optimal solution that contains all 2-essential vertices, by exploiting packing/covering duality. This leads to FPT algorithms to solve these problems where the exponential term in the running time depends only on the number of non-essential vertices in the solution
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