93 research outputs found
Tree Deletion Set has a Polynomial Kernel (but no OPT^O(1) approximation)
In the Tree Deletion Set problem the input is a graph G together with an
integer k. The objective is to determine whether there exists a set S of at
most k vertices such that G-S is a tree. The problem is NP-complete and even
NP-hard to approximate within any factor of OPT^c for any constant c. In this
paper we give a O(k^4) size kernel for the Tree Deletion Set problem. To the
best of our knowledge our result is the first counterexample to the
"conventional wisdom" that kernelization algorithms automatically provide
approximation algorithms with approximation ratio close to the size of the
kernel. An appealing feature of our kernelization algorithm is a new algebraic
reduction rule that we use to handle the instances on which Tree Deletion Set
is hard to approximate
On Approximate Compressions for Connected Minor-Hitting Sets
In the Connected ?-Deletion problem, ? is a fixed finite family of graphs and the objective is to compute a minimum set of vertices (or a vertex set of size at most k for some given k) such that (a) this set induces a connected subgraph of the given graph and (b) deleting this set results in a graph which excludes every F ? ? as a minor. In the area of kernelization, this problem is well known to exclude a polynomial kernel subject to standard complexity hypotheses even in very special cases such as ? = K?, i.e., Connected Vertex Cover.
In this work, we give a (2+?)-approximate polynomial compression for the Connected ?-Deletion problem when ? contains at least one planar graph. This is the first approximate polynomial compression result for this generic problem. As a corollary, we obtain the first approximate polynomial compression result for the special case of Connected ?-Treewidth Deletion
On the Approximate Compressibility of Connected Vertex Cover
The Connected Vertex Cover problem, where the goal is to compute a minimum
set of vertices in a given graph which forms a vertex cover and induces a
connected subgraph, is a fundamental combinatorial problem and has received
extensive attention in various subdomains of algorithmics. In the area of
kernelization, it is known that this problem is unlikely to have efficient
preprocessing algorithms, also known as polynomial kernelizations. However, it
has been shown in a recent work of Lokshtanov et al. [STOC 2017] that if one
considered an appropriate notion of approximate kernelization, then this
problem parameterized by the solution size does admit an approximate polynomial
kernelization. In fact, Lokhtanov et al. were able to obtain a polynomial size
approximate kernelization scheme (PSAKS) for Connected Vertex Cover
parameterized by the solution size. A PSAKS is essentially a preprocessing
algorithm whose error can be made arbitrarily close to 0. In this paper we
revisit this problem, and consider parameters that are strictly smaller than
the size of the solution and obtain the first polynomial size approximate
kernelization schemes for the Connected Vertex Cover problem when parameterized
by the deletion distance of the input graph to the class of cographs, the class
of bounded treewidth graphs, and the class of all chordal graphs.Comment: 1 figure; Revisions from the previous version incorporated based on
the comments from some anonymous reviewer
Polynomial Kernels for Deletion to Classes of Acyclic Digraphs
We consider the problem to find a set X of vertices (or arcs) with |X| <= k in a given digraph G such that D = G-X is an acyclic digraph. In its generality, this is DIRECTED FEEDBACK VERTEX SET or DIRECTED FEEDBACK ARC SET respectively. The existence of a polynomial kernel for these problems is a notorious open problem in the field of kernelization, and little progress has been made.
In this paper, we consider both deletion problems with an additional restriction on D, namely that D must be an out-forest, an out-tree, or a (directed) pumpkin. Our main results show that for each of these three restrictions the vertex deletion problem remains NP-hard, but we can obtain a kernel with k^{O(1)} vertices on general digraphs G. We also show that, in contrast to the vertex deletion problem, the arc deletion problem with each of the above restrictions can be solved in polynomial time
Finding secluded places of special interest in graphs.
Finding a vertex subset in a graph that satisfies a certain property is one of the most-studied topics
in algorithmic graph theory. The focus herein is often on minimizing or maximizing the size
of the solution, that is, the size of the desired vertex set. In several applications, however, we also
want to limit the “exposure” of the solution to the rest of the graph. This is the case, for example,
when the solution represents persons that ought to deal with sensitive information or a segregated
community. In this work, we thus explore the (parameterized) complexity of finding such secluded
vertex subsets for a wide variety of properties that they shall fulfill. More precisely, we study the
constraint that the (open or closed) neighborhood of the solution shall be bounded by a parameter
and the influence of this constraint on the complexity of minimizing separators, feedback vertex
sets, F-free vertex deletion sets, dominating sets, and the maximization of independent sets
Finding secluded places of special interest in graphs
Finding a vertex subset in a graph that satisfies a certain property is one of the most-studied topics in algorithmic graph theory. The focus herein is often on minimizing or maximizing the size of the solution, that is, the size of the desired vertex set. In several applications, however, we also want to limit the “exposure” of the solution to the rest of the graph. This is the case, for example, when the solution represents persons that ought to deal with sensitive information or a segregated community. In this work, we thus explore the (parameterized) complexity of finding such secluded vertex subsets for a wide variety of properties that they shall fulfill. More precisely, we study the constraint that the (open or closed) neighborhood of the solution shall be bounded by a parameter and the influence of this constraint on the complexity of minimizing separators, feedback vertex sets, F-free vertex deletion sets, dominating sets, and the maximization of independent sets
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