3,072 research outputs found
Exploring Subexponential Parameterized Complexity of Completion Problems
Let be a family of graphs. In the -Completion problem,
we are given a graph and an integer as input, and asked whether at most
edges can be added to so that the resulting graph does not contain a
graph from as an induced subgraph. It appeared recently that special
cases of -Completion, the problem of completing into a chordal graph
known as Minimum Fill-in, corresponding to the case of , and the problem of completing into a split graph,
i.e., the case of , are solvable in parameterized
subexponential time . The exploration of this
phenomenon is the main motivation for our research on -Completion.
In this paper we prove that completions into several well studied classes of
graphs without long induced cycles also admit parameterized subexponential time
algorithms by showing that:
- The problem Trivially Perfect Completion is solvable in parameterized
subexponential time , that is -Completion for , a cycle and a path on four
vertices.
- The problems known in the literature as Pseudosplit Completion, the case
where , and Threshold Completion, where , are also solvable in time .
We complement our algorithms for -Completion with the following
lower bounds:
- For , , , and
, -Completion cannot be solved in time
unless the Exponential Time Hypothesis (ETH) fails.
Our upper and lower bounds provide a complete picture of the subexponential
parameterized complexity of -Completion problems for .Comment: 32 pages, 16 figures, A preliminary version of this paper appeared in
the proceedings of STACS'1
Fast Biclustering by Dual Parameterization
We study two clustering problems, Starforest Editing, the problem of adding
and deleting edges to obtain a disjoint union of stars, and the generalization
Bicluster Editing. We show that, in addition to being NP-hard, none of the
problems can be solved in subexponential time unless the exponential time
hypothesis fails.
Misra, Panolan, and Saurabh (MFCS 2013) argue that introducing a bound on the
number of connected components in the solution should not make the problem
easier: In particular, they argue that the subexponential time algorithm for
editing to a fixed number of clusters (p-Cluster Editing) by Fomin et al. (J.
Comput. Syst. Sci., 80(7) 2014) is an exception rather than the rule. Here, p
is a secondary parameter, bounding the number of components in the solution.
However, upon bounding the number of stars or bicliques in the solution, we
obtain algorithms which run in time for p-Starforest
Editing and for p-Bicluster Editing. We
obtain a similar result for the more general case of t-Partite p-Cluster
Editing. This is subexponential in k for fixed number of clusters, since p is
then considered a constant.
Our results even out the number of multivariate subexponential time
algorithms and give reasons to believe that this area warrants further study.Comment: Accepted for presentation at IPEC 201
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
A survey of parameterized algorithms and the complexity of edge modification
The survey is a comprehensive overview of the developing area of parameterized algorithms for graph modification problems. It describes state of the art in kernelization, subexponential algorithms, and parameterized complexity of graph modification. The main focus is on edge modification problems, where the task is to change some adjacencies in a graph to satisfy some required properties. To facilitate further research, we list many open problems in the area.publishedVersio
On the Threshold of Intractability
We study the computational complexity of the graph modification problems
Threshold Editing and Chain Editing, adding and deleting as few edges as
possible to transform the input into a threshold (or chain) graph. In this
article, we show that both problems are NP-complete, resolving a conjecture by
Natanzon, Shamir, and Sharan (Discrete Applied Mathematics, 113(1):109--128,
2001). On the positive side, we show the problem admits a quadratic vertex
kernel. Furthermore, we give a subexponential time parameterized algorithm
solving Threshold Editing in time,
making it one of relatively few natural problems in this complexity class on
general graphs. These results are of broader interest to the field of social
network analysis, where recent work of Brandes (ISAAC, 2014) posits that the
minimum edit distance to a threshold graph gives a good measure of consistency
for node centralities. Finally, we show that all our positive results extend to
the related problem of Chain Editing, as well as the completion and deletion
variants of both problems
- …