19,481 research outputs found
A polynomial algorithm for the k-cluster problem on interval graphs
This paper deals with the problem of finding, for a given graph and a given
natural number k, a subgraph of k nodes with a maximum number of edges. This
problem is known as the k-cluster problem and it is NP-hard on general graphs
as well as on chordal graphs. In this paper, it is shown that the k-cluster
problem is solvable in polynomial time on interval graphs. In particular, we
present two polynomial time algorithms for the class of proper interval graphs
and the class of general interval graphs, respectively. Both algorithms are
based on a matrix representation for interval graphs. In contrast to
representations used in most of the previous work, this matrix representation
does not make use of the maximal cliques in the investigated graph.Comment: 12 pages, 5 figure
Interval scheduling and colorful independent sets
Numerous applications in scheduling, such as resource allocation or steel
manufacturing, can be modeled using the NP-hard Independent Set problem (given
an undirected graph and an integer k, find a set of at least k pairwise
non-adjacent vertices). Here, one encounters special graph classes like 2-union
graphs (edge-wise unions of two interval graphs) and strip graphs (edge-wise
unions of an interval graph and a cluster graph), on which Independent Set
remains NP-hard but admits constant-ratio approximations in polynomial time. We
study the parameterized complexity of Independent Set on 2-union graphs and on
subclasses like strip graphs. Our investigations significantly benefit from a
new structural "compactness" parameter of interval graphs and novel problem
formulations using vertex-colored interval graphs. Our main contributions are:
1. We show a complexity dichotomy: restricted to graph classes closed under
induced subgraphs and disjoint unions, Independent Set is polynomial-time
solvable if both input interval graphs are cluster graphs, and is NP-hard
otherwise.
2. We chart the possibilities and limits of effective polynomial-time
preprocessing (also known as kernelization).
3. We extend Halld\'orsson and Karlsson (2006)'s fixed-parameter algorithm
for Independent Set on strip graphs parameterized by the structural parameter
"maximum number of live jobs" to show that the problem (also known as Job
Interval Selection) is fixed-parameter tractable with respect to the parameter
k and generalize their algorithm from strip graphs to 2-union graphs.
Preliminary experiments with random data indicate that Job Interval Selection
with up to fifteen jobs and 5*10^5 intervals can be solved optimally in less
than five minutes.Comment: This revision does not contain Theorem 7 of the first revision, whose
proof contained an erro
Structural parameterizations for boxicity
The boxicity of a graph is the least integer such that has an
intersection model of axis-aligned -dimensional boxes. Boxicity, the problem
of deciding whether a given graph has boxicity at most , is NP-complete
for every fixed . We show that boxicity is fixed-parameter tractable
when parameterized by the cluster vertex deletion number of the input graph.
This generalizes the result of Adiga et al., that boxicity is fixed-parameter
tractable in the vertex cover number.
Moreover, we show that boxicity admits an additive -approximation when
parameterized by the pathwidth of the input graph.
Finally, we provide evidence in favor of a conjecture of Adiga et al. that
boxicity remains NP-complete when parameterized by the treewidth.Comment: 19 page
Next Generation Cluster Editing
This work aims at improving the quality of structural variant prediction from
the mapped reads of a sequenced genome. We suggest a new model based on cluster
editing in weighted graphs and introduce a new heuristic algorithm that allows
to solve this problem quickly and with a good approximation on the huge graphs
that arise from biological datasets
Spanning trees short or small
We study the problem of finding small trees. Classical network design
problems are considered with the additional constraint that only a specified
number of nodes are required to be connected in the solution. A
prototypical example is the MST problem in which we require a tree of
minimum weight spanning at least nodes in an edge-weighted graph. We show
that the MST problem is NP-hard even for points in the Euclidean plane. We
provide approximation algorithms with performance ratio for the
general edge-weighted case and for the case of points in the
plane. Polynomial-time exact solutions are also presented for the class of
decomposable graphs which includes trees, series-parallel graphs, and bounded
bandwidth graphs, and for points on the boundary of a convex region in the
Euclidean plane. We also investigate the problem of finding short trees, and
more generally, that of finding networks with minimum diameter. A simple
technique is used to provide a polynomial-time solution for finding -trees
of minimum diameter. We identify easy and hard problems arising in finding
short networks using a framework due to T. C. Hu.Comment: 27 page
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
Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs
We study the family of intersection graphs of low density objects in low
dimensional Euclidean space. This family is quite general, and includes planar
graphs. We prove that such graphs have small separators. Next, we present
efficient -approximation algorithms for these graphs, for
Independent Set, Set Cover, and Dominating Set problems, among others. We also
prove corresponding hardness of approximation for some of these optimization
problems, providing a characterization of their intractability in terms of
density
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