1,646 research outputs found
Data Reduction for Graph Coloring Problems
This paper studies the kernelization complexity of graph coloring problems
with respect to certain structural parameterizations of the input instances. We
are interested in how well polynomial-time data reduction can provably shrink
instances of coloring problems, in terms of the chosen parameter. It is well
known that deciding 3-colorability is already NP-complete, hence parameterizing
by the requested number of colors is not fruitful. Instead, we pick up on a
research thread initiated by Cai (DAM, 2003) who studied coloring problems
parameterized by the modification distance of the input graph to a graph class
on which coloring is polynomial-time solvable; for example parameterizing by
the number k of vertex-deletions needed to make the graph chordal. We obtain
various upper and lower bounds for kernels of such parameterizations of
q-Coloring, complementing Cai's study of the time complexity with respect to
these parameters.
Our results show that the existence of polynomial kernels for q-Coloring
parameterized by the vertex-deletion distance to a graph class F is strongly
related to the existence of a function f(q) which bounds the number of vertices
which are needed to preserve the NO-answer to an instance of q-List-Coloring on
F.Comment: Author-accepted manuscript of the article that will appear in the FCT
2011 special issue of Information & Computatio
Enhancing Program Soft Error Resilience through Algorithmic Approaches
The rising count and shrinking feature size of transistors within modern computers is making them increasingly vulnerable to various types of soft faults. This problem is especially acute in high-performance computing (HPC) systems used for scientific computing, because these systems include many thousands of compute cores and nodes, all of which may be utilized in a single large-scale run. The increasing vulnerability of HPC applications to errors induced by soft faults is motivating extensive work on techniques to make these applications more resilient to such faults, ranging from generic techniques such as replication or checkpoint/restart to algorithm-specific error detection and tolerance techniques. Effective use of such techniques requires a detailed understanding of how a given application is affected by soft faults to ensure that (i) efforts to improve application resilience are spent in the code regions most vulnerable to faults, (ii) the appropriate resilience techniques is applied to each code region, and (iii) the understanding be obtained in an efficient manner. This thesis presents two tools: FaultTelescope helps application developers view the routine and application vulnerability to soft errors while ErrorSight helps perform modular fault characteristics analysis for more complex applications. This thesis also illustrates how these tools can be used in the context of representative applications and kernels. In addition to providing actionable insights into application behavior, the tools automatically selects the number of fault injection experiments required to efficiently generation error profiles of an application, ensuring that the information is statistically well-grounded without performing unnecessary experiments
Computing Petaflops over Terabytes of Data: The Case of Genome-Wide Association Studies
In many scientific and engineering applications, one has to solve not one but
a sequence of instances of the same problem. Often times, the problems in the
sequence are linked in a way that allows intermediate results to be reused. A
characteristic example for this class of applications is given by the
Genome-Wide Association Studies (GWAS), a widely spread tool in computational
biology. GWAS entails the solution of up to trillions () of correlated
generalized least-squares problems, posing a daunting challenge: the
performance of petaflops ( floating-point operations) over terabytes
of data.
In this paper, we design an algorithm for performing GWAS on multi-core
architectures. This is accomplished in three steps. First, we show how to
exploit the relation among successive problems, thus reducing the overall
computational complexity. Then, through an analysis of the required data
transfers, we identify how to eliminate any overhead due to input/output
operations. Finally, we study how to decompose computation into tasks to be
distributed among the available cores, to attain high performance and
scalability. With our algorithm, a GWAS that currently requires the use of a
supercomputer may now be performed in matter of hours on a single multi-core
node.
The discussion centers around the methodology to develop the algorithm rather
than the specific application. We believe the paper contributes valuable
guidelines of general applicability for computational scientists on how to
develop and optimize numerical algorithms
Computational study on planar dominating set problem
AbstractRecently, there has been significant theoretical progress towards fixed-parameter algorithms for the DOMINATING SET problem of planar graphs. It is known that the problem on a planar graph with n vertices and dominating number k can be solved in O(2O(k)n) time using tree/branch-decomposition based algorithms. In this paper, we report computational results of Fomin and Thilikos algorithm which uses the branch-decomposition based approach. The computational results show that the algorithm can solve the DOMINATING SET problem of large planar graphs in a practical time and memory space for the class of graphs with small branchwidth. For the class of graphs with large branchwidth, the size of instances that can be solved by the algorithm in practice is limited to about one thousand edges due to a memory space bottleneck. The practical performances of the algorithm coincide with the theoretical analysis of the algorithm. The results of this paper suggest that the branch-decomposition based algorithms can be practical for some applications on planar graphs
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
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