1,295 research outputs found
FPT is Characterized by Useful Obstruction Sets
Many graph problems were first shown to be fixed-parameter tractable using
the results of Robertson and Seymour on graph minors. We show that the
combination of finite, computable, obstruction sets and efficient order tests
is not just one way of obtaining strongly uniform FPT algorithms, but that all
of FPT may be captured in this way. Our new characterization of FPT has a
strong connection to the theory of kernelization, as we prove that problems
with polynomial kernels can be characterized by obstruction sets whose elements
have polynomial size. Consequently we investigate the interplay between the
sizes of problem kernels and the sizes of the elements of such obstruction
sets, obtaining several examples of how results in one area yield new insights
in the other. We show how exponential-size minor-minimal obstructions for
pathwidth k form the crucial ingredient in a novel OR-cross-composition for
k-Pathwidth, complementing the trivial AND-composition that is known for this
problem. In the other direction, we show that OR-cross-compositions into a
parameterized problem can be used to rule out the existence of efficiently
generated quasi-orders on its instances that characterize the NO-instances by
polynomial-size obstructions.Comment: Extended abstract with appendix, as accepted to WG 201
A New Two-Parameter Family of Potentials with a Tunable Ground State
In a previous paper we solved a countably infinite family of one-dimensional
Schr\"odinger equations by showing that they were supersymmetric partner
potentials of the standard quantum harmonic oscillator. In this work we extend
these results to find the complete set of real partner potentials of the
harmonic oscillator, showing that these depend upon two continuous parameters.
Their spectra are identical to that of the harmonic oscillator, except that the
ground state energy becomes a tunable parameter. We finally use these
potentials to analyse the physical problem of Bose-Einstein condensation in an
atomic gas trapped in a dimple potential.Comment: 15 pages, 5 figure
Kelo, Conservation Easements, and Forever: Why Eminent Domain Is Not a Sufficient Check on Conservation Easements\u27 Perpetual Duration
Letter graphs and geometric grid classes of permutations: characterization and recognition
In this paper, we reveal an intriguing relationship between two seemingly
unrelated notions: letter graphs and geometric grid classes of permutations. An
important property common for both of them is well-quasi-orderability,
implying, in a non-constructive way, a polynomial-time recognition of geometric
grid classes of permutations and -letter graphs for a fixed . However,
constructive algorithms are available only for . In this paper, we present
the first constructive polynomial-time algorithm for the recognition of
-letter graphs. It is based on a structural characterization of graphs in
this class.Comment: arXiv admin note: text overlap with arXiv:1108.6319 by other author
Polynomial Kernels for Weighted Problems
Kernelization is a formalization of efficient preprocessing for NP-hard
problems using the framework of parameterized complexity. Among open problems
in kernelization it has been asked many times whether there are deterministic
polynomial kernelizations for Subset Sum and Knapsack when parameterized by the
number of items.
We answer both questions affirmatively by using an algorithm for compressing
numbers due to Frank and Tardos (Combinatorica 1987). This result had been
first used by Marx and V\'egh (ICALP 2013) in the context of kernelization. We
further illustrate its applicability by giving polynomial kernels also for
weighted versions of several well-studied parameterized problems. Furthermore,
when parameterized by the different item sizes we obtain a polynomial
kernelization for Subset Sum and an exponential kernelization for Knapsack.
Finally, we also obtain kernelization results for polynomial integer programs
Streaming Kernelization
Kernelization is a formalization of preprocessing for combinatorially hard
problems. We modify the standard definition for kernelization, which allows any
polynomial-time algorithm for the preprocessing, by requiring instead that the
preprocessing runs in a streaming setting and uses
bits of memory on instances . We obtain
several results in this new setting, depending on the number of passes over the
input that such a streaming kernelization is allowed to make. Edge Dominating
Set turns out as an interesting example because it has no single-pass
kernelization but two passes over the input suffice to match the bounds of the
best standard kernelization
Planar Embeddings with Small and Uniform Faces
Motivated by finding planar embeddings that lead to drawings with favorable
aesthetics, we study the problems MINMAXFACE and UNIFORMFACES of embedding a
given biconnected multi-graph such that the largest face is as small as
possible and such that all faces have the same size, respectively.
We prove a complexity dichotomy for MINMAXFACE and show that deciding whether
the maximum is at most is polynomial-time solvable for and
NP-complete for . Further, we give a 6-approximation for minimizing
the maximum face in a planar embedding. For UNIFORMFACES, we show that the
problem is NP-complete for odd and even . Moreover, we
characterize the biconnected planar multi-graphs admitting 3- and 4-uniform
embeddings (in a -uniform embedding all faces have size ) and give an
efficient algorithm for testing the existence of a 6-uniform embedding.Comment: 23 pages, 5 figures, extended version of 'Planar Embeddings with
Small and Uniform Faces' (The 25th International Symposium on Algorithms and
Computation, 2014
FGF2 AND TGFbeta1 induce precocious post-natal maturation of articular cartilage: implications for repair of osteoarthritic lesions
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