3,200 research outputs found
Kernelization Lower Bounds By Cross-Composition
We introduce the cross-composition framework for proving kernelization lower
bounds. A classical problem L AND/OR-cross-composes into a parameterized
problem Q if it is possible to efficiently construct an instance of Q with
polynomially bounded parameter value that expresses the logical AND or OR of a
sequence of instances of L. Building on work by Bodlaender et al. (ICALP 2008)
and using a result by Fortnow and Santhanam (STOC 2008) with a refinement by
Dell and van Melkebeek (STOC 2010), we show that if an NP-hard problem
OR-cross-composes into a parameterized problem Q then Q does not admit a
polynomial kernel unless NP \subseteq coNP/poly and the polynomial hierarchy
collapses. Similarly, an AND-cross-composition for Q rules out polynomial
kernels for Q under Bodlaender et al.'s AND-distillation conjecture.
Our technique generalizes and strengthens the recent techniques of using
composition algorithms and of transferring the lower bounds via polynomial
parameter transformations. We show its applicability by proving kernelization
lower bounds for a number of important graphs problems with structural
(non-standard) parameterizations, e.g., Clique, Chromatic Number, Weighted
Feedback Vertex Set, and Weighted Odd Cycle Transversal do not admit polynomial
kernels with respect to the vertex cover number of the input graphs unless the
polynomial hierarchy collapses, contrasting the fact that these problems are
trivially fixed-parameter tractable for this parameter.
After learning of our results, several teams of authors have successfully
applied the cross-composition framework to different parameterized problems.
For completeness, our presentation of the framework includes several extensions
based on this follow-up work. For example, we show how a relaxed version of
OR-cross-compositions may be used to give lower bounds on the degree of the
polynomial in the kernel size.Comment: A preliminary version appeared in the proceedings of the 28th
International Symposium on Theoretical Aspects of Computer Science (STACS
2011) under the title "Cross-Composition: A New Technique for Kernelization
Lower Bounds". Several results have been strengthened compared to the
preliminary version (http://arxiv.org/abs/1011.4224). 29 pages, 2 figure
Cross-Composition: A New Technique for Kernelization Lower Bounds
We introduce a new technique for proving kernelization lower bounds, called
cross-composition. A classical problem L cross-composes into a parameterized
problem Q if an instance of Q with polynomially bounded parameter value can
express the logical OR of a sequence of instances of L. Building on work by
Bodlaender et al. (ICALP 2008) and using a result by Fortnow and Santhanam
(STOC 2008) we show that if an NP-complete problem cross-composes into a
parameterized problem Q then Q does not admit a polynomial kernel unless the
polynomial hierarchy collapses. Our technique generalizes and strengthens the
recent techniques of using OR-composition algorithms and of transferring the
lower bounds via polynomial parameter transformations. We show its
applicability by proving kernelization lower bounds for a number of important
graphs problems with structural (non-standard) parameterizations, e.g.,
Chromatic Number, Clique, and Weighted Feedback Vertex Set do not admit
polynomial kernels with respect to the vertex cover number of the input graphs
unless the polynomial hierarchy collapses, contrasting the fact that these
problems are trivially fixed-parameter tractable for this parameter. We have
similar lower bounds for Feedback Vertex Set.Comment: Updated information based on final version submitted to STACS 201
A Complexity Measure for Continuous Time Quantum Algorithms
We consider unitary dynamical evolutions on n qubits caused by time dependent
pair-interaction Hamiltonians and show that the running time of a parallelized
two-qubit gate network simulating the evolution is given by the time integral
over the chromatic index of the interaction graph. This defines a complexity
measure of continuous and discrete quantum algorithms which are in exact
one-to-one correspondence. Furthermore we prove a lower bound on the growth of
large-scale entanglement depending on the chromatic index.Comment: 6 pages, Revte
Performance of distributed mechanisms for flow admission in wireless adhoc networks
Given a wireless network where some pairs of communication links interfere
with each other, we study sufficient conditions for determining whether a given
set of minimum bandwidth quality-of-service (QoS) requirements can be
satisfied. We are especially interested in algorithms which have low
communication overhead and low processing complexity. The interference in the
network is modeled using a conflict graph whose vertices correspond to the
communication links in the network. Two links are adjacent in this graph if and
only if they interfere with each other due to being in the same vicinity and
hence cannot be simultaneously active. The problem of scheduling the
transmission of the various links is then essentially a fractional, weighted
vertex coloring problem, for which upper bounds on the fractional chromatic
number are sought using only localized information. We recall some distributed
algorithms for this problem, and then assess their worst-case performance. Our
results on this fundamental problem imply that for some well known classes of
networks and interference models, the performance of these distributed
algorithms is within a bounded factor away from that of an optimal, centralized
algorithm. The performance bounds are simple expressions in terms of graph
invariants. It is seen that the induced star number of a network plays an
important role in the design and performance of such networks.Comment: 21 pages, submitted. Journal version of arXiv:0906.378
On the density of sets of the Euclidean plane avoiding distance 1
A subset is said to avoid distance if: In this paper we study the number
which is the supremum of the upper densities of measurable
sets avoiding distance 1 in the Euclidean plane. Intuitively, represents the highest proportion of the plane that can be filled by a
set avoiding distance 1. This parameter is related to the fractional chromatic
number of the plane.
We establish that and .Comment: 11 pages, 5 figure
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