306 research outputs found
Faster exponential-time algorithms in graphs of bounded average degree
We first show that the Traveling Salesman Problem in an n-vertex graph with
average degree bounded by d can be solved in O*(2^{(1-\eps_d)n}) time and
exponential space for a constant \eps_d depending only on d, where the
O*-notation suppresses factors polynomial in the input size. Thus, we
generalize the recent results of Bjorklund et al. [TALG 2012] on graphs of
bounded degree.
Then, we move to the problem of counting perfect matchings in a graph. We
first present a simple algorithm for counting perfect matchings in an n-vertex
graph in O*(2^{n/2}) time and polynomial space; our algorithm matches the
complexity bounds of the algorithm of Bjorklund [SODA 2012], but relies on
inclusion-exclusion principle instead of algebraic transformations. Building
upon this result, we show that the number of perfect matchings in an n-vertex
graph with average degree bounded by d can be computed in
O*(2^{(1-\eps_{2d})n/2}) time and exponential space, where \eps_{2d} is the
constant obtained by us for the Traveling Salesman Problem in graphs of average
degree at most 2d.
Moreover we obtain a simple algorithm that counts the number of perfect
matchings in an n-vertex bipartite graph of average degree at most d in
O*(2^{(1-1/(3.55d))n/2}) time, improving and simplifying the recent result of
Izumi and Wadayama [FOCS 2012].Comment: 10 page
Not Just a Theory—The Utility of Mathematical Models in Evolutionary Biology
Models have made numerous contributions to evolutionary biology, but misunderstandings persist regarding their purpose. By formally testing the logic of verbal hypotheses, proof-of-concept models clarify thinking, uncover hidden assumptions, and spur new directions of study. thumbnail image credit: modified from the Biodiversity Heritage Librar
Improved Bounds on Quantum Learning Algorithms
In this article we give several new results on the complexity of algorithms
that learn Boolean functions from quantum queries and quantum examples.
Hunziker et al. conjectured that for any class C of Boolean functions, the
number of quantum black-box queries which are required to exactly identify an
unknown function from C is ,
where is a combinatorial parameter of the class C. We
essentially resolve this conjecture in the affirmative by giving a quantum
algorithm that, for any class C, identifies any unknown function from C using
quantum black-box
queries.
We consider a range of natural problems intermediate between the exact
learning problem (in which the learner must obtain all bits of information
about the black-box function) and the usual problem of computing a predicate
(in which the learner must obtain only one bit of information about the
black-box function). We give positive and negative results on when the quantum
and classical query complexities of these intermediate problems are
polynomially related to each other.
Finally, we improve the known lower bounds on the number of quantum examples
(as opposed to quantum black-box queries) required for -PAC
learning any concept class of Vapnik-Chervonenkis dimension d over the domain
from to . This new lower bound comes
closer to matching known upper bounds for classical PAC learning.Comment: Minor corrections. 18 pages. To appear in Quantum Information
Processing. Requires: algorithm.sty, algorithmic.sty to buil
Density Estimation for Shift-Invariant Multidimensional Distributions
We study density estimation for classes of shift-invariant distributions over R^d. A multidimensional distribution is "shift-invariant" if, roughly speaking, it is close in total variation distance to a small shift of it in any direction. Shift-invariance relaxes smoothness assumptions commonly used in non-parametric density estimation to allow jump discontinuities. The different classes of distributions that we consider correspond to different rates of tail decay.
For each such class we give an efficient algorithm that learns any distribution in the class from independent samples with respect to total variation distance. As a special case of our general result, we show that d-dimensional shift-invariant distributions which satisfy an exponential tail bound can be learned to total variation distance error epsilon using O~_d(1/ epsilon^{d+2}) examples and O~_d(1/ epsilon^{2d+2}) time. This implies that, for constant d, multivariate log-concave distributions can be learned in O~_d(1/epsilon^{2d+2}) time using O~_d(1/epsilon^{d+2}) samples, answering a question of [Diakonikolas et al., 2016]. All of our results extend to a model of noise-tolerant density estimation using Huber\u27s contamination model, in which the target distribution to be learned is a (1-epsilon,epsilon) mixture of some unknown distribution in the class with some other arbitrary and unknown distribution, and the learning algorithm must output a hypothesis distribution with total variation distance error O(epsilon) from the target distribution. We show that our general results are close to best possible by proving a simple Omega (1/epsilon^d) information-theoretic lower bound on sample complexity even for learning bounded distributions that are shift-invariant
Wave-vector dependent intensity variations of the Kondo peak in photoemission from CePd
Strong angle-dependent intensity variations of the Fermi-level feature are
observed in 4d - 4f resonant photoemission spectra of CePd(111), that
reveal the periodicity of the lattice and largest intensity close to the Gamma
points of the surface Brillouin zone. In the framework of a simplified periodic
Anderson model the phenomena may quantitatively be described by a wave-vector
dependence of the electron hopping matrix elements caused by Fermi-level
crossings of non-4f-derived energy bands
Order-Revealing Encryption and the Hardness of Private Learning
An order-revealing encryption scheme gives a public procedure by which two
ciphertexts can be compared to reveal the ordering of their underlying
plaintexts. We show how to use order-revealing encryption to separate
computationally efficient PAC learning from efficient -differentially private PAC learning. That is, we construct a concept
class that is efficiently PAC learnable, but for which every efficient learner
fails to be differentially private. This answers a question of Kasiviswanathan
et al. (FOCS '08, SIAM J. Comput. '11).
To prove our result, we give a generic transformation from an order-revealing
encryption scheme into one with strongly correct comparison, which enables the
consistent comparison of ciphertexts that are not obtained as the valid
encryption of any message. We believe this construction may be of independent
interest.Comment: 28 page
The counterintuitive role of sexual selection in species maintenance and speciation
Sexual selection is generally considered to be an important force in the maintenance of species differentiation. Using population genetic models, we show that when isolated in its purest form of Fisherian sexual selection, sexual selection inhibits rather than assists species maintenance and speciation when isolated populations begin to exchange migrants. The stronger this type of sexual selection becomes, the more it erases any effects of local adaptation that drive trait divergence. Furthermore, if the strength of Fisherian sexual selection itself is allowed to evolve, sexual selection is lost. These results emphasize that additional complications have to be added to sexual selection scenarios for sexual selection to contribute to divergence; Fisherian sexual selection alone has the opposite effect
Families with infants: a general approach to solve hard partition problems
We introduce a general approach for solving partition problems where the goal
is to represent a given set as a union (either disjoint or not) of subsets
satisfying certain properties. Many NP-hard problems can be naturally stated as
such partition problems. We show that if one can find a large enough system of
so-called families with infants for a given problem, then this problem can be
solved faster than by a straightforward algorithm. We use this approach to
improve known bounds for several NP-hard problems as well as to simplify the
proofs of several known results.
For the chromatic number problem we present an algorithm with
time and exponential space for graphs of average
degree . This improves the algorithm by Bj\"{o}rklund et al. [Theory Comput.
Syst. 2010] that works for graphs of bounded maximum (as opposed to average)
degree and closes an open problem stated by Cygan and Pilipczuk [ICALP 2013].
For the traveling salesman problem we give an algorithm working in
time and polynomial space for graphs of average
degree . The previously known results of this kind is a polyspace algorithm
by Bj\"{o}rklund et al. [ICALP 2008] for graphs of bounded maximum degree and
an exponential space algorithm for bounded average degree by Cygan and
Pilipczuk [ICALP 2013].
For counting perfect matching in graphs of average degree~ we present an
algorithm with running time and polynomial
space. Recent algorithms of this kind due to Cygan, Pilipczuk [ICALP 2013] and
Izumi, Wadayama [FOCS 2012] (for bipartite graphs only) use exponential space.Comment: 18 pages, a revised version of this paper is available at
http://arxiv.org/abs/1410.220
The counterintuitive role of sexual selection in species maintenance and speciation
Sexual selection is generally considered to be an important force in the maintenance of species differentiation. Using population genetic models, we show that when isolated in its purest form of Fisherian sexual selection, sexual selection inhibits rather than assists species maintenance and speciation when isolated populations begin to exchange migrants. The stronger this type of sexual selection becomes, the more it erases any effects of local adaptation that drive trait divergence. Furthermore, if the strength of Fisherian sexual selection itself is allowed to evolve, sexual selection is lost. These results emphasize that additional complications have to be added to sexual selection scenarios for sexual selection to contribute to divergence; Fisherian sexual selection alone has the opposite effect
- …