2,563 research outputs found
A new Lenstra-type Algorithm for Quasiconvex Polynomial Integer Minimization with Complexity 2^O(n log n)
We study the integer minimization of a quasiconvex polynomial with
quasiconvex polynomial constraints. We propose a new algorithm that is an
improvement upon the best known algorithm due to Heinz (Journal of Complexity,
2005). This improvement is achieved by applying a new modern Lenstra-type
algorithm, finding optimal ellipsoid roundings, and considering sparse
encodings of polynomials. For the bounded case, our algorithm attains a
time-complexity of s (r l M d)^{O(1)} 2^{2n log_2(n) + O(n)} when M is a bound
on the number of monomials in each polynomial and r is the binary encoding
length of a bound on the feasible region. In the general case, s l^{O(1)}
d^{O(n)} 2^{2n log_2(n) +O(n)}. In each we assume d>= 2 is a bound on the total
degree of the polynomials and l bounds the maximum binary encoding size of the
input.Comment: 28 pages, 10 figure
On the Lattice Distortion Problem
We introduce and study the \emph{Lattice Distortion Problem} (LDP). LDP asks
how "similar" two lattices are. I.e., what is the minimal distortion of a
linear bijection between the two lattices? LDP generalizes the Lattice
Isomorphism Problem (the lattice analogue of Graph Isomorphism), which simply
asks whether the minimal distortion is one.
As our first contribution, we show that the distortion between any two
lattices is approximated up to a factor by a simple function of
their successive minima. Our methods are constructive, allowing us to compute
low-distortion mappings that are within a factor
of optimal in polynomial time and within a factor of optimal in
singly exponential time. Our algorithms rely on a notion of basis reduction
introduced by Seysen (Combinatorica 1993), which we show is intimately related
to lattice distortion. Lastly, we show that LDP is NP-hard to approximate to
within any constant factor (under randomized reductions), by a reduction from
the Shortest Vector Problem.Comment: This is the full version of a paper that appeared in ESA 201
Quantum Algorithm for Computing the Period Lattice of an Infrastructure
We present a quantum algorithm for computing the period lattice of
infrastructures of fixed dimension. The algorithm applies to infrastructures
that satisfy certain conditions. The latter are always fulfilled for
infrastructures obtained from global fields, i.e., algebraic number fields and
function fields with finite constant fields.
The first of our main contributions is an exponentially better method for
sampling approximations of vectors of the dual lattice of the period lattice
than the methods outlined in the works of Hallgren and Schmidt and Vollmer.
This new method improves the success probability by a factor of at least
2^{n^2-1} where n is the dimension. The second main contribution is a rigorous
and complete proof that the running time of the algorithm is polynomial in the
logarithm of the determinant of the period lattice and exponential in n. The
third contribution is the determination of an explicit lower bound on the
success probability of our algorithm which greatly improves on the bounds given
in the above works.
The exponential scaling seems inevitable because the best currently known
methods for carrying out fundamental arithmetic operations in infrastructures
obtained from algebraic number fields take exponential time. In contrast, the
problem of computing the period lattice of infrastructures arising from
function fields can be solved without the exponential dependence on the
dimension n since this problem reduces efficiently to the abelian hidden
subgroup problem. This is also true for other important computational problems
in algebraic geometry. The running time of the best classical algorithms for
infrastructures arising from global fields increases subexponentially with the
determinant of the period lattice.Comment: 52 pages, 4 figure
RKKY Interactions in Graphene: Dependence on Disorder and Gate Voltage
We report the dependence of Ruderman-Kittel-Kasuya-Yoshida\,(RKKY)
interaction on nonmagmetic disorder and gate voltage in grapheme. First the
semiclassical method is employed to reserve the expression for RKKY interaction
in clean graphene. Due to the pseudogap at Dirac point, the RKKY coupling in
undoped grapheme is found to be proportional to . Next, we investigate
how the RKKY interaction depends on nonmagnetic disorder strength and gate
voltage by studying numerically the Anderson tight-binding model on a honeycomb
lattice. We observe that the RKKY interaction along the armchair direction is
more robust to nonmagnetic disorder than in other directions. This effect can
be explained semiclassically: The presence of multiple shortest paths between
two lattice sites in the armchair directions is found to be responsible for the
reduceddisorder sensitivity. We also present the distribution of the RKKY
interaction for the zigzag and armchair directions. We identify three different
shapes of the distributions which are repeated periodically along the zigzag
direction, while only one kind, and more narrow distribution, is observed along
the armchair direction. Moreover, we find that the distribution of amplitudes
of the RKKY interaction crosses over from a non-Gaussian shape with very long
tails to a completely log-normal distribution when increasing the nonmagnetic
disorder strength. The width of the log-normal distribution is found to
linearly increase with the strength of disorder, in agreement with analytical
predictions. At finite gate voltage near the Dirac point, Friedel oscillation
appears in addition to the oscillation from the interference between two Dirac
points. This results in a beating pattern. We study how these beating patterns
are effected by the nonmagnetic disorder in doped graphene
Lattice sparsification and the Approximate Closest Vector Problem
We give a deterministic algorithm for solving the
(1+\eps)-approximate Closest Vector Problem (CVP) on any
-dimensional lattice and in any near-symmetric norm in
2^{O(n)}(1+1/\eps)^n time and 2^n\poly(n) space. Our algorithm
builds on the lattice point enumeration techniques of Micciancio and
Voulgaris (STOC 2010, SICOMP 2013) and Dadush, Peikert and Vempala
(FOCS 2011), and gives an elegant, deterministic alternative to the
"AKS Sieve"-based algorithms for (1+\eps)-CVP (Ajtai, Kumar, and
Sivakumar; STOC 2001 and CCC 2002). Furthermore, assuming the
existence of a \poly(n)-space and -time algorithm for
exact CVP in the norm, the space complexity of our algorithm
can be reduced to polynomial.
Our main technical contribution is a method for "sparsifying" any
input lattice while approximately maintaining its metric structure. To
this end, we employ the idea of random sublattice restrictions, which
was first employed by Khot (FOCS 2003, J. Comp. Syst. Sci. 2006) for
the purpose of proving hardness for the Shortest Vector Problem (SVP)
under norms.
A preliminary version of this paper appeared in the Proc. 24th Annual
ACM-SIAM Symp. on Discrete Algorithms (SODA'13)
(http://dx.doi.org/10.1137/1.9781611973105.78)
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