2,129 research outputs found
Asymptotics of lattice walks via analytic combinatorics in several variables
We consider the enumeration of walks on the two dimensional non-negative
integer lattice with short steps. Up to isomorphism there are 79 unique two
dimensional models to consider, and previous work in this area has used the
kernel method, along with a rigorous computer algebra approach, to show that 23
of the 79 models admit D-finite generating functions. In 2009, Bostan and
Kauers used Pad\'e-Hermite approximants to guess differential equations which
these 23 generating functions satisfy, in the process guessing asymptotics of
their coefficient sequences. In this article we provide, for the first time, a
complete rigorous verification of these guesses. Our technique is to use the
kernel method to express 19 of the 23 generating functions as diagonals of
tri-variate rational functions and apply the methods of analytic combinatorics
in several variables (the remaining 4 models have algebraic generating
functions and can thus be handled by univariate techniques). This approach also
shows the link between combinatorial properties of the models and features of
its asymptotics such as asymptotic and polynomial growth factors. In addition,
we give expressions for the number of walks returning to the x-axis, the
y-axis, and the origin, proving recently conjectured asymptotics of Bostan,
Chyzak, van Hoeij, Kauers, and Pech.Comment: 10 pages, 3 tables, as accepted to proceedings of FPSAC 2016 (without
conference formatting
Parametric shortest-path algorithms via tropical geometry
We study parameterized versions of classical algorithms for computing
shortest-path trees. This is most easily expressed in terms of tropical
geometry. Applications include shortest paths in traffic networks with variable
link travel times.Comment: 24 pages and 8 figure
Antimagic Labelings of Caterpillars
A -antimagic labeling of a graph is an injection from to
such that all vertex sums are pairwise distinct, where
the vertex sum at vertex is the sum of the labels assigned to edges
incident to . We call a graph -antimagic when it has a -antimagic
labeling, and antimagic when it is 0-antimagic. Hartsfield and Ringel
conjectured that every simple connected graph other than is antimagic,
but the conjecture is still open even for trees. Here we study -antimagic
labelings of caterpillars, which are defined as trees the removal of whose
leaves produces a path, called its spine. As a general result, we use
constructive techniques to prove that any caterpillar of order is -antimagic. Furthermore, if is a caterpillar with a
spine of order , we prove that when has at least leaves or consecutive vertices of degree at
most 2 at one end of a longest path, then is antimagic. As a consequence of
a result by Wong and Zhu, we also prove that if is a prime number, any
caterpillar with a spine of order , or is -antimagic.Comment: 13 pages, 4 figure
Spin chain simulations with a meron cluster algorithm
We apply a meron cluster algorithm to the XY spin chain, which describes a
quantum rotor. This is a multi-cluster simulation supplemented by an improved
estimator, which deals with objects of half-integer topological charge. This
method is powerful enough to provide precise results for the model with a
theta-term - it is therefore one of the rare examples, where a system with a
complex action can be solved numerically. In particular we measure the
correlation length, as well as the topological and magnetic susceptibility. We
discuss the algorithmic efficiency in view of the critical slowing down. Due to
the excellent performance that we observe, it is strongly motivated to work on
new applications of meron cluster algorithms in higher dimensions.Comment: 18 pages, 9 figures, published versio
Generating Random Elements of Finite Distributive Lattices
This survey article describes a method for choosing uniformly at random from
any finite set whose objects can be viewed as constituting a distributive
lattice. The method is based on ideas of the author and David Wilson for using
``coupling from the past'' to remove initialization bias from Monte Carlo
randomization. The article describes several applications to specific kinds of
combinatorial objects such as tilings, constrained lattice paths, and
alternating-sign matrices.Comment: 13 page
A Tight Lower Bound for Counting Hamiltonian Cycles via Matrix Rank
For even , the matchings connectivity matrix encodes which
pairs of perfect matchings on vertices form a single cycle. Cygan et al.
(STOC 2013) showed that the rank of over is
and used this to give an
time algorithm for counting Hamiltonian cycles modulo on graphs of
pathwidth . The same authors complemented their algorithm by an
essentially tight lower bound under the Strong Exponential Time Hypothesis
(SETH). This bound crucially relied on a large permutation submatrix within
, which enabled a "pattern propagation" commonly used in previous
related lower bounds, as initiated by Lokshtanov et al. (SODA 2011).
We present a new technique for a similar pattern propagation when only a
black-box lower bound on the asymptotic rank of is given; no
stronger structural insights such as the existence of large permutation
submatrices in are needed. Given appropriate rank bounds, our
technique yields lower bounds for counting Hamiltonian cycles (also modulo
fixed primes ) parameterized by pathwidth.
To apply this technique, we prove that the rank of over the
rationals is . We also show that the rank of
over is for any prime
and even for some primes.
As a consequence, we obtain that Hamiltonian cycles cannot be counted in time
for any unless SETH fails. This
bound is tight due to a time algorithm by Bodlaender et
al. (ICALP 2013). Under SETH, we also obtain that Hamiltonian cycles cannot be
counted modulo primes in time , indicating
that the modulus can affect the complexity in intricate ways.Comment: improved lower bounds modulo primes, improved figures, to appear in
SODA 201
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