1,743 research outputs found
Limit distributions for the problem of collecting pairs
Let . Elements are drawn from the set with
replacement, assuming that each element has probability of being drawn.
We determine the limiting distributions for the waiting time until the given
portion of pairs , , is sampled. Exact distributions of some
related random variables and their characteristics are also obtained.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ114 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
New results on a generalized coupon collector problem using Markov chains
We study in this paper a generalized coupon collector problem, which consists
in determining the distribution and the moments of the time needed to collect a
given number of distinct coupons that are drawn from a set of coupons with an
arbitrary probability distribution. We suppose that a special coupon called the
null coupon can be drawn but never belongs to any collection. In this context,
we obtain expressions of the distribution and the moments of this time. We also
prove that the almost-uniform distribution, for which all the non-null coupons
have the same drawing probability, is the distribution which minimizes the
expected time to get a fixed subset of distinct coupons. This optimization
result is extended to the complementary distribution of that time when the full
collection is considered, proving by the way this well-known conjecture.
Finally, we propose a new conjecture which expresses the fact that the
almost-uniform distribution should minimize the complementary distribution of
the time needed to get any fixed number of distinct coupons.Comment: 14 page
Omnibus Sequences, Coupon Collection, and Missing Word Counts
An {\it Omnibus Sequence} of length is one that has each possible
"message" of length embedded in it as a subsequence. We study various
properties of Omnibus Sequences in this paper, making connections, whenever
possible, to the classical coupon collector problem.Comment: 26 page
Better Approximation Bounds for the Joint Replenishment Problem
The Joint Replenishment Problem (JRP) deals with optimizing shipments of
goods from a supplier to retailers through a shared warehouse. Each shipment
involves transporting goods from the supplier to the warehouse, at a fixed cost
C, followed by a redistribution of these goods from the warehouse to the
retailers that ordered them, where transporting goods to a retailer has
a fixed cost . In addition, retailers incur waiting costs for each
order. The objective is to minimize the overall cost of satisfying all orders,
namely the sum of all shipping and waiting costs.
JRP has been well studied in Operations Research and, more recently, in the
area of approximation algorithms. For arbitrary waiting cost functions, the
best known approximation ratio is 1.8. This ratio can be reduced to 1.574 for
the JRP-D model, where there is no cost for waiting but orders have deadlines.
As for hardness results, it is known that the problem is APX-hard and that the
natural linear program for JRP has integrality gap at least 1.245. Both results
hold even for JRP-D. In the online scenario, the best lower and upper bounds on
the competitive ratio are 2.64 and 3, respectively. The lower bound of 2.64
applies even to the restricted version of JRP, denoted JRP-L, where the waiting
cost function is linear.
We provide several new approximation results for JRP. In the offline case, we
give an algorithm with ratio 1.791, breaking the barrier of 1.8. In the online
case, we show a lower bound of 2.754 on the competitive ratio for JRP-L (and
thus JRP as well), improving the previous bound of 2.64. We also study the
online version of JRP-D, for which we prove that the optimal competitive ratio
is 2
Transition between characters of classical groups, decomposition of Gelfand-Tsetlin patterns and last passage percolation
We study the combinatorial structure of the irreducible characters of the
classical groups , ,
, and the
"non-classical" odd symplectic group , finding new
connections to the probabilistic model of Last Passage Percolation (LPP).
Perturbing the expressions of these characters as generating functions of
Gelfand-Tsetlin patterns, we produce two families of symmetric polynomials that
interpolate between characters of and and between characters of
and . We identify the first family as a
one-parameter specialization of Koornwinder polynomials, for which we thus
provide a novel combinatorial structure; on the other hand, the second family
appears to be new. We next develop a method of Gelfand-Tsetlin pattern
decomposition to establish identities between all these polynomials that, in
the case of characters, can be viewed as describing the decomposition of
irreducible representations of the groups when restricted to certain subgroups.
Through these formulas we connect orthogonal and symplectic characters, and
more generally the interpolating polynomials, to LPP models with various
symmetries, thus going beyond the link with classical Schur polynomials
originally found by Baik and Rains [BR01a]. Taking the scaling limit of the LPP
models, we finally provide an explanation of why the Tracy-Widom GOE and GSE
distributions from random matrix theory admit formulations in terms of both
Fredholm determinants and Fredholm Pfaffians.Comment: 60 pages, 11 figures. Typos corrected and a few remarks adde
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