492,745 research outputs found
New Results for the k-Secretary Problem
Suppose that n numbers arrive online in random order and the goal is to select k of them such that the expected sum of the selected items is maximized. The decision for any item is irrevocable and must be made on arrival without knowing future items. This problem is known as the k-secretary problem, which includes the classical secretary problem with the special case k=1. It is well-known that the latter problem can be solved by a simple algorithm of competitive ratio 1/e which is asymptotically optimal. When k is small, only for k=2 does there exist an algorithm beating the threshold of 1/e [Chan et al. SODA 2015]. The algorithm relies on an involved selection policy. Moreover, there exist results when k is large [Kleinberg SODA 2005].
In this paper we present results for the k-secretary problem, considering the interesting and relevant case that k is small. We focus on simple selection algorithms, accompanied by combinatorial analyses. As a main contribution we propose a natural deterministic algorithm designed to have competitive ratios strictly greater than 1/e for small k >= 2. This algorithm is hardly more complex than the elegant strategy for the classical secretary problem, optimal for k=1, and works for all k >= 1. We explicitly compute its competitive ratios for 2 <= k <= 100, ranging from 0.41 for k=2 to 0.75 for k=100. Moreover, we show that an algorithm proposed by Babaioff et al. [APPROX 2007] has a competitive ratio of 0.4168 for k=2, implying that the previous analysis was not tight. Our analysis reveals a surprising combinatorial property of this algorithm, which might be helpful for a tight analysis of this algorithm for general k
Optimal Algorithms for Free Order Multiple-Choice Secretary
Suppose we are given integer and boxes labeled
by an adversary, each containing a number chosen from an unknown distribution.
We have to choose an order to sequentially open these boxes, and each time we
open the next box in this order, we learn its number. If we reject a number in
a box, the box cannot be recalled. Our goal is to accept the largest of
these numbers, without necessarily opening all boxes. This is the free order
multiple-choice secretary problem. Free order variants were studied extensively
for the secretary and prophet problems. Kesselheim, Kleinberg, and Niazadeh KKN
(STOC'15) initiated a study of randomness-efficient algorithms (with the
cheapest order in terms of used random bits) for the free order secretary
problems.
We present an algorithm for free order multiple-choice secretary, which is
simultaneously optimal for the competitive ratio and used amount of randomness.
I.e., we construct a distribution on orders with optimal entropy
such that a deterministic multiple-threshold algorithm is
-competitive. This improves in three ways the previous
best construction by KKN, whose competitive ratio is .
Our competitive ratio is (near)optimal for the multiple-choice secretary
problem; it works for exponentially larger parameter ; and our algorithm is
a simple deterministic multiple-threshold algorithm, while that in KKN is
randomized. We also prove a corresponding lower bound on the entropy of optimal
solutions for the multiple-choice secretary problem, matching entropy of our
algorithm, where no such previous lower bound was known.
We obtain our algorithmic results with a host of new techniques, and with
these techniques we also improve significantly the previous results of KKN
about constructing entropy-optimal distributions for the classic free order
secretary
The Returning Secretary
In the online random-arrival model, an algorithm receives a sequence of requests that arrive in a random order. The algorithm is expected to make an irrevocable decision with regard to each request based only on the observed history. We consider the following natural extension of this model: each request arrives k times, and the arrival order is a random permutation of the kn arrivals; the algorithm is expected to make a decision regarding each request only upon its last arrival. We focus primarily on the case when k=2, which can also be interpreted as each request arriving at, and departing from the system, at a random time.
We examine the secretary problem: the problem of selecting the best secretary when the secretaries are presented online according to a random permutation. We show that when each secretary arrives twice, we can achieve a competitive ratio of 0.767974... (compared to 1/e in the classical secretary problem), and that it is optimal. We also show that without any knowledge about the number of secretaries or their arrival times, we can still hire the best secretary with probability at least 2/3, in contrast to the impossibility of achieving a constant success probability in the classical setting.
We extend our results to the matroid secretary problem, introduced by Babaioff et al. [3], and show a simple algorithm that achieves a 2-approximation to the maximal weighted basis in the new model (for k=2). We show that this approximation factor can be improved in special cases of the matroid secretary problem; in particular, we give a 16/9-competitive algorithm for the returning edge-weighted bipartite matching problem
Packing Returning Secretaries
We study online secretary problems with returns in combinatorial packing
domains with candidates that arrive sequentially over time in random order.
The goal is to accept a feasible packing of candidates of maximum total value.
In the first variant, each candidate arrives exactly twice. All arrivals
occur in random order. We propose a simple 0.5-competitive algorithm that can
be combined with arbitrary approximation algorithms for the packing domain,
even when the total value of candidates is a subadditive function. For
bipartite matching, we obtain an algorithm with competitive ratio at least
for growing , and an algorithm with ratio at least
for all . We extend all algorithms and ratios to arrivals
per candidate.
In the second variant, there is a pool of undecided candidates. In each
round, a random candidate from the pool arrives. Upon arrival a candidate can
be either decided (accept/reject) or postponed (returned into the pool). We
mainly focus on minimizing the expected number of postponements when computing
an optimal solution. An expected number of is always
sufficient. For matroids, we show that the expected number can be reduced to
, where is the minimum of the ranks of matroid and
dual matroid. For bipartite matching, we show a bound of , where
is the size of the optimum matching. For general packing, we show a lower
bound of , even when the size of the optimum is .Comment: 23 pages, 5 figure
Near-optimal irrevocable sample selection for periodic data streams with applications to marine robotics
We consider the task of monitoring spatiotemporal phenomena in real-time by
deploying limited sampling resources at locations of interest irrevocably and
without knowledge of future observations. This task can be modeled as an
instance of the classical secretary problem. Although this problem has been
studied extensively in theoretical domains, existing algorithms require that
data arrive in random order to provide performance guarantees. These algorithms
will perform arbitrarily poorly on data streams such as those encountered in
robotics and environmental monitoring domains, which tend to have
spatiotemporal structure. We focus on the problem of selecting representative
samples from phenomena with periodic structure and introduce a novel sample
selection algorithm that recovers a near-optimal sample set according to any
monotone submodular utility function. We evaluate our algorithm on a seven-year
environmental dataset collected at the Martha's Vineyard Coastal Observatory
and show that it selects phytoplankton sample locations that are nearly optimal
in an information-theoretic sense for predicting phytoplankton concentrations
in locations that were not directly sampled. The proposed periodic secretary
algorithm can be used with theoretical performance guarantees in many real-time
sensing and robotics applications for streaming, irrevocable sample selection
from periodic data streams.Comment: 8 pages, accepted for presentation in IEEE Int. Conf. on Robotics and
Automation, ICRA '18, Brisbane, Australia, May 201
Simple threshold rules solve explore/exploit trade‐offs in a resource accumulation search task
How, and how well, do people switch between exploration and exploitation to search for and accumulate resources? We study the decision processes underlying such exploration/exploitation trade‐offs using a novel card selection task that captures the common situation of searching among multiple resources (e.g., jobs) that can be exploited without depleting. With experience, participants learn to switch appropriately between exploration and exploitation and approach optimal performance. We model participants' behavior on this task with random, threshold, and sampling strategies, and find that a linear decreasing threshold rule best fits participants' results. Further evidence that participants use decreasing threshold‐based strategies comes from reaction time differences between exploration and exploitation; however, participants themselves report non‐decreasing thresholds. Decreasing threshold strategies that “front‐load” exploration and switch quickly to exploitation are particularly effective in resource accumulation tasks, in contrast to optimal stopping problems like the Secretary Problem requiring longer exploration
Prophet Inequalities with Limited Information
In the classical prophet inequality, a gambler observes a sequence of
stochastic rewards and must decide, for each reward ,
whether to keep it and stop the game or to forfeit the reward forever and
reveal the next value . The gambler's goal is to obtain a constant
fraction of the expected reward that the optimal offline algorithm would get.
Recently, prophet inequalities have been generalized to settings where the
gambler can choose items, and, more generally, where he can choose any
independent set in a matroid. However, all the existing algorithms require the
gambler to know the distribution from which the rewards are
drawn.
The assumption that the gambler knows the distribution from which
are drawn is very strong. Instead, we work with the much simpler
assumption that the gambler only knows a few samples from this distribution. We
construct the first single-sample prophet inequalities for many settings of
interest, whose guarantees all match the best possible asymptotically,
\emph{even with full knowledge of the distribution}. Specifically, we provide a
novel single-sample algorithm when the gambler can choose any elements
whose analysis is based on random walks with limited correlation. In addition,
we provide a black-box method for converting specific types of solutions to the
related \emph{secretary problem} to single-sample prophet inequalities, and
apply it to several existing algorithms. Finally, we provide a constant-sample
prophet inequality for constant-degree bipartite matchings.
We apply these results to design the first posted-price and multi-dimensional
auction mechanisms with limited information in settings with asymmetric
bidders
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