268,360 research outputs found
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
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
Prophet Secretary for Combinatorial Auctions and Matroids
The secretary and the prophet inequality problems are central to the field of
Stopping Theory. Recently, there has been a lot of work in generalizing these
models to multiple items because of their applications in mechanism design. The
most important of these generalizations are to matroids and to combinatorial
auctions (extends bipartite matching). Kleinberg-Weinberg \cite{KW-STOC12} and
Feldman et al. \cite{feldman2015combinatorial} show that for adversarial
arrival order of random variables the optimal prophet inequalities give a
-approximation. For many settings, however, it's conceivable that the
arrival order is chosen uniformly at random, akin to the secretary problem. For
such a random arrival model, we improve upon the -approximation and obtain
-approximation prophet inequalities for both matroids and
combinatorial auctions. This also gives improvements to the results of Yan
\cite{yan2011mechanism} and Esfandiari et al. \cite{esfandiari2015prophet} who
worked in the special cases where we can fully control the arrival order or
when there is only a single item.
Our techniques are threshold based. We convert our discrete problem into a
continuous setting and then give a generic template on how to dynamically
adjust these thresholds to lower bound the expected total welfare.Comment: Preliminary version appeared in SODA 2018. This version improves the
writeup on Fixed-Threshold algorithm
Online Knapsack Problem under Expected Capacity Constraint
Online knapsack problem is considered, where items arrive in a sequential
fashion that have two attributes; value and weight. Each arriving item has to
be accepted or rejected on its arrival irrevocably. The objective is to
maximize the sum of the value of the accepted items such that the sum of their
weights is below a budget/capacity. Conventionally a hard budget/capacity
constraint is considered, for which variety of results are available. In modern
applications, e.g., in wireless networks, data centres, cloud computing, etc.,
enforcing the capacity constraint in expectation is sufficient. With this
motivation, we consider the knapsack problem with an expected capacity
constraint. For the special case of knapsack problem, called the secretary
problem, where the weight of each item is unity, we propose an algorithm whose
probability of selecting any one of the optimal items is equal to and
provide a matching lower bound. For the general knapsack problem, we propose an
algorithm whose competitive ratio is shown to be that is significantly
better than the best known competitive ratio of for the knapsack
problem with the hard capacity constraint.Comment: To appear in IEEE INFOCOM 2018, April 2018, Honolulu H
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