91 research outputs found
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
Single Parameter Combinatorial Auctions with Partially Public Valuations
We consider the problem of designing truthful auctions, when the bidders'
valuations have a public and a private component. In particular, we consider
combinatorial auctions where the valuation of an agent for a set of
items can be expressed as , where is a private single parameter
of the agent, and the function is publicly known. Our motivation behind
studying this problem is two-fold: (a) Such valuation functions arise naturally
in the case of ad-slots in broadcast media such as Television and Radio. For an
ad shown in a set of ad-slots, is, say, the number of {\em unique}
viewers reached by the ad, and is the valuation per-unique-viewer. (b)
From a theoretical point of view, this factorization of the valuation function
simplifies the bidding language, and renders the combinatorial auction more
amenable to better approximation factors. We present a general technique, based
on maximal-in-range mechanisms, that converts any -approximation
non-truthful algorithm () for this problem into
and -approximate truthful
mechanisms which run in polynomial time and quasi-polynomial time,
respectively
Reading Articles Online
We study the online problem of reading articles that are listed in an
aggregated form in a dynamic stream, e.g., in news feeds, as abbreviated social
media posts, or in the daily update of new articles on arXiv. In such a
context, the brief information on an article in the listing only hints at its
content. We consider readers who want to maximize their information gain within
a limited time budget, hence either discarding an article right away based on
the hint or accessing it for reading. The reader can decide at any point
whether to continue with the current article or skip the remaining part
irrevocably. In this regard, Reading Articles Online, RAO, does differ
substantially from the Online Knapsack Problem, but also has its similarities.
Under mild assumptions, we show that any -competitive algorithm for the
Online Knapsack Problem in the random order model can be used as a black box to
obtain an -competitive algorithm for RAO, where
measures the accuracy of the hints with respect to the information profiles of
the articles. Specifically, with the current best algorithm for Online
Knapsack, which is -competitive, we obtain an upper bound
of on the competitive ratio of RAO. Furthermore, we study a
natural algorithm that decides whether or not to read an article based on a
single threshold value, which can serve as a model of human readers. We show
that this algorithmic technique is -competitive. Hence, our algorithms
are constant-competitive whenever the accuracy is a constant.Comment: Manuscript of COCOA 2020 pape
Online Independent Set Beyond the Worst-Case: Secretaries, Prophets, and Periods
We investigate online algorithms for maximum (weight) independent set on
graph classes with bounded inductive independence number like, e.g., interval
and disk graphs with applications to, e.g., task scheduling and spectrum
allocation. In the online setting, it is assumed that nodes of an unknown graph
arrive one by one over time. An online algorithm has to decide whether an
arriving node should be included into the independent set. Unfortunately, this
natural and practically relevant online problem cannot be studied in a
meaningful way within a classical competitive analysis as the competitive ratio
on worst-case input sequences is lower bounded by .
As a worst-case analysis is pointless, we study online independent set in a
stochastic analysis. Instead of focussing on a particular stochastic input
model, we present a generic sampling approach that enables us to devise online
algorithms achieving performance guarantees for a variety of input models. In
particular, our analysis covers stochastic input models like the secretary
model, in which an adversarial graph is presented in random order, and the
prophet-inequality model, in which a randomly generated graph is presented in
adversarial order. Our sampling approach bridges thus between stochastic input
models of quite different nature. In addition, we show that our approach can be
applied to a practically motivated admission control setting.
Our sampling approach yields an online algorithm for maximum independent set
with competitive ratio with respect to all of the mentioned
stochastic input models. for graph classes with inductive independence number
. The approach can be extended towards maximum-weight independent set by
losing only a factor of in the competitive ratio with denoting
the (expected) number of nodes
Buyback Problem - Approximate matroid intersection with cancellation costs
In the buyback problem, an algorithm observes a sequence of bids and must
decide whether to accept each bid at the moment it arrives, subject to some
constraints on the set of accepted bids. Decisions to reject bids are
irrevocable, whereas decisions to accept bids may be canceled at a cost that is
a fixed fraction of the bid value. Previous to our work, deterministic and
randomized algorithms were known when the constraint is a matroid constraint.
We extend this and give a deterministic algorithm for the case when the
constraint is an intersection of matroid constraints. We further prove a
matching lower bound on the competitive ratio for this problem and extend our
results to arbitrary downward closed set systems. This problem has applications
to banner advertisement, semi-streaming, routing, load balancing and other
problems where preemption or cancellation of previous allocations is allowed
Solving Multi-choice Secretary Problem in Parallel: An Optimal Observation-Selection Protocol
The classical secretary problem investigates the question of how to hire the
best secretary from candidates who come in a uniformly random order. In
this work we investigate a parallel generalizations of this problem introduced
by Feldman and Tennenholtz [14]. We call it shared -queue -choice
-best secretary problem. In this problem, candidates are evenly
distributed into queues, and instead of hiring the best one, the employer
wants to hire candidates among the best persons. The quotas are
shared by all queues. This problem is a generalized version of -choice
-best problem which has been extensively studied and it has more practical
value as it characterizes the parallel situation.
Although a few of works have been done about this generalization, to the best
of our knowledge, no optimal deterministic protocol was known with general
queues. In this paper, we provide an optimal deterministic protocol for this
problem. The protocol is in the same style of the -solution for the
classical secretary problem, but with multiple phases and adaptive criteria.
Our protocol is very simple and efficient, and we show that several
generalizations, such as the fractional -choice -best secretary problem
and exclusive -queue -choice -best secretary problem, can be solved
optimally by this protocol with slight modification and the latter one solves
an open problem of Feldman and Tennenholtz [14].
In addition, we provide theoretical analysis for two typical cases, including
the 1-queue 1-choice -best problem and the shared 2-queue 2-choice 2-best
problem. For the former, we prove a lower bound of
the competitive ratio. For the latter, we show the optimal competitive ratio is
while previously the best known result is 0.356 [14].Comment: This work is accepted by ISAAC 201
Constrained Non-Monotone Submodular Maximization: Offline and Secretary Algorithms
Constrained submodular maximization problems have long been studied, with
near-optimal results known under a variety of constraints when the submodular
function is monotone. The case of non-monotone submodular maximization is less
understood: the first approximation algorithms even for the unconstrainted
setting were given by Feige et al. (FOCS '07). More recently, Lee et al. (STOC
'09, APPROX '09) show how to approximately maximize non-monotone submodular
functions when the constraints are given by the intersection of p matroid
constraints; their algorithm is based on local-search procedures that consider
p-swaps, and hence the running time may be n^Omega(p), implying their algorithm
is polynomial-time only for constantly many matroids. In this paper, we give
algorithms that work for p-independence systems (which generalize constraints
given by the intersection of p matroids), where the running time is poly(n,p).
Our algorithm essentially reduces the non-monotone maximization problem to
multiple runs of the greedy algorithm previously used in the monotone case.
Our idea of using existing algorithms for monotone functions to solve the
non-monotone case also works for maximizing a submodular function with respect
to a knapsack constraint: we get a simple greedy-based constant-factor
approximation for this problem.
With these simpler algorithms, we are able to adapt our approach to
constrained non-monotone submodular maximization to the (online) secretary
setting, where elements arrive one at a time in random order, and the algorithm
must make irrevocable decisions about whether or not to select each element as
it arrives. We give constant approximations in this secretary setting when the
algorithm is constrained subject to a uniform matroid or a partition matroid,
and give an O(log k) approximation when it is constrained by a general matroid
of rank k.Comment: In the Proceedings of WINE 201
Secretary Problems: Weights and Discounts
The classical secretary problem studies the problem of selecting online an element (a “secretary”) with maximum value in a randomly ordered sequence. The difficulty lies in the fact that an element must be either selected or discarded upon its arrival, and this decision is irrevocable. Constant-competitive algorithms are known for the classical secretary problems and several variants. We study the following two extensions of the secretary problem:
In the discounted secretary problem, there is a time-dependent “discount” factor , and the benefit derived from selecting an element/secretary e at time t is . For this problem with arbitrary (not necessarily decreasing) functions , we show a constant-competitive algorithm when the expected optimum is known in advance. With no prior knowledge, we exhibit a lower bound and give a nearly matching -competitive algorithm.
In the weighted secretary problem, up to K secretaries can be selected; when a secretary is selected (s)he must be irrevocably assigned to one of K positions, with position k having weight , and assigning object/secretary e to position k has benefit . The goal is to select secretaries and assign them to positions to maximize
the sum of where is the secretary assigned to position k. We give constant-competitive algorithms for this problem.
Most of these results can also be extended to the matroid secretary case for a large family of matroids with a
constant-factor loss, and an O(log rank) loss for general matroids. These results are based on a reduction from various matroids to partition matroids which present a unified approach to many of the upper bounds of Babaioff et al. These problems have connections to online mechanism design. All our algorithms are monotone, and hence lead to truthful mechanisms for the corresponding online auction problems
Pricing Multi-Unit Markets
We study the power and limitations of posted prices in multi-unit markets,
where agents arrive sequentially in an arbitrary order. We prove upper and
lower bounds on the largest fraction of the optimal social welfare that can be
guaranteed with posted prices, under a range of assumptions about the
designer's information and agents' valuations. Our results provide insights
about the relative power of uniform and non-uniform prices, the relative
difficulty of different valuation classes, and the implications of different
informational assumptions. Among other results, we prove constant-factor
guarantees for agents with (symmetric) subadditive valuations, even in an
incomplete-information setting and with uniform prices
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