18,137 research outputs found
Randomized Online Algorithms with High Probability Guarantees
We study the relationship between the competitive ratio and the tail distribution of randomized online problems. To this end, we define a broad class of online problems that includes some of the well-studied problems like paging, k-server and metrical task systems on finite metrics, and show that for these problems it is possible to obtain, given an algorithm with constant expected competitive ratio, another algorithm that achieves the same solution quality up to an arbitrarily small constant error with high probability; the "high probability" statement is in terms of the optimal cost. Furthermore, we show that our assumptions are tight in the sense that removing any of them allows for a counterexample to the theorem
A note on Probably Certifiably Correct algorithms
Many optimization problems of interest are known to be intractable, and while
there are often heuristics that are known to work on typical instances, it is
usually not easy to determine a posteriori whether the optimal solution was
found. In this short note, we discuss algorithms that not only solve the
problem on typical instances, but also provide a posteriori certificates of
optimality, probably certifiably correct (PCC) algorithms. As an illustrative
example, we present a fast PCC algorithm for minimum bisection under the
stochastic block model and briefly discuss other examples
Lower Bounds for Structuring Unreliable Radio Networks
In this paper, we study lower bounds for randomized solutions to the maximal
independent set (MIS) and connected dominating set (CDS) problems in the dual
graph model of radio networks---a generalization of the standard graph-based
model that now includes unreliable links controlled by an adversary. We begin
by proving that a natural geographic constraint on the network topology is
required to solve these problems efficiently (i.e., in time polylogarthmic in
the network size). We then prove the importance of the assumption that nodes
are provided advance knowledge of their reliable neighbors (i.e, neighbors
connected by reliable links). Combined, these results answer an open question
by proving that the efficient MIS and CDS algorithms from [Censor-Hillel, PODC
2011] are optimal with respect to their dual graph model assumptions. They also
provide insight into what properties of an unreliable network enable efficient
local computation.Comment: An extended abstract of this work appears in the 2014 proceedings of
the International Symposium on Distributed Computing (DISC
Budget-Feasible Mechanism Design for Non-Monotone Submodular Objectives: Offline and Online
The framework of budget-feasible mechanism design studies procurement
auctions where the auctioneer (buyer) aims to maximize his valuation function
subject to a hard budget constraint. We study the problem of designing truthful
mechanisms that have good approximation guarantees and never pay the
participating agents (sellers) more than the budget. We focus on the case of
general (non-monotone) submodular valuation functions and derive the first
truthful, budget-feasible and -approximate mechanisms that run in
polynomial time in the value query model, for both offline and online auctions.
Prior to our work, the only -approximation mechanism known for
non-monotone submodular objectives required an exponential number of value
queries.
At the heart of our approach lies a novel greedy algorithm for non-monotone
submodular maximization under a knapsack constraint. Our algorithm builds two
candidate solutions simultaneously (to achieve a good approximation), yet
ensures that agents cannot jump from one solution to the other (to implicitly
enforce truthfulness). Ours is the first mechanism for the problem
where---crucially---the agents are not ordered with respect to their marginal
value per cost. This allows us to appropriately adapt these ideas to the online
setting as well.
To further illustrate the applicability of our approach, we also consider the
case where additional feasibility constraints are present. We obtain
-approximation mechanisms for both monotone and non-monotone submodular
objectives, when the feasible solutions are independent sets of a -system.
With the exception of additive valuation functions, no mechanisms were known
for this setting prior to our work. Finally, we provide lower bounds suggesting
that, when one cares about non-trivial approximation guarantees in polynomial
time, our results are asymptotically best possible.Comment: Accepted to EC 201
Online Bin Covering with Frequency Predictions
We study the discrete bin covering problem where a multiset of items from a
fixed set must be split into disjoint subsets while
maximizing the number of subsets whose contents sum to at least . We study
the online discrete variant, where is finite, and items arrive
sequentially. In the purely online setting, we show that the competitive ratios
of best deterministic (and randomized) algorithms converge to for
large , similar to the continuous setting. Therefore, we consider the
problem under the prediction setting, where algorithms may access a vector of
frequencies predicting the frequency of items of each size in the instance. In
this setting, we introduce a family of online algorithms that perform
near-optimally when the predictions are correct. Further, we introduce a second
family of more robust algorithms that presents a tradeoff between the
performance guarantees when the predictions are perfect and when predictions
are adversarial. Finally, we consider a stochastic setting where items are
drawn independently from any fixed but unknown distribution of . Using
results from the PAC-learnability of probabilities in discrete distributions,
we also introduce a purely online algorithm whose average-case performance is
near-optimal with high probability for all finite sets and all
distributions of .Comment: 27 page
Privacy via the Johnson-Lindenstrauss Transform
Suppose that party A collects private information about its users, where each
user's data is represented as a bit vector. Suppose that party B has a
proprietary data mining algorithm that requires estimating the distance between
users, such as clustering or nearest neighbors. We ask if it is possible for
party A to publish some information about each user so that B can estimate the
distance between users without being able to infer any private bit of a user.
Our method involves projecting each user's representation into a random,
lower-dimensional space via a sparse Johnson-Lindenstrauss transform and then
adding Gaussian noise to each entry of the lower-dimensional representation. We
show that the method preserves differential privacy---where the more privacy is
desired, the larger the variance of the Gaussian noise. Further, we show how to
approximate the true distances between users via only the lower-dimensional,
perturbed data. Finally, we consider other perturbation methods such as
randomized response and draw comparisons to sketch-based methods. While the
goal of releasing user-specific data to third parties is more broad than
preserving distances, this work shows that distance computations with privacy
is an achievable goal.Comment: 24 page
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