14 research outputs found
Submodular Stochastic Probing on Matroids
In a stochastic probing problem we are given a universe , where each
element is active independently with probability , and only a
probe of e can tell us whether it is active or not. On this universe we execute
a process that one by one probes elements --- if a probed element is active,
then we have to include it in the solution, which we gradually construct.
Throughout the process we need to obey inner constraints on the set of elements
taken into the solution, and outer constraints on the set of all probed
elements. This abstract model was presented by Gupta and Nagarajan (IPCO '13),
and provides a unified view of a number of problems. Thus far, all the results
falling under this general framework pertain mainly to the case in which we are
maximizing a linear objective function of the successfully probed elements. In
this paper we generalize the stochastic probing problem by considering a
monotone submodular objective function. We give a -approximation algorithm for the case in which we are given
matroids as inner constraints and matroids as outer constraints.
Additionally, we obtain an improved -approximation
algorithm for linear objective functions
Approximation Algorithms for Correlated Knapsacks and Non-Martingale Bandits
In the stochastic knapsack problem, we are given a knapsack of size B, and a
set of jobs whose sizes and rewards are drawn from a known probability
distribution. However, we know the actual size and reward only when the job
completes. How should we schedule jobs to maximize the expected total reward?
We know O(1)-approximations when we assume that (i) rewards and sizes are
independent random variables, and (ii) we cannot prematurely cancel jobs. What
can we say when either or both of these assumptions are changed?
The stochastic knapsack problem is of interest in its own right, but
techniques developed for it are applicable to other stochastic packing
problems. Indeed, ideas for this problem have been useful for budgeted learning
problems, where one is given several arms which evolve in a specified
stochastic fashion with each pull, and the goal is to pull the arms a total of
B times to maximize the reward obtained. Much recent work on this problem focus
on the case when the evolution of the arms follows a martingale, i.e., when the
expected reward from the future is the same as the reward at the current state.
What can we say when the rewards do not form a martingale?
In this paper, we give constant-factor approximation algorithms for the
stochastic knapsack problem with correlations and/or cancellations, and also
for budgeted learning problems where the martingale condition is not satisfied.
Indeed, we can show that previously proposed LP relaxations have large
integrality gaps. We propose new time-indexed LP relaxations, and convert the
fractional solutions into distributions over strategies, and then use the LP
values and the time ordering information from these strategies to devise a
randomized adaptive scheduling algorithm. We hope our LP formulation and
decomposition methods may provide a new way to address other correlated bandit
problems with more general contexts
The update complexity of selection and related problems
We present a framework for computing with input data specified by intervals, representing uncertainty in the values of the input parameters. To compute a solution, the algorithm can query the input parameters that yield more refined estimates in form of sub-intervals and the objective is to minimize the number of queries. The previous approaches address the scenario where every query returns an exact value. Our framework is more general as it can deal with a wider variety of inputs and query responses and we establish interesting relationships between them that have not been investigated previously. Although some of the approaches of the previous restricted models can be adapted to the more general model, we
require more sophisticated techniques for the analysis and we also obtain improved algorithms for the previous model. We address selection problems in the generalized model and show that there exist 2-update competitive algorithms that do not depend on the lengths or distribution of the sub-intervals and hold against the worst case adversary. We also obtain similar bounds on the competitive ratio for the MST problem
in graphs