3 research outputs found
Submodular Stochastic Probing with Prices
We introduce Stochastic Probing with Prices (SPP), a variant of the
Stochastic Probing (SP) model in which we must pay a price to probe an element.
A SPP problem involves two set systems and
where each is active with probability .
To discover whether is active, it must be probed by paying the price
. If it is probed and active, then it is irrevocably added to the
solution. Moreover, at all times, the set of probed elements must lie in
, and the solution (the set of probed and active elements)
must lie in . The goal is to maximize a set function
minus the cost of the probes. We give a bi-criteria approximation algorithm to
the online version of this problem, in which the elements are shown to the
algorithm in a possibly adversarial order. Our results translate to
state-of-the-art approximations for the traditional (online) stochastic probing
problem
Stochastic Monotone Submodular Maximization with Queries
We study a stochastic variant of monotone submodular maximization problem as
follows. We are given a monotone submodular function as an objective function
and a feasible domain defined on a finite set, and our goal is to find a
feasible solution that maximizes the objective function. A special part of the
problem is that each element in the finite set has a random hidden state,
active or inactive, only the active elements contribute to the objective value,
and we can conduct a query to an element to reveal its hidden state. The goal
is to obtain a feasible solution having a large objective value by conducting a
small number of queries. This is the first attempt to consider nonlinear
objective functions in such a stochastic model. We prove that the problem
admits a good query strategy if the feasible domain has a uniform exchange
property. This result generalizes Blum et al.'s result on the unweighted
matching problem and Behnezhad and Reyhani's result on the weighted matching
problem in both objective function and feasible domain
Online Allocation and Pricing: Constant Regret via Bellman Inequalities
We develop a framework for designing simple and efficient policies for a
family of online allocation and pricing problems, that includes online packing,
budget-constrained probing, dynamic pricing, and online contextual bandits with
knapsacks. In each case, we evaluate the performance of our policies in terms
of their regret (i.e., additive gap) relative to an offline controller that is
endowed with more information than the online controller. Our framework is
based on Bellman Inequalities, which decompose the loss of an algorithm into
two distinct sources of error: (1) arising from computational tractability
issues, and (2) arising from estimation/prediction of random trajectories.
Balancing these errors guides the choice of benchmarks, and leads to policies
that are both tractable and have strong performance guarantees. In particular,
in all our examples, we demonstrate constant-regret policies that only require
re-solving an LP in each period, followed by a simple greedy action-selection
rule; thus, our policies are practical as well as provably near optimal.Comment: To appear in Operations Research, 202