Stochastic Multi-round Submodular Optimization with Budget
arxiv(2024)
摘要
In this work we study the problem of Stochastic Budgeted Multi-round
Submodular Maximization (SBMSm), in which we would like to maximize the sum
over multiple rounds of the value of a monotone and submodular objective
function, subject to the fact that the values of this function depend on the
realization of stochastic events and the number of observations that we can
make over all rounds is limited by a given budget. This problem extends, and
generalizes to multiple round settings, well-studied problems such as
(adaptive) influence maximization and stochastic probing.
We first show that whenever a certain single-round optimization problem can
be optimally solved in polynomial time, then there is a polynomial time dynamic
programming algorithm that returns the same solution as the optimal algorithm,
that can adaptively choose both which observations to make and in which round
to have them. Unfortunately, this dynamic programming approach cannot be
extended to work when the single-round optimization problem cannot be
efficiently solved (even if we allow it would be approximated within an
arbitrary small constant). Anyway, in this case we are able to provide a simple
greedy algorithm for the problem. It guarantees a
(1/2-ϵ)-approximation to the optimal value, even if it non-adaptively
allocates the budget to rounds.
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