Stochastic Submodular Maximization with Performance-Dependent Item Costs

Abstract

We formulate a new stochastic submodular maximization problem by introducing the performance-dependent costs of items. In this problem, we consider selecting items for the case where the performance of each item (i.e., how much an item contributes to the objective function) is decided randomly, and the cost of an item depends on its performance. The goal of the problem is to maximize the objective function subject to a budget constraint on the costs of the selected items. We present an adaptive algorithm for this problem with a theoretical guaran-√ tee that its expected objective value is at least (1−1/ 4 e)/2 times the maximum value attained by any adaptive algorithms. We verify the performance of the algorithm through numerical experiments.

Price of dependence: stochastic submodular maximization with dependent items

“…In this section, we introduce an effective rounding approach that converts the continuous solution to an adaptive policy. Before explaining the rounding phase, we first introduce two important concepts: (β, γ)-balanced contention resolution scheme Chekuri et al (2014) and α-contention resolution scheme Fukunaga et al (2019). 00(0), pp.…”

“…Contention Resolution Scheme (β, γ)-balanced contention resolution scheme ((β, γ)-balanced CRS) is a general framework designed for maximizing set-submodular functions. In Fukunaga et al (2019), the authors extend this concept to the lattice-submodular functions by introducing α-contention resolution scheme (α-CRS).…”

“…Our problem reduces to the stochastic knapsack problem (Gupta et al 2011) when considering linear objective function. Recently, Fukunaga et al (2019), Tang (2021) extended the previous study to the stochastic submodular maximization problem, however, their model does not incorporate outer constraints.…”

“…Hence, our study can be considered as an extension of (Fukunaga et al 2019). Our work is also closely related to submodular probing problem (Adamczyk et al 2016) where they assume each item has binary states, our model allows each item to have multiple states.…”

“…We follow the framework developed in (Fukunaga et al 2019) and introduce the statedependent item costs into the classic stochastic submodular maximization problem. The input of our problem is a set of items, each item has a random state which is drawn from a known probability distribution.…”