The Stochastic Bilevel Continuous Knapsack Problem with Uncertain Follower's Objective

TL;DR

This paper studies a stochastic bilevel continuous knapsack problem where the leader optimizes expected outcomes under uncertain follower profits, providing complexity results and efficient algorithms for specific distributions.

Contribution

It introduces a stochastic model for the bilevel knapsack problem with uncertain follower profits, and develops pseudo-polynomial algorithms and approximation schemes for this setting.

Findings

The problem is tractable with explicit scenario input.

The problem is #P-hard for uniform distributions.

Pseudo-polynomial algorithms are available for certain distributions.

Abstract

We consider a bilevel continuous knapsack problem where the leader controls the capacity of the knapsack, while the follower chooses a feasible packing maximizing his own profit. The leader's aim is to optimize a linear objective function in the capacity and in the follower's solution, but with respect to different item values. We address a stochastic version of this problem where the follower's profits are uncertain from the leader's perspective, and only a probability distribution is known. Assuming that the leader aims at optimizing the expected value of her objective function, we first observe that the stochastic problem is tractable as long as the possible scenarios are given explicitly as part of the input, which also allows to deal with general distributions using a sample average approximation. For the case of independently and uniformly distributed item values, we show that the problem is #P-hard in general, and the same is true even for evaluating the leader's objective function. Nevertheless, we present pseudo-polynomial time algorithms for this case, running in time linear in the total size of the items. Based on this, we derive an additive approximation scheme for the general case of independently distributed item values, which runs in pseudo-polynomial time.