New Bounds for the Stochastic Knapsack Problem

The knapsack problem is a problem in combinatorial optimization that seeks to maximize the objective function \(\sum_{i = 1}^{n} v_ix_i\) subject to the constraints \(\sum_{i = 1}^{n} w_ix_i \leq W\) and \(x_i \in \{0, 1\}\), where \(\mathbf{x}, \mathbf{v} \in \mathbb{R}^{n}\) and \(W\) are provided. We consider the stochastic variant of this problem in which \(\mathbf{v}\) remains deterministic, but \(\mathbf{x}\)is an \(n\)-dimensional vector drawn uniformly at random from \([0, 1]^{n}\). We establish a sufficient condition under which the summation-bound condition is almost surely satisfied. Furthermore, we discuss the implications of this result on the deterministic problem.

New Bounds for the Stochastic Knapsack Problem

The knapsack problem is a problem in combinatorial optimization that seeks to maximize the objective function \(\sum_{i = 1}^{n} v_ix_i\) subject to the constraints \(\sum_{i = 1}^{n} w_ix_i \leq W\) and \(x_i \in \{0, 1\}\), where \(\mathbf{x}, \mathbf{v} \in \mathbb{R}^{n}\) and \(W\) are provided. We consider the stochastic variant of this problem in which \(\mathbf{v}\) remains deterministic, but \(\mathbf{x}\)is an \(n\)-dimensional vector drawn uniformly at random from \([0, 1]^{n}\). We establish a sufficient condition under which the summation-bound condition is almost surely satisfied. Furthermore, we discuss the implications of this result on the deterministic problem.

Logarithmic Regret in the Dynamic and Stochastic Knapsack Problem with Equal Rewards

We study a dynamic and stochastic knapsack problem in which a decision maker is sequentially presented with items arriving according to a Bernoulli process over n discrete time periods. Items have equal rewards and independent weights that are drawn from a known nonnegative continuous distribution F. The decision maker seeks to maximize the expected total reward of the items that the decision maker includes in the knapsack while satisfying a capacity constraint and while making terminal decisions as soon as each item weight is revealed. Under mild regularity conditions on the weight distribution F, we prove that the regret—the expected difference between the performance of the best sequential algorithm and that of a prophet who sees all of the weights before making any decision—is, at most, logarithmic in n. Our proof is constructive. We devise a reoptimized heuristic that achieves this regret bound.