Stochastic Knapsack

Years and Authors of Summarized Original Work

• 2004; Dean, Goemans, Vondrák

• 2011; Bhalgat, Goel, Khanna

Problem Definition

This problem deals with packing a maximum reward set of items into a knapsack of given capacity, when the item sizes are random. The input is a collection of n items, where each item i ∈ [n] : = { 1, …, n} has reward r _i  ≥ 0 and size S _i  ≥ 0, and a knapsack capacity B ≥ 0. In the stochastic knapsack problem, all rewards are deterministic but the sizes are random. The random variables S _i s are independent with known, arbitrary distributions. The actual size of an item is known only when it is placed into the knapsack. The objective is to add items sequentially (one by one) into the knapsack so as to maximize the expected reward of the items that fit into the knapsack. As usual, a subset T of items is said to fit into the knapsack if the total size $\sum _{i\in T}S_{i}$ is at most the knapsack capacity B.

A feasible solution (or policy) to the stochastic knapsack...