Online Knapsack Problem under Expected Capacity Constraint

TL;DR

This paper studies the online knapsack problem with an expected capacity constraint, proposing algorithms with improved competitive ratios for both the secretary and general cases, relevant for modern applications like cloud computing.

Contribution

It introduces novel algorithms for the online knapsack problem under expected capacity constraints, achieving better competitive ratios than existing methods.

Findings

For the secretary case, probability of selecting an optimal item is 1-1/e.

Proposed algorithm for general knapsack achieves a 1/4e competitive ratio.

Results outperform previous best ratio of 1/10e for hard capacity constraints.

Abstract

Online knapsack problem is considered, where items arrive in a sequential fashion that have two attributes; value and weight. Each arriving item has to be accepted or rejected on its arrival irrevocably. The objective is to maximize the sum of the value of the accepted items such that the sum of their weights is below a budget/capacity. Conventionally a hard budget/capacity constraint is considered, for which variety of results are available. In modern applications, e.g., in wireless networks, data centres, cloud computing, etc., enforcing the capacity constraint in expectation is sufficient. With this motivation, we consider the knapsack problem with an expected capacity constraint. For the special case of knapsack problem, called the secretary problem, where the weight of each item is unity, we propose an algorithm whose probability of selecting any one of the optimal items is equal to $1-1/e$ and provide a matching lower bound. For the general knapsack problem, we propose an algorithm whose competitive ratio is shown to be $1/4e$ that is significantly better than the best known competitive ratio of $1/10e$ for the knapsack problem with the hard capacity constraint.