On a reduction for a class of resource allocation problems

TL;DR

This paper introduces a new class of objective functions for resource allocation problems, enabling a reduction to quadratic RAP and improving solution efficiency and complexity bounds in applications like vessel routing and processor scheduling.

Contribution

The authors propose a novel reduction technique for a broad class of resource allocation problems, simplifying their solution by linking them to quadratic RAP.

Findings

Reduction to quadratic RAP enables use of existing algorithms.

Improved worst-case complexity bounds from O(n^2) to O(n log n).

Applicable to vessel routing and processor scheduling problems.

Abstract

In the resource allocation problem (RAP), the goal is to divide a given amount of resource over a set of activities while minimizing the cost of this allocation and possibly satisfying constraints on allocations to subsets of the activities. Most solution approaches for the RAP and its extensions allow each activity to have its own cost function. However, in many applications, often the structure of the objective function is the same for each activity and the difference between the cost functions lies in different parameter choices such as, e.g., the multiplicative factors. In this article, we introduce a new class of objective functions that captures the majority of the objectives occurring in studied applications. These objectives are characterized by a shared structure of the cost function depending on two input parameters. We show that, given the two input parameters, there exists a solution to the RAP that is optimal for any choice of the shared structure. As a consequence, this problem reduces to the quadratic RAP, making available the vast amount of solution approaches and algorithms for the latter problem. We show the impact of our reduction result on several applications and, in particular, we improve the best known worst-case complexity bound of two important problems in vessel routing and processor scheduling from $O(n^2)$ to $O(n \log n)$.