On a Reduction for a Class of Resource Allocation Problems

In the resource allocation problem (RAP), the goal is to divide a given amount of a 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, for example, the multiplicative factors. In this article, we introduce a new class of objective functions that captures a significant number 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 problems in vessel routing and processor scheduling from O ( n 2 ) to O ( n log n ) . Summary of Contribution: The resource allocation problem (RAP) with submodular constraints and its special cases are classic problems in operations research. Because these problems are studied in many different scientific disciplines, many conceptual insights, structural properties, and solution approaches have been reinvented and rediscovered many times. The goal of this article is to reduce the amount of future reinventions and rediscoveries by bringing together these different perspectives on RAPs in a way that is accessible to researchers with different backgrounds. The article serves as an exposition on RAPs and on their wide applicability in many areas, including telecommunications, energy, and logistics. In particular, we provide tools and examples that can be used to formulate and solve problems in these areas as RAPs. To accomplish this, we make three concrete contributions. First, we provide a survey on algorithms and complexity results for RAPs and discuss several recent advances in these areas. Second, we show that many objectives for RAPs can be reduced to a (simpler) quadratic objective function, which makes available the extensive collection of fast and efficient algorithms for quadratic RAPs to solve these problems. Third, we discuss the impact that RAPs and the aforementioned reduction result can make in several application areas.