On a Reduction for a Class of Resource Allocation Problems

Abstract

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 [Formula: see text] to [Formula: see text]. 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.

A Class of Convex Quadratic Nonseparable Resource Allocation Problems with Generalized Bound Constraints

“…A naive approach to solve RAP-DIBC would be to consider each possible combination of intervals for the variables separately and solve the corresponding simple RAP. Since one instance of this simple RAP can be solved in O(n) time [23], this approach has a non-polynomial time complexity of O(m n nF ), where F denotes the number of flops required for one evaluation of the function ϕ. Instead, in this paper, we propose an algorithm for solving all the four special cases (F1,L1), (F1,L2), (F2,L1), and (F2,L2) that runs in O n+m−2 m−2 (n log n + nF ) time.…”

“…This problem has applications in many different fields such as finance, telecommunications, and machine learning (see [20] for a survey). Many efficient methods exist to solve simple RAPs and we refer to [21,23] for recent overviews of such methods and further problem properties.…”

“…The objective is to minimize the peak consumption of the combined EV and static load, which can be expressed by the function T t=1 ϕ(x t + p t ). Common choices for ϕ are the quadratic function, absolute value function, or hinge max functions (see also Section 5.2 of [23]). Summarizing, this leads to the following formulation of the EV scheduling problem: Min-Thres-EV : min x t ∈ {0} ∪ [X min , X max ], t ∈ {1, .…”

ODDO: Online Duality-Driven Optimization

“…Therefore, we refine this bound for the special case where each cost function is of the form a t f t ( x t a t + b t ) where f t is a known increasing strictly convex differentiable function, a := (a t ) t∈T is a known positive-valued vector, and b := (b t ) t∈T is an uncertain vector. As shown in [50], in many applications, the cost functions have this form. In particular, this is the case for the battery charging scheduling problem that we consider in Section 4.1.…”

A fast algorithm for quadratic resource allocation problems with nested constraints

“…Such a structure is present in many applications considered in the literature, in particular in most of the applications surveyed or evaluated in [1,42]. We obtain such efficient algorithms by a reduction result in [37], which states that any optimal solution to an instance of QRAP-NC is also optimal for this instance when we take as objective function i∈N a i f ( xi ai ). As a consequence, our algorithm solves also such problems in O(n log n) time.…”