A Polynomial Time Algorithm for the Resource Allocation Problem with a Convex Objective Function

1979 ◽  
Vol 30 (5) ◽  
pp. 449 ◽  
Author(s):  
N. Katoh ◽  
T. Ibaraki ◽  
H. Mine
Keyword(s):  
Resource Allocation ◽  
Objective Function ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Allocation Problem ◽  
Resource Allocation Problem ◽  
Convex Objective Function

A TWO-RESOURCE ALLOCATION PROBLEM ACCORDING TO AN EXPONENTIAL OBJECTIVE: OPTIMUM DISTRIBUTION OF SEARCHING EFFORT

1994 ◽  
Vol 01 (02) ◽  
pp. 135-146 ◽  
Author(s):  
TETSUO ICHIMORI ◽  
HIROSHI MASUYAMA ◽  
SHIGERU YAMADA
Keyword(s):  
Resource Allocation ◽  
Nonlinear Programming ◽  
Objective Function ◽  
Programming Problem ◽  
Polynomial Time ◽  
Allocation Problem ◽  
Nonlinear Programming Problem ◽  
Resource Allocation Problem ◽  
Optimum Distribution ◽  
Strongly Polynomial

The resource allocation problem has been studied in a variety of applications. This problem usually has only one constraint, i.e., the amount of resource to be allocated is constant. Considering its application areas, however, it is important to treat multiresource problems. In this paper we consider a two-resource allocation problem with an exponential objective function. Though this problem is a nonlinear programming problem, we show that it can be solved in strongly polynomial time.

Optimal Screening of Populations with Heterogeneous Risk Profiles Under the Availability of Multiple Tests

2021 ◽  
Author(s):  
Hrayer Aprahamian ◽  
Hadi El-Amine
Keyword(s):  
Objective Function ◽  
Structural Properties ◽  
Polynomial Time ◽  
Network Flow ◽  
Optimization Problem ◽  
Flow Problem ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Network Flow Problem ◽  
Multiple Tests

We study the design of large-scale group testing schemes under a heterogeneous population (i.e., subjects with potentially different risk) and with the availability of multiple tests. The objective is to classify the population as positive or negative for a given binary characteristic (e.g., the presence of an infectious disease) as efficiently and accurately as possible. Our approach examines components often neglected in the literature, such as the dependence of testing cost on the group size and the possibility of no testing, which are especially relevant within a heterogeneous setting. By developing key structural properties of the resulting optimization problem, we are able to reduce it to a network flow problem under a specific, yet not too restrictive, objective function. We then provide results that facilitate the construction of the resulting graph and finally provide a polynomial time algorithm. Our case study, on the screening of HIV in the United States, demonstrates the substantial benefits of the proposed approach over conventional screening methods. Summary of Contribution: This paper studies the problem of testing heterogeneous populations in groups in order to reduce costs and hence allow for the use of more efficient tests for high-risk groups. The resulting problem is a difficult combinatorial optimization problem that is NP-complete under a general objective. Using structural properties specific to our objective function, we show that the problem can be cast as a network flow problem and provide a polynomial time algorithm.

scholarly journals On a Reduction for a Class of Resource Allocation Problems

2021 ◽  
Author(s):  
Martijn H. H. Schoot Uiterkamp ◽  
Marco E. T. Gerards ◽  
Johann L. Hurink
Keyword(s):  
Resource Allocation ◽  
Cost Function ◽  
Objective Function ◽  
Allocation Problem ◽  
Resource Allocation Problem ◽  
Special Cases ◽  
Input Parameters ◽  
The Cost ◽  
The Impact ◽  
Shared Structure

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.

scholarly journals Two-Agent Preemptive Pareto-Scheduling to Minimize Late Work and Other Criteria

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1517
Author(s):  
Ruyan He ◽  
Jinjiang Yuan
Keyword(s):  
Objective Function ◽  
Polynomial Time ◽  
Single Machine ◽  
Completion Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Pareto Optimal ◽  
Scheduling Problems ◽  
Trade Off ◽  
Late Work

In this paper, we consider three preemptive Pareto-scheduling problems with two competing agents on a single machine. In each problem, the objective function of agent A is the total completion time, the maximum lateness, or the total late work while the objective function of agent B is the total late work. For each problem, we provide a polynomial-time algorithm to characterize the trade-off curve of all Pareto-optimal points.

scholarly journals Equilibrium computation in resource allocation games

2021 ◽  
Author(s):  
Tobias Harks ◽  
Veerle Timmermans
Keyword(s):  
Resource Allocation ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Congestion Games ◽  
Congestion Game ◽  
Equilibrium Computation ◽  
The Difference ◽  
Price Functions ◽  
Resource Allocation Games

AbstractWe study the equilibrium computation problem for two classical resource allocation games: atomic splittable congestion games and multimarket Cournot oligopolies. For atomic splittable congestion games with singleton strategies and player-specific affine cost functions, we devise the first polynomial time algorithm computing a pure Nash equilibrium. Our algorithm is combinatorial and computes the exact equilibrium assuming rational input. The idea is to compute an equilibrium for an associated integrally-splittable singleton congestion game in which the players can only split their demands in integral multiples of a common packet size. While integral games have been considered in the literature before, no polynomial time algorithm computing an equilibrium was known. Also for this class, we devise the first polynomial time algorithm and use it as a building block for our main algorithm. We then develop a polynomial time computable transformation mapping a multimarket Cournot competition game with firm-specific affine price functions and quadratic costs to an associated atomic splittable congestion game as described above. The transformation preserves equilibria in either game and, thus, leads – via our first algorithm – to a polynomial time algorithm computing Cournot equilibria. Finally, our analysis for integrally-splittable games implies new bounds on the difference between real and integral Cournot equilibria. The bounds can be seen as a generalization of the recent bounds for single market oligopolies obtained by Todd (Math Op Res 41(3):1125–1134 2016, 10.1287/moor.2015.0771).

Sign in / Sign up

Export Citation Format

Share Document