Optimal Polynomial Time Algorithm Research Articles

Optimal three-level presort loading of commercial bulk mail in the postal service industry

This study focuses on the presort loading of commercial bulk mail. Here, presort is the process by which a mailer prepares mail such that it is sorted to at least the finest extent required by the postal service provider for a claimed (discounted) price. We formulated this presort loading problem (PLP) as a special case of transportation problem with minimum quantity commitment (MQC) constraints. In addition, we developed a polynomial time optimal solution algorithm for the PLP and performed computational experiments on randomly generated problem instances under various discount structures. Results of the computational experiments show that mailers can potentially reduce their costs by sending mail less frequently, using small-sized mail trays; however, the discount structure does not affect the main results. There is some evidence that smaller cost reductions on mailing fees occur as the variation in the discount rate increases; however, the effects of discount structure are nominal compared with the gains from changing the mailing frequency. Mailing fee savings are more heavily influenced by the discount structure when MQC constraints become tighter.

Scheduling for energy minimization on restricted parallel processors

Scheduling for energy conservation has become a major concern in the field of information technology because of the need to reduce energy use and carbon dioxide emissions. Previous work has focused on the assumption that a task can be assigned to any processor. In contrast, we initially study the problem of task scheduling on restricted parallel processors. The restriction takes account of affinities between tasks and processors; that is, a task has its own eligible set of processors. We adopt the Speed Scaling (SS) method to save energy under an execution time constraint (on the makespan Cmax), and the processors can run at arbitrary speeds in [smin,smax]. Our objective is to minimize the overall energy consumption. The energy-efficient scheduling problem, involving task assignment and speed scaling, is inherently complex as it is proved to be NP-complete for general tasks. We formulate the problem as an Integer Programming (IP) problem. Specifically, we devise a polynomial-time optimal scheduling algorithm for the case in which tasks have a uniform size. Our algorithm runs in O(mn3logn) time, where m is the number of processors and n is the number of tasks. We then present a polynomial-time algorithm that achieves a bounded approximation factor when the tasks have arbitrary-size work. Numerical results demonstrate that our algorithm could provide an energy-efficient solution to the problem of task scheduling on restricted parallel processors.

Total completion time minimization on multiple machines subject to machine availability and makespan constraints

Abstract This paper studies preemptive bi-criteria scheduling on m parallel machines with machine unavailable intervals. The goal is to minimize the total completion time subject to the constraint that the makespan is at most a constant T . We study the unavailability model such that the number of available machines cannot go down by 2 within any period of p max where p max is the maximum processing time among all jobs. We show that there is an optimal polynomial time algorithm.

A Note on Two-Agent Scheduling with Resource Dependent Release Times on a Single Machine

We consider a scheduling problem in which both resource dependent release times and two agents exist simultaneously. Two agents share a common single machine, and each agent wants to minimize a cost function dependent on its own jobs. The release time of eachA-agent’s job is related to the amount of resource consumed. The objective is to find a schedule for the problem of minimizingA-agent’s total amount of resource consumption with a constraint onB-agent’s makespan. The optimal properties and the optimal polynomial time algorithm are proposed to solve the scheduling problem.

Optimal Channel Allocation for Multi-PU and Multi-SU Pairs in Underlay Cognitive Radio Networks

In the underlay cognitive radio networks, this paper defines the joint channel and power allocation problem, which aims to optimise the max-total and max-min throughputs of secondary users (SUs), with the constraints of interference on primary receivers. For the max-total problem, we formulate the problem as a bipartite matching and derive a maximum weighted matching-based sum throughput maximisation algorithm (STMA) to solve this problem. For the max-min problem, on the basis of the optimal relay assignment (ORA) algorithm, we derive a polynomial time optimal channel assignment algorithm (OCAA) to iteratively assign channels to each SU pair under the power constraint. Simulation results demonstrate the effectiveness of our algorithms when compared with random method.

Single machine batch scheduling problem to minimize makespan with controllable setup and jobs processing times

This paper concerns with a single-machine scheduling problem underbatch availability in which both the setup of each batch and the processing times of jobs are controllable by allocating a resource. The completion time ofa job in a batch is that of the last job in the batch. Two batchscheduling problems are investigated. The objective is todetermine the job sequence, the partition of the job sequence intobatches and the resource allocation scheme to minimize makespan,subject to the total amount of resource is bounded by a givenvalue $U$ in the first problem; while in the second problem is tominimize a total cost of makespan and resource without resourcelimitation, respectively. We show that the problems underlying canbe solved in polynomial time and present optimal algorithms.

A polynomial-time algorithm for linear optimization based on a new kernel function with trigonometric barrier term

In this paper, we propose a large-update interior-point algorithm for linear optimization based on a new kernel function. New search directions and proximity measure are defined based on this kernel function. We show that if a strictly feasible starting point is available, then the new algorithm has O(3/4log n/?) iteration complexity.

Single machine total completion time scheduling problem with workload-dependent maintenance duration

A single-machine scheduling problem with workload-dependent maintenance duration is considered. The objective is to minimize total completion time. For the case where the derivation of the maintenance duration function is greater than or equal to 1, a polynomial time optimal algorithm is proposed. For the case where the derivation of the maintenance duration function is less than 1, a polynomial time approximation scheme is proposed.

Topology for Optimal Task Assignment on Multicomputers with Dedicated Resources

One of the geometrical k-cut problem applications is the task assignment in homogeneous networks where processors are homogeneous and some of them have unique or special functionalities. Though the problem is NP-hard in general, there are polynomial-time optimal algorithms if the topology graphs representing networks have regular properties such as tree and general array. We showed that the O(n 3 k) time complexity of existing algorithms for the task assignment problems can be enhanced to O(n 3 logk) in cases of linear array, binary tree, and general array by using the results of [3, 5]. This paper also proposed a new kind of topology graphs on which the geometrical k-cut problem can be solved in polynomial time. The time complexity of the algorithm is O(n 3 k), where n is the number of nodes in a k

Scheduling jobs and a variable maintenance on a single machine with common due-date assignment.

We investigate a common due-date assignment scheduling problem with a variable maintenance on a single machine. The goal is to minimize the total earliness, tardiness, and due-date cost. We derive some properties on an optimal solution for our problem. For a special case with identical jobs we propose an optimal polynomial time algorithm followed by a numerical example.

Enhancing Survivability in Virtualized Data Centers: A Service-Aware Approach

In this paper, we propose a service-aware approach to enhance survivability in virtualized data centers. The idea is to create and maintain a Survivable Virtual Infrastructure (SVI) for each service or tenant, which includes Virtual Machines (VMs) hosting the corresponding application and their backup VMs. A fundamental problem is to determine how to map each SVI to a data center network with minimum operational costs while satisfying each VM's resource requirements and bandwidth demands between VMs before and after failures. This problem can be naturally divided into two subproblems: VM Placement (VMP) and Virtual Link Mapping (VLM). We first present a general optimization framework. Then we propose an efficient algorithm for VMP, and a polynomial-time optimal algorithm for VLM, which can be used as subroutines in the framework. We also present an effective heuristic algorithm that jointly solves two subproblems. It has been shown by extensive simulation results based on the real VM workload traces collected from Syracuse University's green data center that compared to the First Fit Decreasing (FFD) and shortest path routing based baseline algorithm, the proposed algorithms significantly reduce the reserved bandwidth, and yield comparable results in terms of the number of active servers. \begin{keywords}Cloud Computing, Data Center, Service-aware, Survivability, Virtual Machine Management. \end{keywords}

Strong minimum energy $$2$$ 2 -hop rooted topology for hierarchical wireless sensor networks

Given a set of sensors, the strong minimum energy topology (SMET) problem is to assign transmit power to each sensor such that the sum of the transmit powers of all the sensors is minimum subject to the constraint that the resulting topology containing only bidirectional links is strongly connected. This problem is known to be NP-hard. However, most of the wireless sensor networks are hierarchical in nature, where shorter path lengths are preferred for communication between any two nodes. Given a set of sensors and a specified sensor, say $$r$$r, the strong minimum energy $$2$$2-hop rooted topology problem ($$2$$2h-SMERT) is to assign transmit power to each sensor such that the sum of the transmit powers of all the sensors is minimum subject to the constraints that the resulting topology containing only bidirectional links is strongly connected and the length of the path from $$r$$r to any other node is at most $$2$$2. We prove that the $$2$$2h-SMERT problem is NP-hard. We also prove that $$2$$2h-SMERT problem is APX-hard. We then show that $$2$$2h-SMERT problem is not approximable within a factor of $$\frac{1}{2}(1-\epsilon ) \ln n$$12(1-∈)lnn unless NP $$\subseteq DTIME(n^{\log \log n})$$⊆DTIME(nloglogn). On the positive side, we propose a $$2(1+ \ln n)$$2(1+lnn)-approximation algorithm for the $$2$$2h-SMERT problem. We also study two special cases of the $$2$$2h-SMERT problem, namely, the $$d$$d-rest-$$2$$2h-SMERT problem and the $$k$$k-int-$$2$$2h-SMERT problem. We propose a $$2(1+ \ln d)$$2(1+lnd)-approximation algorithm for the $$d$$d-rest-$$2$$2h-SMERT problem. The $$k$$k-int-$$2$$2h-SMERT problem is NP-hard for arbitrary $$k$$k. However, for fixed constant $$k$$k, we propose a $$\frac{k+1}{2}$$k+12-approximation algorithm for the $$k$$k-int-$$2$$2h-SMERT problem and obtain a polynomial time optimal algorithm for the $$k$$k-int-$$2$$2h-SMERT problem for $$k=2$$k=2.

A Polynomial Time Optimal Algorithm for Linear Bin Packing Problem

일반적으로 상자포장 문제는 다항시간으로 근사 해를 찾는 알고리즘만이 존재하여 NP-완전 문제로 알려져 있다. 본 논문은 선형 상자문제의 정확한 해를 다항시간으로 찾을 수 있는 알고리즘을 제안한다. n개 물품의 물품 크기 a<SUB>i</SUB>, (i=1, 2, …)를 내림차순으로 정렬시켜 용량이 C인 필요한 상자 수 k의 약 75% 상자들에 ?0.5C≤a<SUB>i</SUB>?개의 물품을 순서대로 1개씩 넣고, 나머지 물품을 찾아 넣는 방법으로 문제를 25%로 단축시켰다. 나머지 필요한 상자 25%에 대해서도 유사한 방법으로 물품을 채우는 방법을 적용하였다. 제안된 알고리즘을 OR-LIB의 120, 250과 500개의 데이터에 적용한 결과 100% 정확한 해를 얻는데 성공하였다. 제안된 알고리즘은 상자 포장 문제가 P-문제임을 증명하였으며, NP-완전으로 알려진 멀티프로세서 스케쥴링 문제에도 적용할 수 있을 것이다.

Supply Chain Scheduling on a Single Machine with Availability Constraint

In the actual industrial engineering, the machine may be checked to ensure that they can work efficiently. Thus, the machine has an unavailable interval so that the job could be interrupted. When the machine becomes available again, the job can be resumed processing. When the job is completed, it can be delivered in batches to one customer by vehicles with capacity constraint. Our goal is to minimize the sum of arrival time of batches and the total delivery cost. We develop an optimal polynomial time algorithm and give an instance to verify the algorithm.

ESWC: Efficient Scheduling for the Mobile Sink in Wireless Sensor Networks with Delay Constraint

This paper exploits sink mobility to prolong the network lifetime in wireless sensor networks where the information delay caused by moving the sink should be bounded. Due to the combinational complexity of this problem, most previous proposals focus on heuristics and provable optimal algorithms remain unknown. In this paper, we build a unified framework for analyzing this joint sink mobility, routing, delay, and so on. We discuss the induced subproblems and present efficient solutions for them. Then, we generalize these solutions and propose a polynomial-time optimal algorithm for the origin problem. In simulations, we show the benefits of involving a mobile sink and the impact of network parameters (e.g., the number of sensors, the delay bound, etc.) on the network lifetime. Furthermore, we study the effects of different trajectories of the sink and provide important insights for designing mobility schemes in real-world mobile WNNs.

Scheduling two agents with sum-of-processing-times-based deterioration on a single machine

A new scheduling model with two-agent and sum-of-processing-times-based deterioration is proposed in this paper. Two agents compete to perform their respective jobs on a common single machine and each agent has his own criterion to optimize. The sum-of-processing-times-based deterioration of a job is that the actual job processing time is a function of the sum of the normal processing times of the already processed jobs. The objective is to minimize the total completion time of the first agent with the restriction that the makespan of the second agent cannot exceed a given upper bound. The optimal properties of the problems are given and the optimal polynomial time algorithms are proposed to solve the scheduling problems.

On multiprocessor temperature-aware scheduling problems

We study temperature-aware scheduling problems under the model introduced in [Chrobak et al. AAIM 2008], where unit-length jobs of given heat contributions and common release dates are to be scheduled on a set of parallel identical processors. We consider three optimization criteria: makespan, maximum temperature and (weighted) average temperature. On the positive side, we present polynomial time approximation algorithms for the minimization of the makespan and the maximum temperature, as well as, optimal polynomial time algorithms for minimizing the average temperature and the weighted average temperature. On the negative side, we prove that there is no approximation algorithm of absolute ratio $$\frac{4}{3}-\epsilon $$ for the problem of minimizing the makespan for any $$\epsilon > 0$$ , unless $$\mathcal{P}=\mathcal{NP}$$ .

Two-Agent Single-Machine Scheduling with Resource-Dependent Starting Times

We consider several two-agent scheduling problems with resource consumption on a single machine, where each of the agents wants to minimize a measure dependent on its own jobs. The starting time of each job of the first agent is related to the amount of resource consumed. The objective is to minimize the total amount of resource consumption of the first agent with the restriction that the makespan or the total completion time of the second agent cannot exceed a given boundU. The optimal properties and the optimal polynomial time algorithms are proposed to solve the scheduling problems.

Multilayer Bus Optimization for Real-Time Embedded Systems

A major challenge in the design of multicore embedded systems is how to tackle the communications among tasks with performance requirements and precedence constraints. In this paper, we consider the problem of scheduling real-time tasks over multilayer bus systems with the objective of minimizing the communication cost. We show that the problem is NP-hard and determine the best possible approximation ratio of approximation algorithms. First, we propose a polynomial-time optimal algorithm for a restricted case where one multilayer bus, and the unit execution time and communication time are considered. The result is then extended as a pseudopolynomial-time optimal algorithm to consider multiple multilayer buses with arbitrary execution and communication times, as well as different timing constraints and objective functions. We compare the performance of the proposed algorithm with that of some popular heuristics, and provide further insights into the multilayer bus system design.

Disassembly leveling and lot sizing for multiple product types: a basic model and its extension

We consider two interrelated problems that occurred in disassembly systems: disassembly leveling and lot sizing. Disassembly leveling, one of disassembly process planning decisions, is to determine disassembly structures that specify parts and/or subassemblies to be obtained from disassembling used/end-of-life products, and disassembly lot sizing is the problem of determining the amounts of disassembly operations required to satisfy the demands of their parts and/or subassemblies. Unlike the existing studies, this study considers the two problems at the same time for the objective of minimizing the sum of disassembly setup and operation costs. In particular, we consider a generalized version in which disassembly levels may be different even for products of the same type. Two types of the problem are considered in this study. The first one is the basic problem without parts commonality, i.e., products do not share their parts or subassemblies, for which a polynomial time optimal algorithm is suggested after developing a mathematical programming model. The second one is an extended problem with parts commonality. After developing another mathematical programming model for the extension, we prove that it is NP hard. Then, a heuristic algorithm is suggested together with its computational results.