Optimal Parallel Algorithm for the Knapsack Problem Without Memory Conflicts

Abstract The knapsack problem is well known to be NP-complete. Due to its importance in cryptosystem and in number theory, in the past two decades, much effort has been made in order to find techniques that could lead to practical algorithms with reasonable running time. This paper proposes a new parallel algorithm for the knapsack problem where the optimal merging algorithm is adopted. The proposed algorithm is based on an EREW-SIMD machine with shared memory. It is proved that the proposed algorithm is both optimal and the first without memory conflicts algorithm for the knapsack problem. The comparisons of algorithm performance show that it is an improvement over the past researches.     

背包问题的最优并行算法

利用分治策略,提出一种基于SIMD共享存储计算机模型的并行背包问题求解算法.算法允许使用O(2^n/4)^1-ε个并行处理机单元,0≤ε≤1,O(2^n/2)个存储单元,在O(2^n/4(2^n/4)^ε)时间内求解n维背包问题,算法的成本为O(2^n/2).将提出的算法与已有文献结论进行对比表明,该算法改进了已有文献的相应结果,是求解背包问题的成本最优并行算法.同时还指出了相关文献主要结论的错误.     

背包问题无存储冲突的并行三表算法

背包问题属于经典的NP难问题，在信息密码学和数论等研究中具有极重要的应用，将求解背包问题著名的二表算法的设计思想应用于三表搜索中，利用分治策略和无存储冲突的最优归并算法，提出一种基于EREW-SIMD共享存储模型的并行三表算法，算法使用O（2^n/4）个处理机单元和O（2^3n/8）的共享存储空间，在O（2^3n/8）时间内求解n维背包问题．将提出的算法与已有文献结论进行的对比分析表明：文中算法明显改进了现有文献的研究结果，是一种可在小于O（2^n/2）的硬件资源上，以小于O（2n/2）的计算时问求解背包问题的无存储冲突并行算法。     

基于猴群算法求解0-1背包问题

0-1背包问题是一个经典的NP完全问题,该问题在实际生活中具有广泛的应用.针对现有算法在求解0-1背包问题时精度不高的缺点,提出了一种诱导因子猴群算法.所给诱导因子猴群算法的基本思想是,在基本猴群算法的爬过程中引入诱导因子,诱导其向上爬行,从而可以逃逸局部最优解,找到全局最优解.在仿真试验中,与已有方法进行比较,结果说明利用所给诱导因子猴群算法求解0-1背包问题是有效的.     

Dynamic Programming Revisited: Improving Knapsack Algorithms

The contribution of this paper is twofold: At first an improved dynamic programming algorithm for the bounded knapsack problem is given. It decreases the running time for an instance with n items and capacity c from to , which is the same pseudopolynomial complexity as usually given for the 0--1 knapsack problem. In the second part a general approach based on dynamic programming is presented to reduce the storage requirements for combinatorial optimization problems where it is computationally more expensive to compute the explicit solution structure than the optimal solution value. Among other applications of this scheme it is shown that the 0--1 knapsack problem as well as the bounded knapsack problem can be solved in time and space. Received: October 15, 1998; revised March 10, 1999     

Algorithmic aspects of area-efficient hardware/software partitioning

Area efficiency is one of the major considerations in constraint aware hardware/software partitioning process. This paper focuses on the algorithmic aspects for hardware/software partitioning with the objective of minimizing area utilization under the constraints of execution time and power consumption. An efficient heuristic algorithm running in O(n log n) is proposed by extending the method devised for solving the 0-1 knapsack problem. Also, an exact algorithm based on dynamic programming is proposed to produce the optimal solution for small-sized problems. Simulation results show that the proposed heuristic algorithm yields very good approximate solutions while dramatically reducing the execution time.     

Adaptive parallel algorithms for integral knapsack problems

In this paper, the design of a time-efficient and processor-efficient parallel algorithm for the integral knapsack problem is considered. A parallel integral knapsack algorithm is presented, which is adaptive to all parameters, especially to the maximum size of items. The parallel complexity of another important packing problem, the integral exactly-packing problem, is also considered. An optimal O(log n log m) time, parallel integral exactly-packing algorithm is given. Since the partition problem has a constant time, constant processor reduction to the exactly-packing problem, our parallel integral exactly-packing algorithm can be used for job scheduling, task partition, and many other important practical problems. Moreover, the methods and techniques used in this paper can be used for developing processor-efficient and time-efficient parallel algorithms for many other problems. Using the new parallel integral knapsack algorithm, the previously known parallel approximation schemes for the 0–1 knapsack problem and the binpacking problem, by E. W. Mayr and P. S. Gopalkrishnan, are improved upon significantly.     

求解0-1背包问题的二进制狮群算法

针对传统二进制群智能算法求解0-1背包问题易陷入局部最优、收敛速度慢的缺点,提出一种新的解决离散空间问题的二进制狮群算法BLSO。二进制狮群算法对狮王、母狮和幼狮的位置重新定义,引入反置运算、移动算子和学习算子建立全新的位置转移方式和局部搜索规则;加入贪心策略进行解的可行化处理和充分利用,增强局部搜索能力,进一步提高收敛速度。对9个典型的0-1背包算例进行仿真实验,实验结果表明,该算法不仅可以有效求解0-1背包问题,而且还能够以较快的速度搜索到精度较高的次优解甚至全局最优解,具有较好的稳定性;同时,对高维背包问题的求解与参考算法相比,在寻优时间和精度上更具优势。     

An approach to estimating the average-case complexity of postoptimality analysis of discrete optimization problems

It is shown that an algorithm polynomial on average with respect to μ and that determines an optimal solution to a set cover problem that differs from the initial problem in one position of the constraint matrix does not exist if the optimal solution of the original problem is known and DistNP is not a subset of Average-P. A similar result takes place for the knapsack problem.     

An improved monkey algorithm for a 0-1 knapsack problem

The 0-1 knapsack problem is a classic combinational optimization problem. However, many exiting algorithms have low precision and easily fall into local optimal solutions to solve the 0-1 knapsack problem. In order to overcome these problems, this paper proposes a binary version of the monkey algorithm where the greedy algorithm is used to strengthen the local search ability, the somersault process is modified to avoid falling into local optimal solutions, and the cooperation process is adopted to speed up the convergence rate of the algorithm. To validate the efficiency of the proposed algorithm, experiments are carried out with various data instances of 0-1 knapsack problems and the results are compared with those of five metaheuristic algorithms.     

A cost-optimal parallel algorithm for the 0–1 knapsack problem and its performance on multicore CPU and GPU implementations

The 0–1 knapsack problem has been extensively studied in the past years due to its immediate applications in industry and financial management, such as cargo loading, stock cutting, and budget control. Many algorithms have been proposed to solve this problem, most of which are heuristic, as the problem is well-known to be NP-hard. Only a few optimal algorithms have been designed to solve this problem but with high time complexity. This paper proposes the cost-optimal parallel algorithm (COPA) on an EREW PRAM model with shared memory to solve this problem. COPA is scalable and yields optimal solutions consuming less computational time. Furthermore, this paper implements COPA on two scenarios – multicore CPU based architectures using Open MP and GPU based configurations using CUDA. A series of experiments are conducted to examine the performance of COPA under two different test platforms. The experimental results show that COPA could reduce a significant amount of execution time. Our approach achieves the speedups of up to 10.26 on multicore CPU implementations and 17.53 on GPU implementations when the sequential dynamic programming algorithm for KP01 is considered as a baseline. Importantly, GPU implementations outstand themselves in the experimental results.     

求解0-1背包问题的一种新混合算法

用动态规划算法求解0-1背包问题的时空复杂度为O（nC）。这个空间复杂度在求解大规模问题上是不可接受的。从计算0-1背包问题最优值的递归方程出发,给出高效利用内存的动态规划算法。为了克服内存高效的动态规划算法带来的缺点,设计新混合算法求解0-1背包问题。该新混合算法的时间复杂度为O（nC）;它消除了回溯阶段,并且为求得放入背包的物品所使用的空间复杂度仅为O（「n/d?+C）,其中d为计算机字长。实验结果表明,混合算法的工作效率与理论分析相同。     

Approximate algorithms for the knapsack problem on parallel computers

Computing an optimal solution to the knapsack problem is known to be NP-hard. Consequently, fast parallel algorithms for finding such a solution without using an exponential number of processors appear unlikely. An attractive alternative is to compute an approximate solution to this problem rapidly using a polynomial number of processors. In this paper, we present an efficient parallel algorithm for finding approximate solutions to the 0–1 knapsack problem. Our algorithm takes an , 0 < < 1, as a parameter and computes a solution such that the ratio of its deviation from the optimal solution is at most a fraction of the optimal solution. For a problem instance having n items, this computation uses O(n^5/2/^3/2) processors and requires O(log³n + log²nlog(1/)) time. The upper bound on the processor requirement of our algorithm is established by reducing it to a problem on weighted bipartite graphs. This processor complexity is a significant improvement over that of other known parallel algorithms for this problem.     

冲裁件条料最优四块剪切下料方案的生成算法

讨论冲裁件条料剪切下料方案的设计问题。下料方案由一组排样方式组成。首先构造一种生成条料最优四块排样方式的背包算法,然后采用基于列生成的线性规划算法迭代调用上述背包算法,每次都根据生产成本最小的原则改善目标函数并确定各种冲裁件的当前价值,按照当前价值生成一个新的排样方式,最后选择最优的一组排样方式组成下料方案。采用例题将该排样方式生成算法和文献中多段排样方式生成算法进行比较,实验计算结果表明,该算法得到的排样方式排样价值较高。最后通过文献中实例的下料方案求解,可以看出该算法解决实际下料问题是有效的。     

一种求解背包问题的自适应算法

针对二表算法和动态二表算法求解背包问题,提出一个并行自适应算法,能用 个处理机、 的时间、 的空间求解背包问题 ,根据处理机的数目以及存储器的容量来选择参数,充分利用已有的硬件资源,以求得最快的求解速度。实验结果证明了该算法的有效性。     

背包问题的二分网格算法

本文介绍了属于NP难问题的无界背包问题的一种新的精确算法,基于问题的几何结构通过二分搜索方法不断减小解空间,最终直接求出问题的最优效益值和最佳装包方案。当待装入包中的物品数量固定时,算法的时间复杂性为线性时间,初步解决了求解当前呈指数增长背包实例时存在的困难,实验中各种数据实例证明与常用的MTU2和EDU相比,该新算法在理论上是可行的。     

求解0-1背包问题的混沌遗传算法

提出一种改进的混沌遗传算法来求解0-1背包问题。通过利用幂函数载波技术增强混沌搜索的遍历性,把混沌搜索得到的最优解直接作为新群体嵌入遗传算法来改善遗传算法的早熟问题,从而使算法有能力避免陷入局部极值而快速收敛于全局最优解。仿真实验结果表明了该算法求解0-1背包问题的有效性和适用性。     

基于采样和MIMD结构的背包问题并行算法

背包问题属于著名的NP完全问题,在信息密码学和数论研究中有着极其重要的应用。在深入分析背包问题现有并行算法的基础上,本文提出了一种基于采样和MIMD结构的背包问题并行求解算法,并给出了算法性能的理论分析和在IBMP690超级计算机上的实验结果。实验结果表明,当背包实例的维数n≥40时,本算法的并行效率可达60％以上。因此,本并行算法具有较好的可扩展性,能应用于各种MIMD结构的并行机上有效地求解背包问题。     

An effective GRASP and tabu search for the 0–1 quadratic knapsack problem

As a generalization of the classical 0-1 knapsack problem, the 0-1 Quadratic Knapsack Problem (QKP) that maximizes a quadratic objective function subject to a linear capacity constraint is NP-hard in strong sense. In this paper, we propose a memory based Greedy Randomized Adaptive Search Procedures (GRASP) and a tabu search algorithm to find near optimal solution for the QKP. Computational experiments on benchmarks and on randomly generated instances demonstrate the effectiveness and the efficiency of the proposed algorithms, which outperforms the current state-of-the-art heuristic Mini-Swarm in computational time and in the probability to achieve optimal solutions. Numerical results on large-sized instances with up to 2000 binary variables have also been reported.     

An exact constructive algorithm for the knapsack sharing problem

In this paper, we study the knapsack sharing problem (KSP), a variant of the well-known NP-hard single knapsack problem. We propose an exact constructive tree search that combines two complementary procedures: a reduction interval search and a branch and bound. The reduction search has three phases. The first phase applies a polynomial reduction strategy that decomposes the problem into a series of knapsack problems. The second phase is a size reduction strategy that makes the resolution more efficient. The third phase is an interval reduction search that identifies a set of optimal capacities characterizing the knapsack problems. Experimental results provide computational evidence of the better performance of the proposed exact algorithm in comparison to KSPs best exact algorithm, to Cplex and to KSPs latest heuristic approach. Furthermore, they emphasize the importance of the reduction strategies.