背包问题的算法设计与分析研究

背包问题是算法设计分析中的经典问题,本文采用贪婪法、动态规划法及递归法三种方法分别对背包问题、0-1背包问题及简单0-1背包问题进行算法设计和时间复杂度分析,给出具体算法设计和实现过程,并以具体实例详细描述不同方法求解问题解时算法基本思想,总结三种方法实现的优缺点并得出结论。     

背包问题的算法设计与分析研究

背包问题是算法设计分析中的经典问题,本文采用贪婪法、动态规划法及递归法三种方法分别对背包问题、0-1背包问题及简单0-1背包问题进行算法设计和时间复杂度分析,给出具体算法设计和实现过程,并以具体实例详细描述不同方法求解问题解时算法基本思想,总结三种方法实现的优缺点并得出结论。     

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.     

考虑时间因素的0-1背包调度问题

文中提出考虑时间因素的0-1背包调度问题这一具有NP难度的组合优化问题。给定n个物体(每个物体i的重量为w_i,连续加工时间为t_i),以及一个容量为S的背包,要求给出一个调度方案(物品的放入顺序和放入时间),使得任意时刻放入背包的物品总重量不超过背包容量,每个物体需放入背包连续加工时长t_i后才能取出,该问题是求使所有物体均加工完毕的时间尽可能短的调度方案。提出了3种求解算法:迭代动态规划算法、基于分枝限界的完备算法和遗传进化算法。迭代动态规划算法使用动态规划策略放置尽可能多的未加工物体到背包中,然后每次迭代取出加工完成的物品后再使用动态规划放入尽可能多的剩余未加工物品,直至所有物品被加工完成。基于分枝限界的完备算法通过定义上下界及剪枝操作,有效地降低了算法的计算复杂度。遗传进化算法将一个物品装填序列定义为个体,并定义了相应的适应度、选择、交叉与变异操作。在所设计的3组共计36个算例上的实验结果表明,迭代动态规划算法可以很快求出高质量的解,基于分枝限界的完备算法对小规模算例有很好的效果,遗传算法在处理几百个物体的算例时能在1500s内得到比动态规划算法更好的结果。     

0-1背包问题的两种扩展形式及其解法

0-1背包问题是经典的NP-HARD组合优化问题之一,由于其难解性,该问题在信息密码学和数论研究中具有极其重要的应用。首先对01背包问题及其解法进行了分析,然后提出01背包问题的两种扩展形式,并给出了基于动态规划和贪心算法的两种有效算法来解决这两类问题。实验结果验证了所提出方法的有效性。     

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.     

基于0/1背包问题的算法探究

0／1背包问题是计算机科学中的一个经典问题。动态规划法，递归法，回溯法是求解该问题的三种典型方法，使用这三种方法求解0／1背包问题，并对各算法进行了理论分析。用不同规模的0／1背包问题对三种算法进行测试，比较它们的运行时间，发现测试结果与其理论分析结果相符．最后指出就求解不同规模的0／1背包问题而言各算法的优劣。     

A dynamic programming method with lists for the knapsack sharing problem

In this paper, we propose a method to solve exactly the knapsack sharing problem (KSP) by using dynamic programming. The original problem (KSP) is decomposed into a set of knapsack problems. Our method is tested on correlated and uncorrelated instances from the literature. Computational results show that our method is able to find an optimal solution of large instances within reasonable computing time and low memory occupancy.     

An efficient heuristic algorithm for selecting projects

Consider the problem of selecting a set of projects from a large number of available projects such that at least some specified levels of benefits of various types are realized at a minimum cost. This problem can be formulated in terms of the well-known 0–1 multi-dimensional knapsack problem, a special case of the general integer programming problems. In view of the NP-completeness of these problems, this paper proposes a polynomially bounded and efficient heuristic algorithm for its solution. The proposed algorithm proceeds as follows: an initial selection is found by prioritizing the projects according to a computed discard index. This initial selection set is then altered to reduce total costs by using project exchange operations. Computational results indicate that the proposed algorithm is quite effective in finding optimal or near optimal solutions.     

An Optimal Algorithm for Bound and Equality Constrained Quadratic Programming Problems with Bounded Spectrum

An implementation of the recently proposed semi-monotonic augmented Lagrangian algorithm for solving the large convex bound and equality constrained quadratic programming problems is considered. It is proved that if the algorithm is applied to the class of problems with uniformly bounded spectrum of the Hessian matrix, then the algorithm finds an approximate solution at O(1) matrix-vector multiplications. The optimality results are presented that do not depend on conditioning of the matrix which defines the equality constraints. Theory covers also the problems with dependent constraints. Theoretical results are illustrated by numerical experiments.     

Scheduling a Single Server in a Two-machine Flow Shop

We study the problem of scheduling a single server that processes n jobs in a two-machine flow shop environment. A machine dependent setup time is needed whenever the server switches from one machine to the other. The problem with a given job sequence is shown to be reducible to a single machine batching problem. This result enables several cases of the server scheduling problem to be solved in O(n log n) by known algorithms, namely, finding a schedule feasible with respect to a given set of deadlines, minimizing the maximum lateness and, if the job processing times are agreeable, minimizing the total completion time. Minimizing the total weighted completion time is shown to be NP-hard in the strong sense. Two pseudopolynomial dynamic programming algorithms are presented for minimizing the weighted number of late jobs. Minimizing the number of late jobs is proved to be NP-hard even if setup times are equal and there are two distinct due dates. This problem is solved in O(n ³) time when all job processing times on the first machine are equal, and it is solved in O(n ⁴) time when all processing times on the second machine are equal. Received November 20, 2001; revised October 18, 2002 Published online: January 16, 2003     

Polynomial algorithms for protein similarity search for restricted mRNA structures

In this paper we consider the problem of computing an mRNA sequence of maximal similarity for a given mRNA of secondary structure constraints, introduced by Backofen et al. in [R. Backofen, N.S. Narayanaswamy, F. Swidan, On the complexity of protein similarity search under mRNA structure constraints, in: Proceedings of the Annual Symposium of Theoretical Aspects of Computer Science, in: LNCS, vol. 2285, Springer, 2002, pp. 274-286] denoted as the MRSO problem. The problem is known to be NP-complete for planar associated implied structure graphs of vertex degree at most 3. In [G. Blin, G. Fertin, D. Hermelin, S. Vialette, Fixed-parameter algorithms for protein similarity search under mRNA structure constraints, in: Proceedings of Graph-Theoretical Concepts in Computer Science, in: LNCS, vol. 3787, Springer, 2005, pp. 271-282] a first polynomial dynamic programming algorithms for MRSO on implied structure graphs with maximum vertex degree 3 of bounded cut-width is shown.We give a simple but much more general polynomial dynamic programming solution for the MRSO problem for associated implied structure graphs of bounded clique-width. Our result implies that MRSO is polynomial for graphs of bounded tree-width, co-graphs, P₄-sparse graphs, and distance hereditary graphs. Further we conclude that the problem of comparing two solutions for MRSO is hard for the class , which is defined as the set of problems which can be solved in polynomial time with a number of parallel queries to an oracle in NP.     

Algorithms for the boundary selection problem

User-oriented clustering schemes enable the classification of documents based upon the user's perception of the similarity between documents, rather than on some similarity function presumed by the designer to represent the user's criteria. In an earlier paper it was shown that such a classification scheme can be developed in two stages. The first stage involves the accumulation of relevance judgements provided by users,vis-à-vis past query instances, into a suitable structure. The second stage consists of cluster identification. When the structure chosen, in the first stage, for the accumulation of corelevance characteristics of documents is a straight line, the second stage can be formulated as a function optimization problem termed the Boundary Selection Problem (BSP). A branch-and-bound algorithm with a good bounding function is developed for the BSP. Although significant pruning is achieved due to the bounding function, the complexity is still high for a problem of a large size. For such a problem a heuristic that divides it into a number of subproblems, each being solved by a branch-and-bound approach, is developed. Then the overall problem is mapped to an integer knapsack problem and solved by the use of dynamic programming. The tradeoff between accuracy and complexity can be controlled, giving the user a preference of one over the other. Assuming that the heuristic which divides the overall problem introduces no errors and is given sufficient time, the branch and bound with dynamic programming (BBDP) approach will converge to the optimal solution. Two other heuristic approaches, one with the application of a polynomial dynamic programming algorithm and the other which works in a greedy way, are also proposed for the BSP and an experimental comparison of all these approaches is provided. Experimental results indicate that all proposed algorithms show better performance compared with the existing algorithm.     

An incomplete m-exchange algorithm for solving the large-scale multi-scenario knapsack problem

This paper introduces a fast heuristic based algorithm for the max-min multi-scenario knapsack problem. The problem is a variation of the standard 0-1 knapsack problem, in which the profits of the items vary under different scenarios, though the capacity of the knapsack is fixed. The objective of the problem is to find the optimal packing of a set of items so that the minimum total profits of the items in the knapsack over all different scenarios is maximized. For some large-scaled instances, traditional branch-and-bound techniques cannot find an optimal solution within reasonable time, thus we propose a collection of incomplete m-exchange algorithms which are able to produce high quality solutions in just a few minutes of cpu time. Various computational results are also given.     

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.     

基于动态状态树的回溯算法

介绍了背包问题及0-1背包问题,阐述了回溯算法(算法设计的基本方法之一)和状态空间的概念,提出一个基于动态状态空间树的回溯算法.以0-1背包问题为例,说明动态树方法对求解线性规划问题等是非常有用的,且该算法所用时间少于静态状态空间树方法,有助于扩大回溯算法的应用.     

Maintenance routing for train units: The interchange model

Train units need regular preventive maintenance. Given the train units that require maintenance in the forthcoming 1–3 days, the rolling stock schedule must be adjusted so that these urgent units reach the maintenance facility in time. In this paper, we present an integer programming model for solving this problem, give complexity results, suggest solution methods, and report our computational results based on practical instances of NS Reizigers, the main Dutch operator of passenger trains.     

基于动态规划法求解动态o-1背包问题

随机时变背包问题(RTVKP)是一种动态组合优化问题,也是一种典型的NP-hard问题。由于RTVKP问题中物品的价值、重量和背包载重均是动态变化的,导致问题的求解非常困难。在动态规划法基础上,提出了一种求解背包载重随机变化的RTVKP问题的确定性算法,分析了其复杂度和成功求解需要满足的条件。对两个大规模实例的计算表明,该算法是求解RTVKP问题的一种高效算法。     

Dynamo - a portable tool for dynamic load balancing on distributed memory multicomputers

Dynamic load balancing is an important technique when developing applications with unpredictable load distribution on distributed memory multicomputers. A tool, Dynamo, that can be used to utilize dynamic load balancing is presented. This tool separates the application from the load balancer and thus makes it possible to easily exchange the load balancer of a given application and experiment with different load balancing strategies. A prototype of Dynamo has been implemented in the C language on an Intel iPSC/2 Hypercube. Dynamo is demonstrated by two example programs. The first program solves the N queen problem using a backtracking algorithm and the second solves a 0-1 knapsack problem using a depth-first branch and bound algorithm.