A pegging approach to the precedence-constrained knapsack problem

The knapsack problem (KP) is generalized to the case where items are partially ordered through a set of precedence relations. As in ordinary KPs, each item is associated with profit and weight, the knapsack has a fixed capacity, and the problem is to determine the set of items to be packed in the knapsack. However, each item can be accepted only when all the preceding items have been included in the knapsack. The knapsack problem with these additional constraints is referred to as the precedence-constrained knapsack problem (PCKP). To solve PCKP exactly, we present a pegging approach, where the size of the original problem is reduced by applying the Lagrangian relaxation followed by a pegging test. Through this approach, we are able to solve PCKPs with thousands of items within a few minutes on an ordinary workstation.     

Heuristic and Exact Algorithms for the Precedence-Constrained Knapsack Problem

The knapsack problem (KP) is generalized taking into account a precedence relation between items. Such a relation can be represented by means of a directed acyclic graph, where nodes correspond to items in a one-to-one way. As in ordinary KPs, each item is associated with profit and weight, the knapsack has a fixed capacity, and the problem is to determine the set of items to be included in the knapsack. However, each item can be adopted only when all of its predecessors have been included in the knapsack. The knapsack problem with such an additional set of constraints is referred to as the precedence-constrained knapsack problem (PCKP). We present some dynamic programming algorithms that can solve small PCKPs to optimality, as well as a preprocessing method to reduce the size of the problem. Combining these, we are able to solve PCKPs with up to 2000 items in less than a few minutes of CPU time.     

Extending Dantzig's bound to the bounded multiple-class binary Knapsack problem

 The bounded multiple-class binary knapsack problem is a variant of the knapsack problem where the items are partitioned into classes and the item weights in each class are a multiple of a class weight. Thus, each item has an associated multiplicity. The constraints consists of an upper bound on the total item weight that can be selected and upper bounds on the total multiplicity of items that can be selected in each class. The objective is to maximize the sum of the profits associated with the selected items. This problem arises as a sub-problem in a column generation approach to the cutting stock problem. A special case of this model, where item profits are restricted to be multiples of a class profit, corresponds to the problem obtained by transforming an integer knapsack problem into a 0-1 form. However, the transformation proposed here does not involve a duplication of solutions as the standard transformation typically does. The paper shows that the LP-relaxation of this model can be solved by a greedy algorithm in linear time, a result that extends those of Dantzig (1957) and Balas and Zemel (1980) for the 0-1 knapsack problem. Hence, one can derive exact algorithms for the multi-class binary knapsack problem by adapting existing algorithms for the 0-1 knapsack problem. Computational results are reported that compare solving a bounded integer knapsack problem by transforming it into a standard binary knapsack problem versus using the multiple-class model as a 0-1 form. Received: May 1998 / Accepted: February 2002-09-04 Published online: December 9, 2002 Key Words. Knapsack problem – integer programming – linear programming relaxation     

Knapsack with variable weights satisfying linear constraints

We introduce a variant of the knapsack problem, in which the weights of items are also variables but must satisfy a system of linear constraints, and the capacity of knapsack is given and known. We discuss two models: (1) the value of each item is given; (2) the value-to-weight ratio of each item is given. The goal is to determine the weight of each item, and to find a subset of items such that the total weight is no more than the capacity and the total value is maximized. We provide several practical application scenarios that motivate our study, and then investigate the computational complexity and corresponding algorithms. In particular, we show that if the number of constraints is a fixed constant, then both problems can be solved in polynomial time. If the number of constraints is an input, then we show that the first problem is NP-Hard and cannot be approximated within any constant factor unless \(\mathrm{P}= \mathrm{NP}\), while the second problem is NP-Hard but admits a polynomial time approximation scheme. We further propose approximation algorithms for both problems, and extend the results to multiple knapsack cases with a fixed number of knapsacks and identical capacities.     

The static stochastic knapsack problem with normally distributed item sizes

This paper develops exact and heuristic algorithms for a stochastic knapsack problem where items with random sizes may be assigned to a knapsack. An item’s value is given by the realization of the product of a random unit revenue and the random item size. When the realization of the sum of selected item sizes exceeds the knapsack capacity, a penalty cost is incurred for each unit of overflow, while our model allows for a salvage value for each unit of capacity that remains unused. We seek to maximize the expected net profit resulting from the assignment of items to the knapsack. Although the capacity is fixed in our core model, we show that problems with random capacity, as well as problems in which capacity is a decision variable subject to unit costs, fall within this class of problems as well. We focus on the case where item sizes are independent and normally distributed random variables, and provide an exact solution method for a continuous relaxation of the problem. We show that an optimal solution to this relaxation exists containing no more than two fractionally selected items, and develop a customized branch-and-bound algorithm for obtaining an optimal binary solution. In addition, we present an efficient heuristic solution method based on our algorithm for solving the relaxation and empirically show that it provides high-quality solutions.     

Sensitivity analysis to perturbations of the weight of a subset of items: The knapsack case study

In this paper, we study the sensitivity of the optimum to perturbations of the weight of a subset of items of both the knapsack problem (denoted KP) and knapsack sharing problem (denoted KSP). The sensitivity interval of the weight associated to an item is characterized by two limits, called lower and upper values, which guarantee the optimality of the solution at hand whenever the new weight’s value belongs to such an interval. For each perturbed weight, we try to establish approximate values of the sensitivity interval whenever the original problem is solved. We do it by applying a dynamic programming method where all established results require a negligible runtime. First, two cases are studied when considering an optimal solution of KP: (i) the case in which all perturbations are (non)negatives and (ii) the general case in which the set of the perturbed items is divided into two disjoint subsets (the first subset contains the nonnegative perturbations and the second one represents the subset of negative perturbations). Second, we show how we can adapt the results of KP to the KSP. All established results require a negligible runtime which grows the interest of such a study. Finally, for each of these problems, we will see the impact of the established results on an example while considering the various cases.     

Time-Constrained Restless Bandits and the Knapsack Problem for Perishable Items (Extended Abstract)

Motivated by a food promotion problem, we introduce the Knapsack Problem for Perishable Items (KPPI) to address a dynamic problem of optimally filling a knapsack with items that disappear randomly. The KPPI naturally bridges the gap and elucidates the relation between the PSPACE-hard restless bandit problem and the NP-hard knapsack problem. Our main result is a problem decomposition method resulting in an approximate transformation of the KPPI into an associated 0-1 knapsack problem. The approach is based on calculating explicitly the marginal productivity indices in a generic finite-horizon restless bandit subproblem.     

多背包问题的半定松驰算法

The multiple knapsack problem denoted by MKP (B,S,rn,n) can be defined as follows. A set B of n items and a set S of rn knapsacks are given such that each item j has a profit pi and weight wj,and each knapsack i has a capacity Ci. The goal is to find a subset of items of maximum profit such that they have a feasible packing in the knapsacks. MKP (B,S,m,n) is strongly NP-Complete and no polynomial time approximation algorithm can have an approximation ratio better than 0.5. In the last ten years,semi-definite programming has been empolyed to solve some combinatorial problems successfully. This paper firstly presents a semi-definite relaxation algorithm (MKPS) for MKP (B,S,rn,n). It is proved that MKPS have a approximation ratio better than 0. 5 for a subclass of MKP (B,S,m,n) with n≤100, m≤5 and max^nj=1{wj}/min^mi=1={Ci}≤2/3.     

AnO(n 3 logn) strong polynomial algorithm for an isotonic regression knapsack problem

Journal of Optimization Theory and Applications - We introduce the isotonic regression knapsack problem $$\begin{gathered} \min (1/2)\sum\limits_{i = 1}^n {\{ d_i x_i^2 - 2\alpha _i x_i \} } ,...     

0-1背包问题的蜂群优化算法

在项目决策与规划、资源分配、货物装载、预算控制等工作中,提出了0-1背包问题.0-1背包问题是组合优化中的典型NP难题,根据群集智能原理,给出一种基于蜂群寻优思想的新算法—蜂群算法,并针对0-1背包问题进行求解.经实验仿真并与蚁群算法计算结果作对比,验证了算法在0-1背包问题求解上的有效性和更快的收敛速度.     

Knapsack problems with setups

Knapsack problems with setups find their application in many concrete industrial and financial problems. Moreover, they also arise as subproblems in a Dantzig–Wolfe decomposition approach to more complex combinatorial optimization problems, where they need to be solved repeatedly and therefore efficiently. Here, we consider the multiple-class integer knapsack problem with setups. Items are partitioned into classes whose use implies a setup cost and associated capacity consumption. Item weights are assumed to be a multiple of their class weight. The total weight of selected items and setups is bounded. The objective is to maximize the difference between the profits of selected items and the fixed costs incurred for setting-up classes. A special case is the bounded integer knapsack problem with setups where each class holds a single item and its continuous version where a fraction of an item can be selected while incurring a full setup. The paper shows the extent to which classical results for the knapsack problem can be generalized to these variants with setups. In particular, an extension of the branch-and-bound algorithm of Horowitz and Sahni is developed for problems with positive setup costs. Our direct approach is compared experimentally with the approach proposed in the literature consisting in converting the problem into a multiple choice knapsack with pseudo-polynomial size.     

An improved branch and bound algorithm for a strongly correlated unbounded knapsack problem

An unbounded knapsack problem (KP) was investigated that describes the loading of items into a knapsack with a finite capacity, W, so as to maximize the total value of the loaded items. There were n types of an infinite number of items, each type with a distinct weight and value. Exact branch and bound algorithms have been successfully applied previously to the unbounded KP, even when n and W were very large; however, the algorithms are not adequate when the weight and the value of the items are strongly correlated. An improved branching strategy is proposed that is less sensitive to such a correlation; it can therefore be used for both strongly correlated and uncorrelated problems.     

Unbounded knapsack problem with controllable rates: the case of a random demand for items

This paper focuses on a dynamic, continuous-time control generalization of the unbounded knapsack problem. This generalization implies that putting items in a knapsack takes time and has a due date. Specifically, the problem is characterized by a limited production horizon and a number of item types. Given an unbounded number of copies of each type of item, the items can be put into a knapsack at a controllable production rate subject to the available capacity. The demand for items is not known until the end of the production horizon. The objective is to collect items of each type in order to minimize shortage and surplus costs with respect to the demand. We prove that this continuous-time problem can be reduced to a number of discrete-time problems. As a result, solvable cases are found and a polynomial-time algorithm is suggested to approximate the optimal solution with any desired precision.     

A reduction dynamic programming algorithm for the bi-objective integer knapsack problem

This paper presents a backward state reduction dynamic programming algorithm for generating the exact Pareto frontier for the bi-objective integer knapsack problem. The algorithm is developed addressing a reduced problem built after applying variable fixing techniques based on the core concept. First, an approximate core is obtained by eliminating dominated items. Second, the items included in the approximate core are subject to the reduction of the upper bounds by applying a set of weighted-sum functions associated with the efficient extreme solutions of the linear relaxation of the multi-objective integer knapsack problem. Third, the items are classified according to the values of their upper bounds; items with zero upper bounds can be eliminated. Finally, the remaining items are used to form a mixed network with different upper bounds. The numerical results obtained from different types of bi-objective instances show the effectiveness of the mixed network and associated dynamic programming algorithm.     

The knapsack problem with neighbour constraints

We study a constrained version of the knapsack problem in which dependencies between items are given by the adjacencies of a graph. In the 1-neighbour knapsack problem, an item can be selected only if at least one of its neighbours is also selected. In the all-neighbours knapsack problem, an item can be selected only if all its neighbours are also selected.We give approximation algorithms and hardness results when the vertices have both uniform and arbitrary weight and profit functions, and when the dependency graph is directed and undirected.     

Global Regularity of 3D Nonhomogeneous Incompressible Micropolar Fluids

This paper is concerned with the global well-posedness of strong and classical solutions for the 3D nonhomogeneous incompressible micropolar equations with vacuum. We prove that the problem (1.1)–(1.5) has a unique global strong/classical solution \((\rho,u,w)\), provided \(\mu_{1}\) is sufficiently large, or \(\|\rho_{0}\|_{L^{\infty}}\) or \(\|\rho_{0}^{1/2}u_{0}\| ^{2}_{L^{2}}+\|\rho_{0}^{1/2}w_{0}\|^{2}_{L^{2}}\) is small enough.

     

Some nodes matrices appearing in the numerical analysis for singular integral equations

Let {p _m (w)} be the sequence of Jacobi polynomials corresponding to the weightw(x)=(1–x)(1+x), 0, <1. denote=">x _k (w)=cos _m,k (w),k=1,...,m, the zeros ofp _m (w). If +=0, then the estimates

一类奇异三阶三点边值问题的正解

In this paper,we study the existence of positive solutions for the nonlinear singular third-order three-point boundary value problemu （t） = λa（t）f（t,u（t））,u（0） = u （1） = u （η） = 0,where λ is a positive parameter and 0 ≤ η 1 2 .By using the classical Krasnosel’skii’s fixed point theorem in cone,we obtain various new results on the existence of positive solution,and the solution is strictly increasing.Finally we give an example.     

求解0-1背包问题的细菌觅食算法

0-1背包问题是组合优化中的一个典型NP难题,介于其具有广泛的实际应用,有效的解决该问题具有非常重要的意义.给出了一种新的群智能算法—细菌觅食算法,对0-1背包问题进行求解.经模拟仿真验证了该算法的有效性,并将其结果与其他方法进行对比分析.     

Global Smooth Solutions to a System of Dissipative Nonlinear Evolution Equations

The existence and uniqueness are proved for global classical solutions of the following initial-boundary problem for the system of parabolic equations which is proposed by Hsieh as a substitute for the Rayleigh-Benard equation and can lead to Lorenz equations: {ψ_t = -(σ - α)ψ - σθ_x, + αψ_{xx} θ_t = -(1- β)θ + vψ_x + (ψθ)_x + βθ_{xx} ψ(0,t) = ψ(1,t) = 0, θ_x(0,t) = θ_x(1,t) = 0 ψ(x,0) = ψ_0(x), θ(x,0) = θ_0(x)