A strongly polynomial FPTAS for the symmetric quadratic knapsack problem

The symmetric quadratic knapsack problem (SQKP), which has several applications in machine scheduling, is NP-hard. An approximation scheme for this problem is known to achieve an approximation ratio of (1 + ?) for any ? > 0. To ensure a polynomial time complexity, this approximation scheme needs an input of a lower bound and an upper bound on the optimal objective value, and requires the ratio of the bounds to be bounded by a polynomial in the size of the problem instance. However, such bounds are not mentioned in any previous literature. In this paper, we present the first such bounds and develop a polynomial time algorithm to compute them. The bounds are applied, so that we have obtained for problem (SQKP) a fully polynomial time approximation scheme (FPTAS) that is also strongly polynomial time, in the sense that the running time is bounded by a polynomial only in the number of integers in the problem instance.     

On the continuous quadratic knapsack problem

We introduce a new algorithm for the continuous bounded quadratic knapsack problem. This algorithm is motivated by the geometry of the problem, is based on the iterative solution of a series of simple projection problems, and is easy to understand and implement. In practice, the method compares favorably to other well-known algorithms (some of which have superior worst-case complexity) on problem sizes up ton = 4000.     

The symmetric quadratic traveling salesman problem

In the quadratic traveling salesman problem a cost is associated with any three nodes traversed in succession. This structure arises, e.g., if the succession of two edges represents energetic conformations, a change of direction or a possible change of transportation means. In the symmetric case, costs do not depend on the direction of traversal. We study the polyhedral structure of a linearized integer programming formulation of the symmetric quadratic traveling salesman problem. Our constructive approach for establishing the dimension of the underlying polyhedron is rather involved but offers a generic path towards proving facetness of several classes of valid inequalities. We establish relations to facets of the Boolean quadric polytope, exhibit new classes of polynomial time separable facet defining inequalities that exclude conflicting configurations of edges, and provide a generic strengthening approach for lifting valid inequalities of the usual traveling salesman problem to stronger valid inequalities for the symmetric quadratic traveling salesman problem. Applying this strengthening to subtour elimination constraints gives rise to facet defining inequalities, but finding a maximally violated inequality among these is NP-complete. For the simplest comb inequality with three teeth the strengthening is no longer sufficient to obtain a facet. Preliminary computational results indicate that the new cutting planes may help to considerably improve the quality of the root relaxation in some important applications.     

Partially ordered knapsack and applications to scheduling

In the partially ordered knapsack problem (POK) we are given a set N of items and a partial order ?_P on N. Each item has a size and an associated weight. The objective is to pack a set N^′⊆N of maximum weight in a knapsack of bounded size. N^′ should be precedence-closed, i.e., be a valid prefix of ?_P. POK is a natural generalization, for which very little is known, of the classical Knapsack problem. In this paper we present both positive and negative results. We give an FPTAS for the important case of a two-dimensional partial order, a class of partial orders which is a substantial generalization of the series-parallel class, and we identify the first non-trivial special case for which a polynomial-time algorithm exists. Our results have implications for approximation algorithms for scheduling precedence-constrained jobs on a single machine to minimize the sum of weighted completion times, a problem closely related to POK.     

可分离的二次背包问题的一种直接算法

研究了可分离二次背包问题的一种直接算法.此类背包问题的目标函数是二次的,且含有严格的一次项,其不等式约束是线性的.给出所求模型的一般形式,经过预处理该模型,最终归为求解两类问题（P1）和（P2）.重点是求解（P2）问题的最优解,通过分析（P2）问题的结构特点,假设固定一次项后问题的最优解和相应不等式的拉格朗日乘子已求出,通过比较拉格朗日乘子和（P2）问题的一次项系数来调节λ的大小,从而求出（P2）问题的最优解.对于（P1）问题,改进了Bretthauer和Shetty给出的算法（Bretthauer K M,Shetty B.A pegging algorithm for the nonlinear resource allocation problem.Computers and Operations Research,2002,29（5）：505-527）.此算法的计算复杂性为O（n）.数值算例表明,将这种固定变量算法和文中的定理5结合起来,能够快速有效地求解此类更一般的二次背包问题.     

Strategic oscillation for the quadratic multiple knapsack problem

The quadratic multiple knapsack problem (QMKP) consists in assigning a set of objects, which interact through paired profit values, exclusively to different capacity-constrained knapsacks with the aim of maximising total profit. Its many applications include the assignment of workmen to different tasks when their ability to cooperate may affect the results. Strategic oscillation (SO) is a search strategy that operates in relation to a critical boundary associated with important solution features (such as feasibility). Originally proposed in the context of tabu search, it has become widely applied as an efficient memory-based methodology. We apply strategic oscillation to the quadratic multiple knapsack problem, disclosing that SO effectively exploits domain-specific knowledge, and obtains solutions of particularly high quality compared to those obtained by current state-of-the-art algorithms.     

Breakpoint searching algorithms for the continuous quadratic knapsack problem

We give several linear time algorithms for the continuous quadratic knapsack problem. In addition, we report cycling and wrong-convergence examples in a number of existing algorithms, and give encouraging computational results for large-scale problems.      

Simple solution methods for separable mixed linear and quadratic knapsack problem

Quadratic knapsack problem has a central role in integer and nonlinear optimization, which has been intensively studied due to its immediate applications in many fields and theoretical reasons. Although quadratic knapsack problem can be solved using traditional nonlinear optimization methods, specialized algorithms are much faster and more reliable than the nonlinear programming solvers. In this paper, we study a mixed linear and quadratic knapsack with a convex separable objective function subject to a single linear constraint and box constraints. We investigate the structural properties of the studied problem, and develop a simple method for solving the continuous version of the problem based on bi-section search, and then we present heuristics for solving the integer version of the problem. Numerical experiments are conducted to show the effectiveness of the proposed solution methods by comparing our methods with some state of the art linear and quadratic convex solvers.     

An approximation scheme for the two-stage,two-dimensional knapsack problem

We present an approximation scheme for the two-dimensional version of the knapsack problem which requires packing a maximum-area set of rectangles in a unit square bin, with the further restrictions that packing must be orthogonal without rotations and done in two stages. Achieving a solution which is close to the optimum modulo a small additive constant can be done by taking wide inspiration from an existing asymptotic approximation scheme for two-stage two-dimensional bin packing. On the other hand, getting rid of the additive constant to achieve a canonical approximation scheme appears to be widely nontrivial.     

The quadratic knapsack problem—a survey

The binary quadratic knapsack problem maximizes a quadratic objective function subject to a linear capacity constraint. Due to its simple structure and challenging difficulty it has been studied intensively during the last two decades. The present paper gives a survey of upper bounds presented in the literature, and show the relative tightness of several of the bounds. Techniques for deriving the bounds include relaxation from upper planes, linearization, reformulation, Lagrangian relaxation, Lagrangian decomposition, and semidefinite programming. A short overview of heuristics, reduction techniques, branch-and-bound algorithms and approximation results is given, followed by an overview of valid inequalities for the quadratic knapsack polytope. The paper is concluded by an experimental study where the upper bounds presented are compared with respect to strength and computational effort.     

Tabu-enhanced iterated greedy algorithm: A case study in the quadratic multiple knapsack problem

Iterated greedy search is a simple and effective metaheuristic for combinatorial problems. Its flexibility enables the incorporation of components from other metaheuristics with the aim of obtaining effective and powerful hybrid approaches. We propose a tabu-enhanced destruction mechanism for iterated greedy search that records the last removed objects and avoids removing them again in subsequent iterations. The aim is to provide a more diversified and successful search process with regards to the standard destruction mechanism, which selects the solution components for removal completely at random.     

A surrogate relaxation based algorithm for a general quadratic multi-dimensional knapsack problem

In this paper, we develop a framework to solve a General Quadratic Multi-dimensional Knapsack Problem using surrogate relaxation. This paper exploits the fact that a continuous single constraint quadratic knapsack problem can be solved by inspection by the procedure given by Mathur et al. [6]. A preliminary computational study indicates that the proposed algorithm is much more efficient than some of the alternate procedures.     

A new upper bound for the 0-1 quadratic knapsack problem

The 0-1 quadratic knapsack problem (QKP) consists in maximizing a positive quadratic pseudo-Boolean function subject to a linear capacity constraint. We present in this paper a new method, based on Lagrangian decomposition, for computing an upper bound of QKP. We report computational experiments which demonstrate the sharpness of the bound (relative error very often less than 1%) for large size instances (up to 500 variables).     

A continuous knapsack problem with separable convex utilities: Approximation algorithms and applications

We study a continuous knapsack problem with separable convex utilities. We show that the problem is NP-hard, and provide two simple algorithms that have worst-case performance guarantees. We consider as an application a novel subsidy allocation problem in the presence of market competition, subject to a budget constraint and upper bounds on the amount allocated to each firm, where the objective is to minimize the market price of a good.     

The multiple-choice multi-period knapsack problem

This paper introduces the multiple-choice multi-period knapsack problem in the interface of multiple-choice programming and knapsack problems. We first examine the properties of the multiple-choice multi-period knapsack problem. A heuristic approach incorporating both primal and dual gradient methods is then developed to obtain a strong lower bound. Two branch-and-bound procedures for special-ordered-sets type 1 variables that incorporate, respectively, a special algorithm and the multiple-choice programming technique are developed to locate the optimal solution using the above lower bound as the initial solution. A computer program written in IBM's APL2 is developed to assess the quality of this lower bound and to evaluate the performance of these two branch-and-bound procedures.     

A fully polynomial approximation algorithm for the 0–1 knapsack problem

A fully polynomial ?-approximation algorithm is developed for the 0–1 knapsack problem. The algorithm uses results of Lawler and Ibarra and Kim. A pseudo-polynomial dynamic programming algorithm is first suggested which solves the problem in O(nb log n) time and O(b) space.     

An inverse eigenvalue problem and a matrix approximation problem for symmetric skew-hamiltonian matrices

Based on the theory of inverse eigenvalue problem, a correction of an approximate model is discussed, which can be formulated as NX=XΛ, where X and Λ are given identified modal and eigenvalues matrices, respectively. The solvability conditions for a symmetric skew-Hamiltonian matrix N are established and an explicit expression of the solutions is derived. For any estimated matrix C of the analytical model, the best approximation matrix to minimize the Frobenius norm of C − N is provided and some numerical results are presented. A perturbation analysis of the solution is also performed, which has scarcely appeared in existing literatures. Supported by the National Natural Science Foundation of China(10571012, 10771022), the Beijing Natural Science Foundation (1062005) and the Beijing Educational Committee Foundation (KM200411232006, KM200611232010).     

On the quadratic finite element approximation to the obstacle problem

Summary. In this paper, we obtain the error bound for any , for the piecewise quadratic finite element approximation to the obstacle problem, without the hypothesis that the free boundary has finite length (see [3]). Received October 31, 2000 / Revised version received July 23, 2001 / Published online October 17, 2001 The project was supported by the National Natural Science Foundation of China