Solving knapsack sharing problems with general tradeoff functions

A knapsack sharing problem is a maximin or minimax mathematical programming problem with one or more knapsack constraints (an inequality constraint with all non-negative coefficients). All knapsack sharing algorithms to date have assumed that the objective function is composed of single variable functions called tradeoff functions which are strictly increasing continuous functions. This paper develops optimality conditions and algorithms for knapsack sharing problems with any type of continuous tradeoff function including multiple-valued and staircase functions which can be increasing, decreasing, unimodal, bimodal, or even multi-modal. To do this, optimality conditions are developed for a special type of multiple-valued, nondecreasing tradeoff function called an ascending function. The optimal solution to any knapsack sharing problem can then be found by solving an equivalent problem where all the tradeoff functions have been transformed to ascending functions. Polynomial algorithms are developed for piecewise linear tradeoff functions with a fixed number of breakpoints. The techniques needed to construct efficient algorithms for any type of tradeoff function are discussed.     

Stochastic on-line knapsack problems

Different classes of on-line algorithms are developed and analyzed for the solution of {0, 1} and relaxed stochastic knapsack problems, in which both profit and size coefficients are random variables. In particular, a linear time on-line algorithm is proposed for which the expected difference between the optimum and the approximate solution value isO(log^3/2 n). An(1) lower bound on the expected difference between the optimum and the solution found by any on-line algorithm is also shown to hold.Corresponding author.Partially supported by the Basic Research Action of the European Communities under Contract 3075 (Alcom).Partially supported by research project Models and Algorithms for Optimization of the Italian Ministry of University and Scientific and Technological Research (MURST 40%).     

Revisiting surrogate relaxation for the multidimensional knapsack problem

The multidimensional knapsack problem (MKP) is a classic problem in combinatorial optimisation. Several authors have proposed to use surrogate relaxation to compute upper bounds for the MKP. In their papers, the surrogate dual is solved heuristically. We show that, using a modern dual simplex solver as a subroutine, one can solve the surrogate dual exactly in reasonable computing times. On the other hand, the resulting upper bound tends to be strong only for relatively small MKP instances.     

Hybrid rounding techniques for knapsack problems

We address the classical knapsack problem and a variant in which an upper bound is imposed on the number of items that can be selected. We show that appropriate combinations of rounding techniques yield novel and more powerful ways of rounding. Moreover, we present a linear-storage polynomial time approximation scheme (PTAS) and a fully polynomial time approximation scheme (FPTAS) that compute an approximate solution, of any fixed accuracy, in linear time. These linear complexity bounds give a substantial improvement of the best previously known polynomial bounds [A. Caprara, et al., Approximation algorithms for knapsack problems with cardinality constraints, European J. Oper. Res. 123 (2000) 333-345].     

Multi-constrained matroidal knapsack problems

We consider multi-constrained knapsack problems where the sets of elements to be selected are subject to combinatorial constraints of matroidal nature. For this important class of NP-hard combinatorial optimization problems we prove that Lagrangean relaxation techniques not only provide good bounds to the value of the optimum, but also yield approximate solutions, which are asymptotically optimal under mild probabilistic assumptions.Partially supported by research projects Analisi e progetto di algoritmi and Modelli ed algoritmi per l'ottimizzazione of the Italian Ministry of Education (MPI 40%), and by NATO Grant RG 85/0240. Orally presented at the 12th International Symposium on Mathematical Programming, Boston, August 1985.     

Lifting the knapsack cover inequalities for the knapsack polytope

Valid inequalities for the knapsack polytope have proven to be very useful in exact algorithms for mixed-integer linear programming. In this paper, we focus on the knapsack cover inequalities, introduced in 2000 by Carr and co-authors. In general, these inequalities can be rather weak. To strengthen them, we use lifting. Since exact lifting can be time-consuming, we present two fast approximate lifting procedures. The first procedure is based on mixed-integer rounding, whereas the second uses superadditivity.     

一类具有上下界的均衡问题

本文获得一类具有上、下界的抽象均衡问题解的存在性条件, 回答了Isac, Sehgal和Singh提出的一个公开问题,进一步研究了具上、下界及约束条件的抽象均 衡问题,获得了更为深刻的结果. 最后,把上述公开问题推广到两个空间的情形.     

A dynamic programming algorithm for the bilevel knapsack problem

We propose an efficient dynamic programming algorithm for solving a bilevel program where the leader controls the capacity of a knapsack, and the follower solves the resulting knapsack problem. We propose new recursive rules and show how to solve the problem as a sequence of two standard knapsack problems.     

An upper bound for SLP using first and total second moments

In 1987, J. Dulá considered the problem of finding an upper bound for the expectation of a so-called simplicial function of a random vector and used for this purpose first and total second moments. Under the same moment conditions we consider some different cases of recourse functions and demonstrate how the related moment problems can be solved by solving nonsmooth (unconstrained) optimization problems and thereafter satisfying simple linear constraint systems.     

Fractional knapsack problems

The fractional knapsack problem to obtain an integer solution that maximizes a linear fractional objective function under the constraint of one linear inequality is considered. A modification of the Dinkelbach's algorithm [3] is proposed to exploit the fact that good feasible solutions are easily obtained for both the fractional knapsack problem and the ordinary knapsack problem. An upper bound of the number of iterations is derived. In particular it is clarified how optimal solutions depend on the right hand side of the constraint; a fractional knapsack problem reduces to an ordinary knapsack problem if the right hand side exceeds a certain bound.     

On lower and upper bounds of matrices

Using an approach of Bergh, we give an alternate proof of Bennett's result on lower bounds for non-negative matrices acting on non-increasing non-negative sequences in lp when p?1 and its dual version, the upper bounds when 0<p?1. We also determine such bounds explicitly for some families of matrices.     

A robust approach to the chance-constrained knapsack problem

In this paper, the chance-constrained knapsack problem (CKP) is addressed. Relying on robust optimization, a tractable combinatorial algorithm is proposed to solve approximately CKP. For two specific classes of uncertain knapsack problems, it is proved to solve CKP at optimality.     

Solving the 0–1 proportional knapsack problem by sampling

In this paper, we deal with the proportional knapsack problem that is a variation on the ordinary knapsack problem. In the proportional knapsack problem, we look at filling an urn with objects having two characteristics: color and weight. The colors of the objects in the urn should be proportional to the distribution of the colors in the object universe, and the total weight of the objects in the urn should be as close as possible to the capacity of the urn. The formulation of the problem was motivated by a real-life application from the area of finance, called a dollar roll. We show that the proportional knapsack problem is NP-hard, and then, using sampling, develop a heuristic procedure for solving the problem.Partial support from the Fund for the Promotion of Research and from the Alexander Goldberg Memorial Fund at Technion is gratefully acknowledged.     

A faster algorithm for the continuous bilevel knapsack problem

We construct a fast algorithm with time complexity $O(nlogn)$ for a continuous bilevel knapsack problem with interdiction constraints for $n$ items. This improves on a recent algorithm from the literature with quadratic time complexity $O\left(n^2\right)$.     

Solving the bi-objective multi-dimensional knapsack problem exploiting the concept of core

This paper deals with the bi-objective multi-dimensional knapsack problem. We propose the adaptation of the core concept that is effectively used in single-objective multi-dimensional knapsack problems. The main idea of the core concept is based on the “divide and conquer” principle. Namely, instead of solving one problem with n variables we solve several sub-problems with a fraction of n variables (core variables). The quality of the obtained solution can be adjusted according to the size of the core and there is always a trade off between the solution time and the quality of solution. In the specific study we define the core problem for the multi-objective multi-dimensional knapsack problem. After defining the core we solve the bi-objective integer programming that comprises only the core variables using the Multicriteria Branch and Bound algorithm that can generate the complete Pareto set in small and medium size multi-objective integer programming problems. A small example is used to illustrate the method while computational and economy issues are also discussed. Computational experiments are also presented using available or appropriately modified benchmarks in order to examine the quality of Pareto set approximation with respect to the solution time. Extensions to the general multi-objective case as well as to the computation of the exact solution are also mentioned.     

The 0-1 knapsack problem with fuzzy data

The 0-1 knapsack problem with imprecise profits and imprecise weights of items is considered. The imprecise parameters are modeled as fuzzy intervals. A method of choosing a solution under the uncertainty is proposed and two methods for solving the constructed models are provided.     

A knapsack intersection hierarchy

We introduce a knapsack intersection hierarchy for strengthening linear programming relaxations of packing integer programs. In level t of the hierarchy, all valid cuts are added for the integer hull of the intersection of all t-row relaxations. This model captures the maximum possible strength of t-row cuts, an approach often used by solvers for small t. We investigate the integrality gap of the strengthened formulations on the all-or-nothing flow problem in trees (also called unsplittable flow on trees).     

A new lower bound for the linear knapsack problem with general integer variables

It is well known that the linear knapsack problem with general integer variables (LKP) is NP-hard. In this paper we first introduce a special case of this problem and develop an O(n) algorithm to solve it. We then show how this algorithm can be used efficiently to obtain a lower bound for a general instance of LKP and prove that it is at least as good as the linear programming lower bound. We also present the results of a computational study that show that for certain classes of problems the proposed bound on average is tighter than other bounds proposed in the literature.     

Polynomial size IP formulations of knapsack may require exponentially large coefficients

A desirable property of integer formulations is to consist of few inequalities having small coefficients. We show that these targets are conflicting by proving the existence of knapsack sets that need exponentially many inequalities or exponentially large coefficients in any integer formulation. Moreover, we show that there exist undirected graphs such that (in a natural model) every integer formulation of stable sets requires exponentially large coefficients if the number of non-trivial inequalities is bounded by a constant.