Approximation algorithms for the generalized incremental knapsack problem

A note on a QPTAS for maximum weight triangulation of planar point sets

While the classical knapsack problem has been the object to be solved by optimization algorithm proposals for many years, another version of this problem, discounted {0-1} knapsack problem, is gaining a lot of attention recently. The original knapsack problem requires selecting specific items from an item set to maximize the total benefit while ensuring that the total weight does not exceed the knapsack capacity. Meanwhile, discounted {0-1} knapsack problem has more stringent requirements in which items are divided into groups, and only up to one item from a particular group can be selected. This constraint, which does not exist in the original knapsack problem, makes discounted {0-1} knapsack problem even more challenging. In this paper, we propose a new algorithm based on salp swarm algorithm in the form of four different variants to resolve the discounted {0-1} knapsack problem. In addition, we also make use of an effective data modeling mechanism and a greedy repair operator that helps overcome local optima when finding the global optimal solution. Experimental and statistical results show that our algorithm is superior to currently available algorithms in terms of solution quality, convergence, and other statistical criteria.

Many optimization problems encountered in practice are multi-objective by nature, i.e., different objectives are conflicting and equally important. Many times, it is not desirable to drop some of them or to optimize them in a composite single objective or hierarchical manner. Furthermore, cost parameters change over time which makes optimization problems harder. For instance, in the transport sector, travel costs are a function of travel time which changes depending on the time of the day a vehicle is travelling (e.g., due to road congestion). Road congestion results in tremendous delays which lead to a decrease in the service quality and the responsiveness of logistic service providers. In Chapter 2, we develop a generic approach to deal with Multi-Objective Scheduling Problems (MOSPs) with State-Dependent Cost Parameters. The aim is to determine the set of Pareto solutions that capture the trade offs between the different conflicting objectives. Due to the complexity of MOSPs, an efficient approximation based on dynamic programming is developed. The approximation has a provable worse case performance guarantee. Even though the generated approximate Pareto front consist of fewer solutions, it still represents a good coverage of the true Pareto front. Furthermore, considerable gains in computation times are achieved. In Chapter 3, the developed methodology is validated on the multi-objective timedependent knapsack problem. In the classical knapsack problem, the input consists of a knapsack with a finite capacity and a set of items, each with a certain weight and a cost. A feasible solution to the knapsack problem is a selection of items such that their total weight does not exceed the knapsack capacity. The goal is to maximize the single objective function consisting of the total pro t of the selected items. We extend the classical knapsack problem in two ways. First, we consider time-dependent profits (e.g., in a retail environment profit depends on whether it is Christmas or not).

We consider a natural generalization of the knapsack problem and the multiple knapsack problem, which has two phases of packing decisions. In this problem, we have a set of items, several small knapsacks called boxes, and a large knapsack called container. Each item has a size and profit, each box has a size and the container has a capacity. The first phase is to select some items to pack into the boxes, and the second phase is to select the boxes (each includes some packed items) to pack into the container. The total profit of the problem is determined by the items that are selected and packed into the container within some packed box, and the objective is to maximize the total profit. This problem is motivated by various practical applications, e.g., in logistics. It is a generalization of the multiple knapsack problem, and hence is strongly NP-hard. We mainly propose three approximation algorithms for it. Particularly, the first one is a \(\frac{1}{4}\)-approximation algorithm based on its linear programming relaxation; the second one is based on applying the algorithms for the multiple knapsack problem and the knapsack problem, and has an approximation ratio \(\frac{1}{3} - \epsilon \) for any small enough \(\epsilon >0\). We finally provide a polynomial time approximation scheme for this problem.

We analyze the competitive ratio and the advice complexity of the online unbounded knapsack problem. An instance is given as a sequence of n items with a size and a value each, and an algorithm has to decide whether or not and how often to pack each item into a knapsack of bounded capacity. The items are given online and the total size of the packed items must not exceed the knapsack’s capacity, while the objective is to maximize the total value of the packed items. While each item can only be packed once in the classical knapsack problem (also called the 0-1 knapsack problem), the unbounded version allows for items to be packed multiple times. We show that the simple unbounded knapsack problem, where the size of each item is equal to its value, allows for a competitive ratio of 2. We also analyze randomized algorithms and show that, in contrast to the 0-1 knapsack problem, one uniformly random bit cannot improve an algorithm’s performance. More randomness lowers the competitive ratio to less than 1.736, but it can never be below 1.693. In the advice complexity setting, we measure how many bits of information (so-called advice bits) the algorithm has to know to achieve some desired solution quality. For the simple unbounded knapsack problem, one advice bit lowers the competitive ratio to 3/2. While this cannot be improved with fewer than log2n advice bits for instances of length n, a competitive ratio of 1+ε can be achieved with O(ε-1·log(nε-1)) advice bits for any ε>0}\\varvec{0}$$\\end{document}]]>. We further show that no amount of advice bounded by a function f(n) allows an algorithm to be optimal. We also study the online general unbounded knapsack problem and show that it does not allow for any bounded competitive ratio for both deterministic and randomized algorithms, as well as for algorithms using fewer than log2n advice bits. We also provide a surprisingly simple algorithm that uses O(ε-1·log(nε-1)) advice bits to achieve a competitive ratio of 1+ε for any ε>0}\\varvec{0}$$\\end{document}]]>.

Knapsack problem is a classical optimization problem. It is widely used in resource allocation problems. We are aware of 0-1 knapsack and fractional knapsack problem. Here, we will discuss about a newer version, unified knapsack problem, which is a combination of above two versions. In 0-1 knapsack problem each item is either be taken or left behind but in fractional knapsack problem a fraction of any item can be taken, while in unified knapsack problem some items can be fractioned and some cannot.KeywordsResource allocationfractional knapsack problemunified knapsack problem0-1 knapsack problem

O Problema da Mochila Compartimentada é uma variação do clássico problema da mochila e pode ser enunciado considerando-se a seguinte situação hipotética: um alpinista deve carregar sua mochila compossíveis itens de seu interesse. A cada item atribui-se o seu peso e um valor de utilidade (até aqui, o problema coincide com o clássico Problema da Mochila). Entretanto, os itens são de agrupamentos distintos (alimentos, medicamentos, utensílios, etc.) e devem estar em compartimentos separados na mochila. Os compartimentos da mochila são flexíveis e têm capacidades limitadas. A inclusão de um compartimento tem um custo fixo que depende do agrupamento com que foi preenchido, além de introduzir uma perda da capacidade da mochila. O problema consiste em determinar as capacidades adequadas de cada compartimento e como esses devem ser carregados, maximizando o valor de utilidade total, descontado o custo de incluir compartimentos. Neste trabalho propomos um modelo de otimização não linear inteiro para o problema e algumas heurísticas para sua resolução. Uma aplicação prática de relevância deste problema aparece no corte de bobinas de aço, sujeito à laminação, detalhado em apêndice.

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A classical knapsack problem with two objectives is considered in which concepts of supported and non supported efficient solutions are pointed out.

This paper is dedicated to a study of different extensions of the classical knapsack problem to the case when different elements of the problem formulation are subject to a degree of uncertainty described by random variables. This brings the knapsack problem into the realm of stochastic programming. Two different model formulations are proposed, based on the introduction of probability constraints. The first one is a static quadratic knapsack with a probability constraint on the capacity of the knapsack. The second one is a two-stage quadratic knapsack model, with recourse, where we introduce a probability constraint on the capacity of the knapsack in the second stage. As far as we know, this is the first time such a constraint has been used in a two-stage model. The solution techniques are based on the semidefinite relaxations. This allows for solving large instances, for which exact methods cannot be used. Numerical experiments on a set of randomly generated instances are discussed below.

[EN]This paper proposes a modified iterated greedy
\nalgorithm for solving an NP-hard knapsack problem known as
\nthe knapsack problem with setup. In contrast to the classical
\n0-1 knapsack problem, items belong to different classes, and
\nchoosing at least one item from a specific class causes a
\nclass setup capacity to be consumed and a reduction of the
\ntotal profit in terms of a penalty that has to be paid. The
\nperformance of our technique was compared with those of
\nboth a sophisticated, tree-search based, heuristic from the
\nliterature and the application of CPLEX to the respective ILP
\nmodel. Experimental results show that the proposed technique
\noutperforms the heuristic from the literature and yields better
\nperformance than CPLEX for medium and large size problem
\ninstances.

In this paper, we propose a multiperiod single-sourcing problem (MPSSP), which takes both transportation and inventory into consideration, suitable for evaluating the performance of a logistics distribution network in a dynamic environment. We reformulate the MPSSP as a Generalized Assignment Problem (GAP) with a convex objective function. We then extend a branch-and-price algorithm that was developed for the GAP to this problem. The pricing problem is a so-called Penalized Knapsack Problem (PKP), which is a knapsack problem where the objective function includes an additional convex term penalizing the total use of capacity of the knapsack. The optimal solution of the relaxation of the integrality constraints in the PKP shows a similar structure to the optimal solution of the knapsack problem, that allows for an efficient solution procedure for the pricing problem. We perform an extensive numerical study of the branch-and-price algorithm.

The knapsack problem is one of the classical NP-hard problems in operations research. It has been thoroughly studied in the last few decades and several exact algorithms for its solution can be found in the literature. In this paper, we propose a new ant colony optimization (ACO) algorithm for solving the knapsack problem. Comparing with the basic ACO, this improved algorithm combines inner mutation and outer mutation that make it more effective and efficient in solving the knapsack problem. Numerical example is presented to illustrate the model

sland operation is a special operation mode for service restoration for the smart distribution system with distributed generation. In this paper, a model of island reconfiguration of the smart distribution system is proposed. The objection of this model is maximizing the total power supply profit. Meanwhile, load priority, controllable and uncontrollable load, and the restrictions of power balance, voltage and equipment capability are all taken into account. The island configuration problem is first simplified as a special knapsack problem, and based on the solution of this knapsack problem an initial island scheme is gained. By feasibility checking and adjusting, a island configuration is finally proposed. The results of the samples demonstrate that the model and method presented in this paper is feasible and valid.

The 0-1 knapsack problem (KP) is a classical NP-hard problem with binary decision variables. The traditional differential evolution (DE) is an effective stochastic parallel search evolutionary algorithm and customized to continuous function optimization. To solve KPs, based on DE, a discrete binary version of differential evolution (DBDE) was employed, where each component of a mutated vector component changes with the probability and will take on a zero or one value. Moreover, a heuristic operator was employed to handle the constraint and to enhance local search. The approach was implemented to 3 cases. By comparisons with the other evolutionary algorithms, DBDE have shown the feasibility and effectiveness.