О рекурсивно-параллельном алгоритме решения задачи о рюкзаке

In this paper, we offer an efficient parallel algorithm for solving the NP-complete Knapsack Problem in its basic, so-called 0-1 variant. To find its exact solution, algorithms belonging to the category ”branch and bound methods” have long been used. To speed up the solving with varying degrees of efficiency, various options for parallelizing computations are also used. We propose here an algorithm for solving the problem, based on the paradigm of recursive-parallel computations. We consider it suited well for problems of this kind, when it is difficult to immediately break up the computations into a sufficient number of subtasks that are comparable in complexity, since they appear dynamically at run time. We used the RPM ParLib library, developed by the author, as the main tool to program the algorithm. This library allows us to develop effective applications for parallel computing on a local network in the .NET Framework. Such applications have the ability to generate parallel branches of computation directly during program execution and dynamically redistribute work between computing modules. Any language with support for the .NET Framework can be used as a programming language in conjunction with this library. For our experiments, we developed some C# applications using this library. The main purpose of these experiments was to study the acceleration achieved by recursive-parallel computing. A detailed description of the algorithm and its testing, as well as the results obtained, are also given in the paper.Предлагается эффективный параллельный алгоритм решения NP-полной задачи о рюкзаке в ее исходном, так называемом 0-1 варианте. Для нахождения ее точного решения издавна применяются алгоритмы, относящиеся к категории "методов ветвей и границ". Для ускорения получения результата с разной степенью эффективности применяются также различные варианты организации параллельных вычислений. Мы предлагаем здесь алгоритм решения задачи, основанный на парадигме рекурсивно-параллельных вычислений. Он представляется нам хорошо пригодным для задач такого рода, когда трудно сразу разбить вычисления на достаточное количество сравнимых по трудоемкости подзадач, поскольку она проявляется динамически во время вычислений. В качестве основного инструмента для программной реализации алгоритма использовалась разработанная автором библиотека RPM_ParLib, позволяющая создавать эффективные приложения для вычислений на локальной сети в среде .NET Framework. Такие приложения обладают способностью порождать параллельные ветви вычислений непосредственно во время выполнения программы и динамически перераспределять работу между вычислительными модулями. При этом в качестве языка программирования может использоваться любой язык с поддержкой .NET Framework. Для проведения экспериментов было написано несколько программ на языке C# с использованием упомянутой библиотеки. Основной целью этих экспериментов было исследование ускорения, достигаемого за счет рекурсивно-параллельной организации вычислений. Подробное описание алгоритма и эксперимента, а также полученные результаты приводятся в работе.         

An exact approach for the bilevel knapsack problem with interdiction constraints and extensions

We consider the bilevel knapsack problem with interdiction constraints, an extension of the classic 0–1 knapsack problem formulated as a Stackelberg game with two agents, a leader and a follower, that choose items from a common set and hold their own private knapsacks. First, the leader selects some items to be interdicted for the follower while satisfying a capacity constraint. Then the follower packs a set of the remaining items according to his knapsack constraint in order to maximize the profits. The goal of the leader is to minimize the follower’s total profit. We derive effective lower bounds for the bilevel knapsack problem and present an exact method that exploits the structure of the induced follower’s problem. The approach strongly outperforms the current state-of-the-art algorithms designed for the problem. We extend the same algorithmic framework to the interval min–max regret knapsack problem after providing a novel bilevel programming reformulation. Also for this problem, the proposed approach outperforms the exact algorithms available in the literature

A Fast Algorithm for Strongly Correlated Knapsack Problems

AbstractWe consider a variant of the 0–1 Knapsack Problem, where the profit of each item corresponds to its weight plus a fixed constant. These so-called Strongly Correlated Knapsack Problems have attained much interest due to their apparent hardness and wide applicability in several fixed-charge problems.A specialized algorithm for the problem is presented, where the main approach is to derive an additional constraint from an extended cover. By surrogate relaxation with optimal multipliers, we obtain a Subset-sum Problem defined in the profits of the items. It is proved that an optimal solution to the Subset-sum Problem is also an optimal solution to the original problem provided that the largest possible number of items is chosen. Based on this observation, a 2-optimal heuristic is derived which solves the problem to optimality for several large-sized problems. In those cases where the heuristic fails, we solve the problem to optimality by restricting the problem to a fixed number of chosen items β. For each value of β the problem is solved through dynamic programming.Extensive computational experiments are provided showing that we are able to solve strongly correlated instances faster than other algorithms solve uncorrelated instances