A class of knapsack problems which generalizes the classical ones is studied. Algorithms based on the dynamical programming and Branch and Bound methods are proposed. The correctness and time estimation of the algorithm are given. 1 Problem formulation We study a class of knapsack problems which generalizes problems from [1-2]. The main mathematical model we shall use is formulated as follows. ...

The Lasserre/Sum-of-Squares hierarchy is a systematic procedure to strengthen LP relaxations by constructing a sequence of increasingly tight formulations. For a wide variety of optimization problems, this approach captures the convex relaxations used in the best available approximation algorithms. The capacitated covering IP is an integer program of the form min{cx : Ux ≥ d, 0 ≤ x ≤ b, x ∈ Z+}...

The Temporal Knapsack Problem (TKP) is a generalization of the standard Knapsack Problem where a time horizon is considered, and each item consumes the knapsack capacity during a limited time interval only. In this paper we solve the TKP using what we call a Recursive Dantzig-Wolfe Reformulation (DWR) method. The generic idea of Recursive DWR is to solve a Mixed Integer Program (MIP) by recursi...

Constraints arising in practice often contain many 0-1 variables and one or a small number of continuous variables. Existing knapsack separation routines cannot be used on such constraints. Here we study such constraint sets, and derive valid inequalities that can be used as cuts for such sets, as well for more general mixed 0-1 constraints. Specifically we investigate the polyhedral structure ...

In this paper, we investigate the parametric knapsack problem, in which the item profits are affine functions depending on a real-valued parameter. The aim is to provide a solution for all values of the parameter. It is well-known that any exact algorithm for the problemmay need to output an exponential number of knapsack solutions. We present a fully polynomial-time approximation scheme (FPTAS...

This paper presents a new heuristic algorithm for the MultipleChoice Multi-Dimension Knapsack Problem (MMKP) in PRAM model. MMKP is a variant of the classical 0-1 knapsack problem, has a knapsack with multidimensional capacity constraints, groups of items, each item having a utility value and multidimensional resource constraints. The problem is to maximize the total value of the items in the k...

RÉSUMÉ : We consider two variants of knapsack problems with setups arising as subproblems in a DantzigWolfe decomposition approach to more complex combinatorial optimization problems. In the multiple-class binary 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 ...