Abstract Scalarization is a common technique to transform a multiobjective optimization problem into a scalar-valued optimization problem. This article deals with the weighted Tchebycheff scalarization applied to multiobjective discrete optimization problems. This scalarization consists of minimizing the weighted maximum distance of the image of a feasible solution to some desirable reference point. By choosing a suitable weight, any Pareto optimal image can be obtained. In this article, we provide a comprehensive theory of this set of eligible weights. In particular, we analyze the polyhedral and combinatorial structure of the set of all weights yielding the same Pareto optimal solution as well as the decomposition of the weight set as a whole. The structural insights are linked to properties of the set of Pareto optimal solutions, thus providing a profound understanding of the weighted Tchebycheff scalarization method and, as a consequence, also of all methods for multiobjective optimization problems using this scalarization as a building block.

Abstract This article is dedicated to the weight set decomposition of a multiobjective (mixed-)integer linear problem with three objectives. We propose an algorithm that returns a decomposition of the parameter set of the weighted sum scalarization by solving biobjective subproblems via Dichotomic Search which corresponds to a line exploration in the weight set. Additionally, we present theoretical results regarding the boundary of the weight set components that direct the line exploration. The resulting algorithm runs in output polynomial time, i.e. its running time is polynomial in the encoding length of both the input and output. Also, the proposed approach can be used for each weight set component individually and is able to give intermediate results, which can be seen as an “approximation” of the weight set component. We compare the running time of our method with the one of an existing algorithm and conduct a computational study that shows the competitiveness of our algorithm. Further, we give a state-of-the-art survey of algorithms in the literature.

We consider the (linear) parametric 0–1 knapsack problem in which the profits of the items are affine-linear functions of a real-valued parameter and the task is to compute a solution for all values of the parameter. For this problem, it is known that the piecewise linear convex function mapping the parameter to the optimal objective value of the corresponding instance (called the optimal value function ) can have exponentially many breakpoints (points of slope change), which implies that every optimal algorithm for the problem must output a number of solutions that is exponential in the number of items. We provide the first (parametric) polynomial time approximation scheme (PTAS) for the parametric 0–1 knapsack problem. Moreover, we exploit the connection between the parametric problem and the bicriteria problem in order to show that the parametric 0–1 knapsack problem admits a parametric FPTAS when the parameter is restricted to the positive real line and the slopes and intercepts of the affine-linear profit functions of the items are nonnegative. The method used to obtain this result applies to many linear parametric optimization problems and provides a general connection between bicriteria and linear parametric optimization problems.