Reducing the 0-1 Knapsack Problem with a Single Continuous Variable to the Standard 0-1 Knapsack Problem

Abstract

The 0-1 knapsack problem with a single continuous variable (KPC) is a natural extension of the binary knapsack problem (KP), where the capacity is not any longer fixed but can be extended which is expressed by a continuous variable. This variable might be unbounded or restricted by a lower or upper bound, respectively. This paper concerns techniques in order to reduce several variants of KPC to KP which enables the authors to employ approaches for KP. The authors propose both, an equivalent reformulation and a heuristic one bringing along less computational effort. The authors show that the heuristic reformulation can be customized in order to provide solutions having an objective value arbitrarily close to the one of the original problem.

Exploring search space trees using an adapted version of Monte Carlo tree search for combinatorial optimization problems

“…Although this algorithm was invented more than twenty years ago, it still represents the current state-of-the-art (see e.g. [68], [69], [70] and [71] for relatively recent articles that support this claim). It is able to exactly solve many problem instances containing several thousands of items in a matter of (milli)seconds.…”

On the integration of Dantzig-Wolfe and Fenchel decompositions via directional normalizations

“…Studies have been carried out on the structure, the cut selection, and the strengthening of the linear relaxation of this type of polyhedron by Marchand and Wolsey [36] as well as in the more general framework of linear programs in mixed variables [21,24,16]. Moreover, optimization methods on this polyhedron have been studied by Büther and Briskorn [14], Lin, Zhu, and Ali [34], Zhao and Li [46], He et al [28], and Liu [35].…”

“…This problem can be solved as a sequence of two 0-1 knapsack problems using a case disjunction. This method was presented by Büther and Briskorn [14] and is recalled in B. In our experiments, we use the MINKNAP algorithm proposed by Pisinger [43] to solve the two associated knapsack problems.…”

“…All the decomposition methods presented in this work assume that there exists an efficient algorithm capable of optimizing a linear function on the polyhedron Q 2 which in our version of the unsplittable flow problem can be written as: problem can be addressed by solving two classic 0-1 knapsack problems using a case disjunction. This method was presented by Büther and Briskorn [14] but we recall it here. Consider the following disjunction: either the flow of the commodities respects the capacity of the arc a, or the flow of the commodities exceeds the capacity of the arc a.…”

Data-Driven Optimization of Incentive-based Demand Response System with Uncertain Responses of Customers

“…With this simple assumption, Equation (2) can be rewritten as Equation (3) by adding the continuous non-negative variables y and z. In the literature, the knapsack problem [42] with unlimited boundary was solved by adding a single continuous variable [43], and this concept is applied to our problem, in order to make the equation linear. In our problem, 2K continuous variables are added, instead of one variable, because K scenarios are included in the calculation.…”