Many systems have to be maintained while the underlying constraints, costs and or profits change over time. Although the state of a system may evolve during time, a non-negligible transition cost is incured for transitioning from one state to another. In order to model such situations, (ICALP 2014) and (ICALP 2014) introduced a multistage model where the input is a sequence of instances (one for each time step), and the goal is to find a sequence of solutions (one for each time step) that are both (i) near optimal for each time step and (ii) as stable as possible. We focus on the multistage version of the Knapsack problem where we are given a time horizon t=1,2,...,T, and a sequence of knapsack instances I_1,I_2,...,I_T, one for each time step, defined on a set of n objects. In every time step t we have to choose a feasible knapsack S_t of I_t, which gives a knapsack profit. To measure the stability similarity of two consecutive solutions S_t and S_ t+1 , we identify the objects for which the decision, to be picked or not, remains the same in S_t and S_ t+1 , giving a transition profit. We are asked to produce a sequence of solutions S_1,S_2,...,S_T so that the total knapsack profit plus the overall transition profit is maximized. We propose a PTAS for the Multistage Knapsack problem. Then, we prove that there is no FPTAS for the problem even in the case where T=2, unless P=NP. Furthermore, we give a pseudopolynomial time algorithm for the case where the number of steps is bounded by a fixed constant and we show that otherwise the problem remains NP-hard even in the case where all the weights, profits and capacities are 0 or 1.
It is well known that for the usual Knapsack problem, in the continuous relaxation (variables in @math ), at most one variable is fractional. @cite_11 showed that this can be generalized for @math .
@cite_11
2031666601
Abstract We address a variant of the classical knapsack problem in which an upper bound is imposed on the number of items that can be selected. This problem arises in the solution of real-life cutting stock problems by column generation, and may be used to separate cover inequalities with small support within cutting-plane approaches to integer linear programs. We focus our attention on approximation algorithms for the problem, describing a linear-storage Polynomial Time Approximation Scheme (PTAS) and a dynamic-programming based Fully Polynomial Time Approximation Scheme (FPTAS). The main ideas contained in our PTAS are used to derive PTAS's for the knapsack problem and its multi-dimensional generalization which improve on the previously proposed PTAS's. We finally illustrate better PTAS's and FPTAS's for the subset sum case of the problem in which profits and weights coincide.