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
1989921461 1985985425 2076163541 2065139435
Given @math elements with non-negative integer weights @math and an integer capacity @math , we consider the counting version of the classic knapsack problem: find the number of distinct subsets whose weights add up to at most @math . We give the first deterministic, fully polynomial-time approximation scheme (FPTAS) for estimating the number of solutions to any knapsack constraint (our estimate has relative error @math ). Our algorithm is based on dynamic programming. Previously, randomized polynomial-time approximation schemes (FPRAS) were known first by Morris and Sinclair via Markov chain Monte Carlo techniques, and subsequently by Dyer via dynamic programming and rejection sampling. In addition, we present a new method for deterministic approximate counting using read-once branching programs. Our approach yields an FPTAS for several other counting problems, including counting solutions for the multidimensional knapsack problem with a constant number of constraints, the general integer knapsack problem, and the contingency tables problem with a constant number of rows. Given @math elements with nonnegative integer weights @math and an integer capacity @math , we consider the counting version of the classic knapsack problem: find the number of distinct subsets whose weights add up to at most the given capacity. We give a deterministic algorithm that estimates the number of solutions to within relative error @math in time polynomial in @math and @math (fully polynomial approximation scheme). More precisely, our algorithm takes time @math . Our algorithm is based on dynamic programming. Previously, randomized polynomial-time approximation schemes were known first by Morris and Sinclair via Markov chain Monte Carlo techniques and subsequently by Dyer via dynamic programming and rejection sampling. Given a set of items, a set of scenarios, and a knapsack of fixed capacity, a nonnegative weight is associated with each item; and a value is associated with each item under each scenario. The max-min Knapsack MNK problem is defined as filling the knapsack with a selected set of items so that the minimum total value gained under all scenarios is maximized. The MNK problem is a generalization of the conventional knapsack problem to situations with multiple scenarios. This extension significantly enlarges its scope of applications, especially in the application of recent robust optimization developments. In this paper, the MNK problem is shown to be strongly NP-hard for an unbounded number of scenarios and pseudopolynomially solvable for a bounded number of scenarios. Effective lower and upper bounds are generated by surrogate relaxation. The ratio of these two bounds is shown to be bounded by a constant for situations where the data range is limited to be within a fixed percentage from its mean. This result leads to an approximation algorithm for MNK in the special case. A branch-and-bound algorithm has been implemented to efficiently solve the MNK problem to optimality. Extensive computational results are presented. We show that for each 0<@e@?1 there exists an extended formulation for the knapsack problem, of size polynomial in the number of variables, whose value is at most (1+@e) times the value of the integer program.