Approximation Schemes for Multiperiod Binary Knapsack Problems

Abstract

An instance of the multiperiod binary knapsack problem (MPBKP) is given by a horizon length T, a non-decreasing vector of knapsack sizes (c1, …, cT) where ct denotes the cumulative size for periods 1, …, t, and a list of n items. Each item is a triple (r, q, d) where r denotes the reward or value of the item, q its size, and d denotes its time index (or, deadline). The goal is to choose, for each deadline t, which items to include to maximize the total reward, subject to the constraints that for all t= 1, …, T, the total size of selected items with deadlines at most t does not exceed the cumulative capacity of the knapsack up to time t. We also consider the multiperiod binary knapsack problem with soft capacity constraints (MPBKP-S) where the capacity constraints are allowed to be violated by paying a penalty that is linear in the violation. The goal of MPBKP-S is to maximize the total profit, which is the total reward of the selected items less the total penalty. Finally, we consider the multiperiod binary knapsack problem with soft stochastic capacity constraints (MPBKP-SS), where the non-decreasing vector of knapsack sizes (c1, …, cT) follows an arbitrary joint distribution with the set of possible sample paths (realizations) of knapsack sizes and the probability of each sample path given to the algorithm. For MPBKP, we exhibit a fully polynomial-time approximation scheme with runtime O~(min{n+T3.25ϵ2.25,n+T2ϵ3,nTϵ2,n2ϵ}) that achieves (1 + ϵ) approximation; for MPBKP-S, the (1 + ϵ) approximation can be achieved in O(nlognϵ·min{Tϵ,n}). To the best of our knowledge, our algorithms are the first FPTAS for any multiperiod version of the Knapsack problem since its study began in 1980s. For MPBKP-SS, we prove that a natural greedy algorithm is a 2-approximation when all items have the same size. Our algorithms also provide insights on how other multiperiod versions of the knapsack problem may be approximated.