An instance of the multiperiod binary knapsack problem (MPBKP) is given by a horizon length $T$, a non-decreasing vector of knapsack sizes $(c_1, \ldots, c_T)$ where $c_t$ denotes the cumulative size for periods $1,\ldots,t$, and a list of $n$ items. Each item is a triple $(r, q, d)$ where $r$ denotes the reward of the item, $q$ its size, and $d$ 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 $t=1,\ldots,T$, 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 is to maximize the total profit, i.e., the total reward of 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 $(c_1, \ldots, c_T)$ follow some arbitrary joint distribution but we are given access to the profit as an oracle, and we choose a subset of items to maximize the total expected profit, i.e., the total reward less the total expected penalty. For MPBKP, we exhibit a fully polynomial-time approximation scheme with runtime $\tilde{\mathcal{O}}\left(\min\left\{n+\frac{T^{3.25}}{\epsilon^{2.25}},n+\frac{T^{2}}{\epsilon^{3}},\frac{nT}{\epsilon^2},\frac{n^2}{\epsilon}\right\}\right)$ that achieves $(1+\epsilon)$ approximation; for MPBKP-S, the $(1+\epsilon)$ approximation can be achieved in $\mathcal{O}\left(\frac{n\log n}{\epsilon}\cdot\min\left\{\frac{T}{\epsilon},n\right\}\right)$; for MPBKP-SS, a greedy algorithm is a 2-approximation when items have the same size.

翻译：多周期二进制问题( MPBKP) 的示例。 每个项目都是三美元( r, q, d), 其中美元表示项目奖励 $T$, $Q3, 一个非减制的 knpsack 规模的矢量 $( c_ 1,\ ldot, c_ T) 美元, 其中美元表示周期累积规模 1,\ ldot, t美元, 和一个美元项目清单。 每个项目都是三美元( r, q, d), 其中美元表示项目奖励 $, $Q, 其规模和 美元时间指数( 或, 截止日期) 。 目标是为每个期限选择一个非减量的矢量矢量矢量矢量的矢量 $t, 其中项目包括最大限度的奖励, 其中所有$t=1,\ldot, 美元, 选定项目总规模不超过 knackackack, 其中我们完全可以支付罚款。