#### Approximation Schemes for Multiperiod Binary Knapsack Problems

##### Zuguang Gao, John R. Birge, Varun Gupta

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.

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