Approximation Schemes for Multiperiod Binary Knapsack Problems

An instance of the multiperiod binary knapsack problem (MPBKP) is given by a horizon length T, non-decreasing vector sizes $$(c_1, \ldots , c_T)$$ where $$c_t$$ denotes cumulative size for periods $$1,\ldots ,t$$ and list n items. Each item triple (r, q, d) r reward or value item, q its size, d time index (or, deadline). The goal to choose, each deadline t, which items include maximize total reward, subject constraints that all $$t=1,\ldots ,T$$ selected with deadlines at most t does not exceed capacity up t. We also consider soft (MPBKP-S) are allowed be violated paying penalty linear in violation. MPBKP-S profit, less penalty. Finally, we stochastic (MPBKP-SS), follows an arbitrary joint distribution set possible sample paths (realizations) probability path algorithm. For MPBKP, exhibit fully polynomial-time approximation scheme runtime $$\tilde{\mathcal {O}}\left( \min \left\{ n+\frac{T^{3.25}}{\epsilon ^{2.25}},n+\frac{T^{2}}{\epsilon ^{3}},\frac{nT}{\epsilon ^2},\frac{n^2}{\epsilon }\right\} \right) $$ achieves $$(1+\epsilon )$$ approximation; MPBKP-S, can achieved $$\mathcal {O}\left( \frac{n\log n}{\epsilon }\cdot \frac{T}{\epsilon },n\right\} . To best our knowledge, algorithms first FPTAS any version Knapsack since study began 1980s. MPBKP-SS, prove natural greedy algorithm 2-approximation when have same size. Our provide insights on how other versions may approximated.