Note: On the set-union Knapsack problem.

*(English)*Zbl 0831.90088The authors have defined a generalization of the 0-1 knapsack problem called the set-union knapsack problem (SKP):
\[
\max\Biggl\{ \sum_{i\in K} v_i \Biggl|\sum_{j\in P_K} s_j\leq b,\;K\subseteq\{1,\dots, m\}\Biggr\},
\]
where \(P_i\subseteq \{1,\dots, n\}\) so that \(\bigcup^\infty_{i= 1} P_i= \{1,\dots, n\}\) and \(P_K= \bigcup_{i\in K} P_i\). They show that the SKP remains NP- hard even in very restricted cases and present an exact dynamic programming algorithm that runs in polynomial time only for special cases. Two applications of SKP are also presented: machine loading in flexible manufacturing systems and the allocations of storage to fields in a data base.

Reviewer: D.Marinescu (Braşov)

##### MSC:

90C10 | Integer programming |

90C35 | Programming involving graphs or networks |

90C39 | Dynamic programming |

##### Keywords:

generalization of the 0-1 knapsack problem; set-union knapsack; NP-hard; machine loading in flexible manufacturing systems
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\textit{O. Goldschmidt} et al., Nav. Res. Logist. 41, No. 6, 833--842 (1994; Zbl 0831.90088)

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