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I am searching for information on a variant of the 0-1 Knapsack Problem I will call the Vector Knapsack Problem (VKP), which is basically the same as the standard KP except that the values being summed are vectors (and them sum is the Euclidean norm of the sum of the vectors). Formally, given a set of $m$ vectors $v \in \Re^{n}$, a set of non-negative weights $\alpha \in \Re_{0+}$, and a non-negative maximum Euclidean norm $W \in \Re_{0+}$,

$$Maximize: ||\sum_{i=1}^{m} v_{i}x_{i}||_{2}$$ $$S.T.:\sum_{i=1}^{m}\alpha_{i}x_{i}\leq W, x \in \{0, 1\}$$

What, if anything is known about the complexity of this problem? It seems like it should be NP-Hard at minimum (as a generalization of the 0-1 Knapsack Problem), but it is unclear to me whether the decision problem variant is NP-Complete.

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Yes, the decision problem is NP-complete. If $n=1$ it is exactly the knapsack problem, which is already known to be NP-complete.

Possibly related Given a set of 2D vectors, find the furthest reachable point.

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