Problem: Double$-$Knapsack

Input: Item values v$1,$v$2,,$vn$,$ item sizes s$1,$s$2,,$sn$,$ and capacities C$1$and C$2$ of two knapsacks. $($All positive integers.$)$

Output: Two disjoint subsets S$1,$ S$2\{1,2,,$n$\}$ of items with the maximum possible total value $$i in S$1$S$2$vi$,$ subject to$$

$$i in S$1$C$1$ and $$i in S$2$C$2.$

Here are two possible algorithmic approaches:

$(1)$ use the single Knapsack$$algorithm to pick a maximum$-$value solution S$1$ that fits in the first knapsack, and then use it again on the remaining items to pick a maximum$-$value solution S$2$ that fits in the second knapsack.$$

$(2)$ use the single Knapsack algorithm to pick a maximum$-$value solution S that would fit in a knapsack with capacity C$1+$C$2,$ then partition S arbitrarily into two sets S$1$and S$2$ with total sizes at most C$1$and C$2,$ respectively.

$$

Which of the following statements are true? $($Choose all that apply.$)$

Group of answer choices

Algorithm $(1)$ is guaranteed to produce an optimal solution to the double$-$knapsack problem but algorithm $(2)$ is not.$$

$$

Algorithm $(2)$ is guaranteed to produce an optimal solution to the double$-$knapsack problem but algorithm $(1)$ is not.$$

Algorithm $(1)$ is guaranteed to produce an optimal solution to the double$-$knapsack problem when C$1=$C$2.$

Neither algorithm is guaranteed to produce an optimal solution to the double$-$knapsack problem.