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I had an online round at a company where I was asked this question.

There are $N$ items, and you have to choose some items from them such that the total weight does not exceed $W$.

Each item has three properties, weight, profit and type. There are only two types of objects.

Type 0 item can be selected independently own their own.

A type 1 item cannot the selected on its own and needs another item of type 1.

Note: The problem does not say if this means what we have to select at least two type 1 items to select any at all, or that all the type 1 items must be in pairs.

Constraints:

  • $N < 10^3$

  • $W < 10^5$

  • $\mathit{weight}, \mathit{profit} < 10^5$

This is obviously (?) related to the knapsack problem, as without the restriction on the type 1 objects, it would actually be the 0-1 knapsack problem. How do I go about doing this? Any help and general ideas are extremely appreciated.

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I think this can be solved the same way that knapsack problem is solved using dynamic programming.

I present the problem for the case where type 1 items are to be selected in pairs, but the other case is solvable in a similar way.

Suppose the weights are $\{w_1, …, w_n\}$, the profits are $\{p_1, …, p_n\}$ and the types are $\{t_1, …, t_n\}$. Let's define, for $w\in [\![0, W]\!], i \in [\![1, n]\!], j\in\{0,1\}$, $K(w, i, j)$ representing the maximum profit using a knapsack of capacity $w$, using items $1, …, i$, with a parity of type 1 items equal to $j$.

We want to determine $K(W, n, 0)$. Using this definition, we can calculate $K(w, i, j)$ by distinguishing if we take or not the $i$-th item:

$K(w, i, j) = \max(K(w-w_i, i - 1, j \oplus t_i) + p_i, K(w, i - 1, j))$ (where $\oplus$ is the addition modulo 2)

With this formula, we can then compute $K(W, n, 0)$ in time complexity $O(nW)$.

In the case you just need to pick at least two items of type 1 if you want any, $K(w, i, j)$ could represent the maximum profit with $j\in \{0, 1,2\}$ representing respectively exactly zero type 1 item, exactly one type 1 item or at least 2 type 1 items.

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  • $\begingroup$ Hey, thank you so much! That really clears it up. $\endgroup$ – Soumya Pattanayak Apr 3 at 18:34

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