# Questions tagged [knapsack-problems]

A problem in combinatorial optimization. Given a set of items with both weight and value, determine the number of each item to include in a collection so that the total weight is at most a given limit and the value of the collection is maximized.

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### Knapsack with a fixed number of weights

Consider a special case of the knapsack problem in which all weights are integers, and the number of different weights is fixed. For example, the weight of every item is either 1k or 2k or 4k. There ...
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### Determine aproximation factor in a greedy algorithm

Suppose we have n food dishes associated to a cost c, and we have i guests such that each one of them has a certain number of preferences. We want to choose a menu such that we minimize the cost and ...
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### Still not understanding why the Knapsack Problem does NOT have a polynomial-time solution

All the explanations for why the $O(nW)$ DP algorithm that solves the Knapsack Problem is NOT polynomial repeat the same thing: it is the length (in bits) of the input that matters, not its value/...
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### How to put elements in allowed bags?

Let's say I have a list of "Items" I also have a list of "Bags". Each bag is a set of "Items" which gives what item can be placed in that bag. But only one item can go in each bag. I want to place ...
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### Shrinking Item 0-1 Knapsack problem

I have encountered a variant of the knapsack problem with shrinking items. Effectively, it is a 0-1 knapsack problem where the initial weight of each item is $W(n)+V(n)$ and their value is $V(N)$, ...
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### Closest Value instead of Max Value in Knapsack problem

I have a problem like the knapsack problem, except instead of finding the max value, I'm trying to find the closest value to a given value. Anyone know where to start, have a name for this problem, ...
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### Multiple Knapsack Problem, but with exact weight/capacity? (All knapsacks have to be filled completely)

I have a few knapsacks of given capacities. Also, much more items of given weights and values. As usual, items have to be put into knapsacks to maximize profit. What is different from a standard ...
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### Multi Knapsack each with different constraints

It seems that there's no end to knapsack variations… here's the one I bumped into (at work): There are: N items, each with the usual value and weight properties. M bins, each with an upper ...
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### What's the time complexity bound for the Knapsack with real weights?

Let's start with the formalism: Let $W$ be the total weight of our bag, $1,...,n$ be our elements, $w_1,...,w_n$ their corresponding weights, and $v_1,...,v_n$ their corresponding values. As is ...
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### DP recurrence relations: Coin change vs Knapsack

Take: KP recurrence relation $max { [v + f(k-1,g-w ), f(k-1,g)] }$ if w <= g and k>0 CCP recurrence relation $min {[1 + f(r,c-v), f(r-1,c)]}$ if v <= c and r>0 I don't ...
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### Oven cooking queue problem: need help to determine its nature

CS community! I'm more of a practical programmer than a computer scientist, so I need your help to find a solution (or at least point me into the right direction) to an extremely practical problem ...
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### Solving the Knapsack problem in $O(v^*n^2)$, where $v^*$ is the maximum value of all $n$ items

For my study I am supposed to develop an dynamic programming algorithm which solves the following simplification of the Knapsack problem in $O(v^*n^2)$ time ($v^*$ is the maximum value of all elements)...
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### Help writing a dynamic programming algorithm in English

It’s Friday night and you have $n$ parties to go to. Party $i$ has a start time $s_i$ end time $t_i$ and a value $v_i \ge 0$. Think of the value as an indicator of how excited you are about that party....
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### Knapsack problem: equal profits

I am looking for references to efficient algorithms that solve knapsack problem where all profits are equal. More formal definition of the problem from a Wikipedia article on KPs: If all the ...
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### Variant of the Knapsack Problem

I've got problem on integer programming, specifically with the following knapsack problem. I'd be happy to get some suggestions on how to solve the problem in a time efficient way. There are 120 ...
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### Calculate maximum sum of nodes property with limit on distance being traversed between nodes in a given graph

Given is an undirected weighted graph with N nodes, with each node having a property/Value. Aim is to find the bestpath which maximizes the sum of the nodes property which can be visited given a ...
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### Knapsack-type problem where the objective function is a ratio

I have a problem where I have a number of proposed initiatives each with a cost and payoff. I need to select a subset of these initiatives in order to maximize the ROI for the selected set as a whole ...
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### Specific quadratic 0-1 knapsack problem solvable in linear time?

I am interested in a simple variant of the quadratic knapsack problem. Let $\{w_1, \ldots, w_n\} \in \{0,1\}$ be $n$ weights and $\{v_1, \ldots, v_n\} \in \mathbb{R}$ be $n$ values. Furthermore, ...
Given two arrays of the same length $n$: $A = \{a_1,a_2,\dots,a_n\}$, $B = \{b_1,b_2,\dots,b_n\}$, I have to maximize the following expression: \frac{a_{i_1} + a_{i_2} + a_{i_3}+ \dots + a_{i_k}}...