# 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 Greedy Approximation: Worst Case

I am currently studying approximation algorithms and I have run into an issue with a study problem. The approximation algorithm is for the general Knapsack problem, and it proposes a greedy approach, ...
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### What algorithms exist for solving natural number linear systems?

I'm looking at the following problem: Given $n$-dimensional vectors of natural numbers $v_1, \ldots, v_m$ and some input vector $u$, is $u$ a linear combination of the $v_i$'s with natural number ...
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### Algorithms for two and three dimensional Knapsack

I know that the 2D and 3D Knapsack problems are NPC, but is there any way to solve them in reasonable time if the instances are not very complicated? Would dynamic programming work? By 2D (3D) ...
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### Find non-overlapping scheduled jobs with maximum cost

Given a set of n jobs with [start time, end time, cost] find a subset so that no 2 jobs overlap and the cost is maximum. Now I'm not sure if a greedy algorithm will do the trick. That is, sort by ...
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### Balanced Weight Distribution in Bins/Buckets

Let $W = \{w_1,w_2,...w_n\}$ be a set of integer weights. Let $B = \{b_1,b_2,...b_m\}$ be a set of buckets, with $m \leq n$. Let $T(b_j)$ represent the total weight present in bucket $b_j$, which is ...
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### Knapsack with same value

I'm wondering if there's a name/reference for the variant of knapsack problem where all items have the same value (so we only care about maximizing the number of items), but there are multiple weight ...
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### Relations between the knapsack problem, the bin packing problem, and the set packing problem?

I wonder what relations are between the knapsack problem, the bin packing problem and the set packing problem? From their mathematical formulations, I don't see the first two belong to the third one ...
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### Dynamic Programming Solution to 0,1 KnapSack Problem

I am trying to understand the DP solution to the basic knapsack problem.However even after reading through a variety of tutorials ,its still beyond my comprehension.I am taking an algorithmics course ...
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### Minimize a sum with a weight constraint

We are given N sets, each of which has a finite number of pairs $(x_i,y_i)$. $M_1=\{(0,0), (x_{1,1},y_{1,1}), ...\}$ ... $M_N=\{(0,0), (x_{1,N},y_{1,N}), ...\}$ ...
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### Dynamic Programming - Thief Variation Probem

I've encountered a Dynamic Programming problem which is a variation of the thief one. Say you are a thief and you are given a number of houses in a row you should rob : House_1,House_2 \dots ...
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### FPT algorithm for Knapsack

I tried to search whether Knapsack (the decision version) has a FPT algorithm, but didn't find anything on the topic. Can someone help with a reference?
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### Finding all feasible solutions to a multiple-knapsack program

How can I find all feasible solutions to a 0-1 integer program that I have based on a knapsack-style problem? I have $n$ items and $m$ knapsacks. Each knapsack has a space limitation and each item ...
I'm trying to find a method to randomly split a connected planar graph $G$ into two connected components, such that the sum of the weights of vertices in each component are relatively close. (If there ...
I'm trying to understand why an alternate formula for finding the best $p$ items in a 0/1 knapsack with $n$ items isn't working. The formula was proposed by @Carlos Linares López in this answer: ...