# Questions tagged [packing]

Packing problems are a class of optimization problems in which one has to pack objects together as densely as possible. One could be for example packing rectangles inside a rectangle.

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### Proof of lemma for competitive ratio on subsequence - Strip Packing

In this paper by Azar and Epstein in Section 4 they consider online strip packing of rectangles without rotation in a setting where when a rectangle arrives it has to be lowered monotonically ...
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### Bin packing problem and optimality proof

Let $W$ be an array of weights. Store all the weights of $W$ in bins such that in each bin a heavier weight always go before a lighter weight (if $w_i\in W$ is stored before $w_j\in W$ then ...
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### Looking for algorithm to pack rectangular squares while considering locality

I have a couple of rectangles (~10'000 - 1'000'000) with various integer dimensions where each side is something between 1-10 I got to arrange these (without rotating) into a block of fixed width ...
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### What is the approximation ratio of this bin-backing algorithm?

Consider the following algorithm for bin packing: Initially, sort the items by their size. Put the largest item in a new bin. Fill the bin with small items in ascending order of size, up to the ...
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### Can this special case of bin packing be solved in polynomial time?

Consider a multiset of $n$ integers, where each integer is between $1$ and $3 M$. The sum of all integers is $3 S$. There are three bins. The capacity of each bin is $C = S + M$. Is there a polynomial-...
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### What's the name of this packing problem?

I'm trying to pack sets of intervals, to find distinct buckets of intervals. The buckets should not be overlapping. For example if I have these intervals: ...
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### Optimize stacking time series by offsetting start times (feels like a backpack problem?)

Given a time-series of data collected from a single running process that takes 8 hours to complete: Minute GB of Disk Space Used 0 0 1 8 2 15 3 22 ...Etc. It is sampled every minute, for 8 ...
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### Bin packing when items can be broken

In the bin packing problem, there are some $m$ items of size less than $1$, and they have to be packed into as few as possible bins of size $1$. The problem is NP-hard, but if we are allowed to break ...
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### Packing a sphere with cuboids

This question on the Mathematics SE addresses how to pack a sphere with unit cubes. This addresses how to pack a 2D grid with rectangles. We can pack a sphere with the minimum number of unit cubes $m$ ...
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### One-dimensional packing problem: Optimal decomposition of music structure

I am currently working on my Master thesis on the visualization of music structure and I'm looking to find an optimal description of repetitions found in a piece of music. Problem Description Given a ...
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### Bin packing variant for maximizing value in a bin

I'm creating a tool where, given a list of N items that have a volume (v) and a price (p), ...
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### Optimal way to pack items with multidimensional weight such that the number of items is minimized?

I am given a set of items S = {a1,a2,a3,...,an}. Each item has a corresponding M dimensional bit vector indicating the properties of that item. For example, if item x has corresponding vector: {0, 1, ...
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### Additive approximation to bin packing

The bin packing problem is an NP-hard optimization problem that has many constant-factor approximation algorithms. I am looking for an additive approximation. I.e., given a set $I$ of items and bin ...
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### offline Bin Packing problem with multiple size bins

As per my research on stack overflow communities, This is probably known as cutting stock problem / multiple Knapsack problem (a variant of the bin packing problem) which is NP hard. here are the ...
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### Why we can't have some algorithm to be polynomial if there are generic conditions that make them so?

I explain it better: There are some algorithms that is clearly in NP, also NP-complete, but that under certain conditions they can be solved in polynomial time. An example is Bin Packing, the decision ...
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### Approximate bin-packing?

Let $X_1,...X_n$ denote some bins, and $w_1,...w_m$ some positive real numbers, where $m \in \mathbb{N}$, and the order matters, so e.g. we can't switch the position of $w_n$ and $w_1$. The goal is to ...
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### Pack Paths [Concave and Convex]

I would like to design an algorithm to pack closed paths into a rectangle. An example of one of these paths is below: The rectangle will have a fixed width, but the height will expand to accommodate ...
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### Optimal distribution of N points in non euclidean volume, where each point is furthest away from the others

Given N points, I want to find the optimal configuration for which all the points are as far away from each other as possible. The metric I'm considering is an approximation to the perceived distance ...
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### packing with time-variant weights

This appears to be a knapsack / bin-packing problem, but I seem to have got stuck and could appreciate contributions. Scenario 1: Tough (for me!) There is a one day conference with a set of (4 or ...
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### Hexagon packing algorithm

I'm trying to pack hexagons, within bigger hexagons, as shown here: For this example, I have 5 "children" to put in my "father" hexagon. Each time I've got too many children, I would like to reduce ...
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### Reducing 3 SAT to 3 SET PACKING

I'm trying to prove NP-hardness of 3 SET PACKING, which is a following problem: given a family of sets where each set contains 3 elements, decide whether the family contains k sets that are pairwise ...
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### Greedy algorithm Packing problem

Assume that $A$ is the set of objects such that each object $x_i \in A$ has value $w_i$. We wish to pack these objects into group, each pack containing at least $k$ objects. Our goal is to minimize ...
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### A scheduling problem on an oriented graph with multiple constraints

The problem is the following : Data An oriented graph $(V, E)$ : to be understood as a set of partially ordered tasks A map $d: V -> \mathbb{N}$ : to be understood a function mapping tasks to a ...
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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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### How are the prime numbers encoded in Knuth's example of fitting primes into memory cache?

Could somebody please help me understand what is going on here (in plain English)? I think that $(k \mathbin{\&} 63)$ has the effect of modular division. Is that right? How are the primes encoded /...
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### Finding the maximum possible size of S, where S is a set of pairwise-disjoint subsets of the list, and every element of S sums to k

Say I had a list of numbers in the range of 1-20 for example: [5, 16, 17, 3, 2, 14, 4, 9, 11, 19], and an integer k (let's say k = 40) How would I find the maximum possible size of S, where S is a ...
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### For a given shape, find set of points with the maximum average distance

Within some shape, I want to find a set of points where the distance between each point is maximized. This seems similar to sphere packing to me except that part of the sphere can be outside the shape....
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### Algorithm for packing various shapes inside of a rectangle

Say I am given a rectangle of width $W$ and length $L$. I now have to fit as many regular shapes of area $A$ into this rectangle as possible. For example, if the shape is a circle, I need to fit as ...
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### Non-trivial bin-packing instance with 5 objects

Bin packing problem is a problem, where one has to find the minimum number of bins of size $v$ required to store $n$ objects of sizes $s_1, \ldots, s_n$. Object sizes are never greater than $v$. For ...
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### A problem similar to the Bin packing problem?

I'm working on a problem that is very similar to the bin packing problem, but for me, the objective is to minimize the maximum weight given m bins. The problem statement is: Given n items, each with ...
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### Job scheduling and packing algorithm

I was thinking about developing a daily production work plan algorithm for an enterprise. The problem is as following: There are various tasks that needs to be completed, each has a deadline, a ...
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### Schedule repeating jobs of fixed length and different weights

I have a scheduling problem that I am not sure is a known variant with known algorithms. It does appear like bin packing problem at first, but after imposing the constraints, I am not sure what it ...
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### What does a solution to the Rectangle-Fit problem look like?

I've been learning about NP-Complete problems, and came across the rectangle fit problem. Basically, the rectangle-fit problem is the problem of whether or not a set of 2d rectangles can fit in a ...
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### How to show that there are $\binom{M+K}{M}$ different number of type bins?

In textbook by Vazirani's textbook, chapter 9 about Bin Packing. He give the following lemma. Lemma 9.4 Let $\epsilon >1$ be fixed, and let K be a fixed nonnegative integer. Consider the ...
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### route and/or packing optimization algorithm

What type of problem would this question fall under, are there known algorithms/heuristics for it, what would be good resources to learn more about solving it? Given: a list of items each with a ...
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### How can i fill shelves with products so that I have the maximum amount of sales?

I have to do a project where I write a greedy algorithm to maximize a company's sales. There are 6 shelves, each with 8m length. I have to position 100 items whose length, value and max sales ...