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.
Questions tagged [packing]
84 questions
21
votes
1 answer
How to pack polygons inside another polygon?
I have ordered a few leather sheets from which I would like to build juggling balls by sewing edges together. I'm using the Platonic solids for the shape of the balls.
I can scan the leather sheets and generate a polygon that approximates the shape…
alecail
- 422
- 2
- 10
20
votes
2 answers
How to devise an algorithm to arrange (resizable) windows on the screen to cover as much space as possible?
I would like to write a simple program that accepts a set of windows (width+height) and the screen resolution and outputs an arrangement of those windows on the screen such that the windows take the most space. Therefore it is possible to resize a…
daniel.jackson
- 441
- 2
- 7
19
votes
1 answer
guillotine cuts versus general cuts
Cutting problems are problems where a certain large object should be cut to several small objects. For example, imagine you have a factory that works with large sheets of raw glass, of width $W$ and length $L$. There are several buyers, each of…
Erel Segal-Halevi
- 6,088
- 1
- 25
- 60
12
votes
1 answer
Is packing a bag of presents easier for Rupert than Santa?
Or: Do we need Rupert in order to get presents at all?
Routing issues aside, Santa faces the following problem (many, many times over):
Given a bag with capacity¹ $C$ and a set of presents $\{p_1, \dots, p_n\}$, each with size $s_i$, he wants to…
Raphael
- 73,212
- 30
- 182
- 400
12
votes
1 answer
Is the 0-1 Knapsack problem where value equals weight NP-complete?
I have a problem which I suspect is NP-complete. It is easy to prove that it is NP. My current train of thought revolves around using a reduction from knapsack but it would result in instances of 0-1-Knapsack with the value of every item being equal…
Zeta Two
- 255
- 2
- 6
9
votes
3 answers
How to solve an arrangement problem at the Archive Nationale of France using graph theory?
Good evening! I'm actually doing an internship at the Archives Nationales of France and I encountered a situation I wanted to solve using graphs...
I. The dusty situation
We want to optimize the arrangement of books of my library according to their…
Revolucion for Monica
- 695
- 1
- 7
- 20
9
votes
2 answers
Equivalence of independent set and set packing
According to Wikipedia, the Independent Set problem is a special case of the Set Packing problem. But, it seems to me that these problems are equivalent.
The Independent Set search problem is: given a graph $G(V,E)$ and an integer $n$, find $n$…
Erel Segal-Halevi
- 6,088
- 1
- 25
- 60
8
votes
1 answer
Fitting different rectangles inside a rectangle
I have a fixed rectangle of size X x Y.
I also have a bunch of rectangles of different sizes.
I want to check if these rectangles can fit in the larger X x Y rectangles knowing that one can rotate any one of them by 90°.
Any great algorithm to do…
Gary
- 81
- 1
- 2
8
votes
1 answer
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 largest item that fits.
Close the bin. If some items…
Erel Segal-Halevi
- 6,088
- 1
- 25
- 60
8
votes
1 answer
Relaxed Bin Packing Problem
The problem I have is like this bin packing problem, but instead I have $n$ bins and a collection of items with discrete masses. I need to put at least $m$ kg of stuff in each bin.
Is there an efficient way of doing this? Is there a way that will…
Lucas
- 201
- 1
- 2
7
votes
0 answers
What's the complexity of solving a packing LP?
Linear Programming is in polynomial time weakly
(when numbers are encoded in unary).
AFAIK it remains open if it is possible to solve LP
in polynomial time strongly (when numbers are encoded in binary).
My questions is about a special subclass of…
user1742364
- 642
- 4
- 12
6
votes
0 answers
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 i.e. the set packing problem, although I feel the…
Tim
- 5,035
- 5
- 37
- 71
6
votes
1 answer
Dividing bins into segments
This may be a question with a well known answer, but I've been thinking on it for two days, and can't quite come up with a satisfactory answer.
Consider the problem of dividing $p n$ bins numbered $1$ through $pn$ into $p m + 1$ segments by placing…
notwatson
- 63
- 3
6
votes
1 answer
Bin packing with twin items
Assume we are given $k$ bins of capacity $b$ and $n$ items with integral sizes $x_1,\dots,x_n$. The bin packing problem is to decide whether there exists an assignment of items to bins such that no bin exceeds its capacity. In this standard from,…
Dennis Kraft
- 618
- 5
- 12
6
votes
2 answers
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-time algorithm to decide whether all integers…
Erel Segal-Halevi
- 6,088
- 1
- 25
- 60