# Questions tagged [packing]

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### Packing rectangles into convex polygons but without rotations

I am interested in the problem of packing identical copies of (2 dimensional) rectangles into a convex (2 dimensional) polygon without overlaps. In my problem you are not allowed to rotate the ...
938 views

### What is the complexity of rectangle packing when rotations are allowed?

In the rectangle packing problem, one is given a set of rectangles $\{r_1,\dots,r_n\}$ and bounding rectangle $R$. The task is to find a placement of $r_1,\ldots,r_n$ inside $R$ such that none of ...
1k views

### Is the following problem NP hard?

Consider a collection of sets $F=\{F_1,F_2,\dotsc,F_n\}$ over a base set $U=\{e_1,e_2,\dotsc,e_n\}$ where $|F_i|$ $\ll$ $n$ and $e_i \in F_i$, and let $k$ be a positive integer. The goal is to find ...
750 views

### Fitting minimum number of rectangles of width/height 1 into a 2D grid

Consider a problem in which you are given a 2D grid (e.g. a chessboard) where certain squares are occupied and you need to put the minimum number of non-overlapping rectangles of any size w x h where ...
812 views

### NP-Hardness of a special case of orthogonal packing problem

Let $V$ be a set of $D$-dimensional rectangular shapes. For $d \in \{1,...,D\}$ and $v \in V$, $w_d(v) \in \mathbb{Q}^{+}$ describes the length of $v$ in the dimension $d$. The same notation is used ...
441 views

### Does this bin packing problem have a name?

My problem is related to the standard bin packing problem (where you have bins of capacity $1$, items of capacity $(0,1]$, and want to pack the items into as few bins as possible), but there are a ...
205 views

### What is the complexity of pallet loading for identical non-rectangular objects?

In the pallet loading problem, we are asked to place a set of small identical 2-D rigid objects into a large bounding rectangle such that no two objects overlap. This problem is a special case of the ...
683 views

### Hardness of Approximation results for Special Set Packing Problem Wanted

Is there any inapproximability result for the following NP-hard problem, which is a special case of the weighted Set Packing Problem? The general Set Packing Problem would be: Given A Collection of ...
693 views

### Bin Packing with uniform size constraints

Consider the following version of the Bin Packing problem: We are given $k$ unit-size bins and $n$ items with sizes $\epsilon < a_i \le 1$ for $1 \le i \le n$. Is it possible to pack items in bins? ...
184 views

### Does the following type of hitting problem have a name?

Given a ground set, say $[n]=\{1,2,\dots,n\}$, and a collection of subset families $\mathcal F_i\subseteq 2^{[n]}$, $i=1,2,\dots,m$, I want to select $m$ sets $B_i\in\mathcal F_i$ such that the ...
231 views

### $\rho OPT + k$ approximation for bin packing (Unpublished result of David P. Williamson)

I am currently stuck on Exercise 5.12 in this book, which is apparently an unpublished result of David P. Williamson according to the book notes. The problem asks to use randomized rounding and first ...
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### Is set packing easier when the sets are squares?

I am interested in the following problem: ...
212 views

### What is the current "state-of-the-art" solver for quadratic knapsack problems?

New to this forum, so please let me know if my question format is incorrect. For linear KP with $n$ items and $c$ capacity, dynamic programming can find exact solutions in $\mathcal{O}(nc)$. I have ...
107 views

### Reduction from a geometric decision problem to its maximization problem

I am interested in the following NP-complete decision problem: ...
69 views

### Complexity of Finding Optimal Synergistic Set Packings

Motivation: While developing tools for fast execution of machine learning workflows, we realized that many workflows require intermediate results -- sometimes we should cache these results, and ...
69 views

### Bin packing where each item must occur in $k$ bins

I am looking for information on a generalization of bin-packing in which each item should appear in exactly $k$ different bins, for some positive integer $k$. The standard bin packing problem ...
264 views

### Variation of bin packing

My problem is related to the standard bin packing problem, but in my case each item has a value, and the objective is to minimise the number of bins used to pack all the items PLUS the sum of the ...
221 views

### Is $\{0,1\}$-Vector bin packing NP-Hard when vectors have constant dimension?

The paper https://cs.brown.edu/people/seny/pubs/vbponline.pdf discusses $\{0,1\}$-Vector Bin packing in the online setting and give lower bounds. However, they do not mention anything about the ...
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### Using bin-packing algorithms to approximate maximum-makespan

Bin-packing (BP) and maximum-makespan (MM) are dual problems. In both problems, the input can be defined as a set $S$ of positive integers, and the output is a partition of $S$. In BP, there is a ...
34 views

### Bin Packing And Preemptive Multi-Core Scheduling

I am trying to solve a preemptive multi-core scheduling problem, where the input is the tasks. The number of cores M can be decided after seeing the input tasks. I ...
164 views

### Is pooling-aware bin packing NP-Hard?

I am unable to prove whether the following problem is NP-Hard. It seems like a bin-packing or a partition problem, without being close enough to either of them (at least I do not see the reduction to ...
126 views

### Maximize number of bins and minimize cost of elements chosen from a set

I am considering the following problem: there is a set of elements $S$ where each element is assigned to a bin $B$. The bins are disjoint and their union is $S$. There is also a cost function ...
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### Partitioning based on distribution

Having a set of numbers $S={s_i}$, I want to assign them to bins, $b_i$, such that the sum of items on bins follow a specific distribution. For two bins and uniform distribution, this problem is ...
There are $n$ bins and $m$ balls, $b_i$ where $0<i\le m$. Balls are with different weights $w_i$ and have dependency between them. ball $b_1$ depends on $b_2$, $b_2$ depends on $b_3$, and so on. It ...