Questions tagged [partitions]
A partition or partitioning of a set A is a collection of disjoint sets whose union yields A.
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Find the minimum sum of distances between sets of points to a straight line in a plane
Given $n$ dots on a plane, such as: n couples ($x_i$,$y_i$)
I would like to find a line parallel to y-axis ( $x=b$ ), such that the sum of all of the point's distances from that line will be minimal
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Is this solution for the Partition Problem right?
I've tried to solve the Partition Problem for 2 subsets minimizing the difference in the sum between the two subsets. I've coded the following:
...
1
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1answer
35 views
Bounds on graph partitioning
I am trying to find some generic upper bounds on the "standard" graph partitioning problem. Say I have a graph with $|V|$ vertices and $|E|$ edges I want to partition it into two equal $N/...
2
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2answers
79 views
Path graph partitioning, minimizing cut while minimizing maximum total node weight in each part
Suppose there is a path (linear) graph $G = (V, E)$ where $V = \{0, \ldots, n - 1\}$ and $E=\{(0, 1), (1, 2), \ldots, (n - 2, n - 1)\}$, with edge weights $w_e : E \to \mathbb{N}$ and vertex weights $...
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2answers
51 views
Maximise the value of the minimum weight intra edge
I've been doing review problems for a midterm and I came across this one
problem that I haven't been able to solve. The problem essentially says
that given a complete graph $G=(V,E)$ partition the ...
3
votes
1answer
74 views
If a solution to Partition is known to exist, can it be found in polynomial time?
In the Partition problem, there is a set of integers, and the goal is to decide whether it can be partitioned into two sets of equal sum. This problem is known to be NP-complete.
Suppose we are given ...
2
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2answers
215 views
Partitioning a graph into subgraphs with overlapping nodes
I'd like to partition a graph into subgraphs with overlapping nodes.
To do a simple partition into two, I could use kernighan_lin_bisection algorithm available in ...
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2answers
70 views
How to reduce the original partition problem to one of its variation?
Here's a statement of the set partition problem:
The set partition problem takes as input a set $S = \{ a_1, a_2, ..., a_n \}$(all positive integers). Can $S$ be partitioned into two sets $A$ and $B$ ...
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16 views
Showing that Subset Sum Reduces to 3-Partition [duplicate]
Given a set of integers S (positive and negative, may contain duplicates) can S be divided into three disjoint subsets that all sum to the same value? Prove this problem is NP-complete.
This is a ...
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34 views
Optimal partitioning of n-arrays
You're given N integer arrays. Each array can have different size and contains unique values. However same integers can be found in different arrays.
The goal is to partition those arrays into K ...
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23 views
Partition graph in a way that minimizes inter-partition edges
I have a graph in which certain vertices are labeled. I need to assign labels to all of the other vertices in a way that minimizes the number of edges between different-label vertices.
How can I do ...
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102 views
Reduction from 3-partition to ABC-partition
The ABC-partition problem is a variant of 3-partition in which, instead of a single set $S$ with $3 m$ positive integers, there are three sets $A, B, C$ with $m$ positive integers in each. The goal is ...
2
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93 views
3-partition problem without the restriction to triplets [closed]
In the standard 3-partition problem, there are $3 m$ integers, their sum is $m T$, and they have to be partitioned into $m$ subsets of sum $T$ and size $3$.
Consider the variant without the ...
2
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1answer
57 views
Minimum cardinality graph partition that removes specific edges
I have a computational problem that I want to solve. I'm not sure if it has already been studied in literature, or if so under what name. I'd appreciate any pointers to literature or suggestions for ...
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1answer
32 views
Multi-dimensional Knapsack with Minimum Value constraints for Dimensions
In MDK, we have a vector $W = \{W_1, W_2, ..., W_d\}$ where each element corresponds to the maximum weight for the respective dimension in the knapsack.
I want to add a conditional constraint: $V = {...
2
votes
1answer
141 views
Finding all partitions of a grid into k connected components
I am working on floor planing on small orthogonal grids. I want to partition a given $m \times n$ grid into $k$ (where $k \leq nm$, but usually $k \ll nm$) connected components in all possible ways so ...
1
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1answer
143 views
Polynomial-Time reduction from Partition to MakeSpan
Partition Problem:
Input: $A:=$ {$a_{1}, ..., a_{n} $}. $a_{i} \in \mathbb{N}$ $\forall i \in$ $\{1, \ldots, n\}$.
Question: Exists a subset $A_{1} \subset A$ with: $\sum_{a_{i} \in A_{1}} a_{i} = \...
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1answer
172 views
How can I improve my algorithm for finding optimally balanced P-way partitioning of array
I have an array of $N$ weights $w_i$, say $w_i=\{4, 5, 12, 16, 3, 10, 1\}$, and I need to divide this array into $P$ partitions such that partitions are optimally balanced, i.e. that maximum sum of ...
2
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1answer
29 views
Partitioning tuples
Given are tuples $(a_{11},\dots,a_{1k}), (a_{21},\dots,a_{2k}), \dots, (a_{n1},\dots,a_{nk})$. We want to know if there is a partition of the tuples into two parts, so that for every coordinate $i=1,\...
1
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1answer
48 views
For which values of $m$ is $m$-partition hard?
Consider the following reduction: $m$-colouring to $m$-partition.
Define an $m$-partition as: Given an undirected graph $G = (V, E)$ and an integer $j$. Does there exist a partition of the vertices ...
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2answers
53 views
Number partitioning targeting ratio of subset sums and equal size
I've seen a number of questions and answers related to the partitioning problem of dividing a set into 2 subsets of equal size and sum that use greedy or dynamic programming solutions to get ...
1
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1answer
49 views
Computing the partition function
Are there in any relatively efficient known algorithms to compute the partition function $P(n)$ for a given $n$? What's known about the complexity class of this problem as a function of $n$?
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1answer
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How to prove the NP-completeness of MOD-PARTITION
MOD-PARTITION: Given a set of integers $A={a_1,...,a_n}$, their weights $w = \{w_1, w_2, \dots, w_n\}$ and the number $k$, does there exist a subset $X$ of $A$ such that: $(\sum_{x \in X} w(x) * x) \...
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Prove Product Partition is NP-complete in the strong sense
I am trying to understand how to prove that the Product Partition problem is NP-complete in the strong sense. The problem is similar to the normal Partition problem, except in this case the product of ...
2
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1answer
86 views
Schedule X Classes In N Classrooms
I would really appreciate any thought on this, or under which category does this problem fall (Interval scheduling, Interval partitioning,...)
I am really out of thoughts
I have X number of classes ...
1
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1answer
42 views
Amount of k-partitions of a number
I'm stuck on writing an algorithm for getting the amount of distinct partitions for a number $n$ with the partition being size $k$. It's important that there isn't any repetition in the partitions.
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1answer
76 views
Does Kernighan-Lin algorithm guarantee its partitions to be a connected graph?
Currently I am experimenting with Kernighan-Lin algorithm to produce coarse representation of navigation mesh for hierarchical pathfinding.
Based on the use case, my requirement is that partitions ...
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0answers
21 views
Find the max partition of unique elements where each element corresponds to the set pool containing that element
Given a list of sets:
a b c -> _
c d -> d
b d -> b
a c -> a
a c -> c
The objective is to find the max partition of unique elements with ...
6
votes
1answer
94 views
Partitioning bag of sets such that each set in a group has a unique element
Suppose I have a bag (or multiset) of sets $S = \{s_1, s_2, \dots, s_n\}$ and $\emptyset\notin S$. I wish to partition $S$ into groups of sets such that within each group each set has at least one ...
3
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1answer
32 views
Smoothed analysis of the Partition problem
I am studying smoothed analysis and trying to apply it to the Partition decision problem: given $n$ real numbers with a sum of $2 S$, decide whether there exists a subset with a sum of exactly $S$.
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1answer
177 views
Conditions under which the 3-partition problem is not strongly NP-complete?
I'm a bit confused about the 3-partition problem. More specifically I'm confused about this from the Wikipedia article:
Let B denote the (desired) sum of each subset Si, or equivalently, let the ...
3
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1answer
39 views
Computing minimum partition of poset of $N$ intervals into chains in $o(N^{2.5})$ time?
Consider a set $P$ of $N$ intervals $\{I_i = (l_i, r_i)\}$ partially ordered according the standard interval order: $I_i < I_j$ iff $r_i \le l_j$.
I want to find a minimum cardinality partition of ...
4
votes
2answers
99 views
Equal partition up to one integer
In the partition problem, the task is to partition $n$ given integers into two subsets $A$ and $B$ with equal sum. This problem is known to be NP-hard, but it becomes easy if the "equal sum" ...
2
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0answers
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Seeking an algorithm for finding the partition of data on an interval that maximizes the minimum fitness among the blocks
In the paper "An algorithm for optimal partitioning of data on an interval" (link) the authors describe an algorithm for partitioning data on an interval to maximize a fitness function. The fitness ...
0
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2answers
133 views
How to partition disagreeable people into compatible groups
We have a number of people that must be partitioned into groups, but there may be people that dislike other individuals. Partition the people into the minimum number of groups such that no person is ...
3
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1answer
136 views
Finding partition with maximum number of edges between sets
Given a graph (say in adjacency list form), is there an algorithm to find a partition of vertices such that the number of edges between the two sets of the partition is the maximum possible?
For ...
0
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1answer
138 views
Weighted graph clustering with maximum size constraint
I'm currently trying to solve a clustering problem. I need to cluster/partition an undirected weighted graph into groups that are restricted to size n.
I have ...
2
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1answer
76 views
Divide grid so that each box has only one object
Is there an efficient algorithm to divide a 2D space, which contains several different sized rectangles, such that each partition has only a single object. Please see the attached image.
I have come ...
0
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0answers
77 views
Directed Acyclic Graph partition into minimum subgraphs with a constraint
I have this problem, not sure there is a name for it, wherein a Directed Acyclic Graph has different colored nodes. The idea is to partition it into minimum number of subgraphs with the following 2 ...
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2answers
74 views
Sequential subsequence removal with arbitrary predicate
I want to extract sub-sequences from a sequence of float values.
The "scale" and range of these values is arbitrary (as I can manipulate it at will) but the "shape" is consistent.
For a visual ...
0
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0answers
19 views
Evaluating clustering/partitioning quality
I'm wondering what are the most common/recognized methods to assess the quality of a clustering.
That is because I have developed a tool that can cluster/partition a network (in this case, a public ...
5
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1answer
69 views
NP-hardness even with perturbations
Consider the following problem, which can be called "2-SET-PARTITION":
Given two sets of positive numbers, $a_1,\ldots,a_n$ and $b_1,\ldots,b_n$, where $\sum_{i\in[n]}a_i = \sum_{i\in[n]}b_i = 2 S$,...
4
votes
1answer
54 views
Partitioning a set so both parts have sum at least $c$ times the total sum
Let $c\in(0,1/2]$ be a constant. Given a set of positive integers with sum $S$, is there a partition into two subsets such that both subsets have sum at least $cS$?
If $c=1/2$, this is the famous ...
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0answers
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Balanced $\epsilon$-separated partitioning by a hyperplane
Suppose we have $m$ points in $R^n$ and $\epsilon>0$ is a given constant. How can we find a hyperplane that the number of points that are $\epsilon$-close to it is minimum, with the constraint that ...
2
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1answer
43 views
Partitioning a graph with specific constraints
We have an exercise where we need to find the partitions G[V1] and G[V2] of a graph G=(V,E), that fulfill the following constraints. We also know that there exists at least one partition that fulfills ...
3
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2answers
507 views
Is finding Kth largest element using selection algorithm taking O(n) only if K is fixed?
Wikipedia here https://en.m.wikipedia.org/wiki/Selection_algorithm shows an algorithm using sort of quicksort.. in order to find Kth largest or smallest element taking O(n) time only on average. The ...
1
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1answer
49 views
Partitioning a boolean circuit for automatic parallelization
tl;dr: I have a problem where I have a Boolean circuit and need to implement it with very specific single-thread primitives, such that SIMD computation is significantly cheaper after a threshold. I'm ...
0
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1answer
1k views
Partition array into k subsets
We are given an array and a number K. Partition array into K subsets such that
let MaxSum be the maximum sum of among subsets.
We have to minimize summation =$$\sum_{i=1}^{k}MaxSum-sum(i) $$
Is ...
3
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0answers
138 views
Split a graph into 2 components with known distribution?
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 ...
2
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0answers
18 views
Is there a more commonly-used term for “multi-pivoting”?
Consider the following computational problem (or rather, task):
Given:
An array of $A$ of (not necessarily distinct) elements from a fully-ordered finite domain
An ordered sequence pivot ...
