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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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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30 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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0answers
21 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 ...
3
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0answers
90 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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83 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
55 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
16 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
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1answer
69 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 ...
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1answer
73 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} = \...
1
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1answer
84 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
21 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,\...
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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
29 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 ...
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0answers
26 views
Quicksort with lomuto partition - how many repeating elements are too many?
I know that quicksort with Lomuto's partition method faces worst case run-time $\Theta(n^2)$ when there are many repeating elements in the array.
However, I'm trying to figure out - how many ...
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1answer
33 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
63 views
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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0answers
109 views
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
77 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 ...
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1answer
40 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
59 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
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1answer
90 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
30 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
146 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
34 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
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2answers
93 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
24 views
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 ...
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2answers
68 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
votes
1answer
79 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 ...
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1answer
101 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
72 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 ...
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0answers
71 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
71 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 ...
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0answers
16 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
66 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
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1answer
53 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
16 views
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
37 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
399 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 ...
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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 ...
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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
115 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 ...
5
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1answer
151 views
Optimal partitioning of n-tuples
Motivation
Recently I was trying to optimize some API calls and reduced the problem to optimization of a cumulative number of identifiers across all the requests. I put some considerable effort into ...
2
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1answer
966 views
Solve PARTITION-INTO-THREE-SETS in pseudo-polynomial time
Let PARTITION-INTO-THREE-SETS be defined as following:
Input: Positive integers $a_1, ..., a_n$
Problem: Are there three pairwise disjoint sets $I, J, K \subseteq \{1, ..., n\}$ with $I \cup J \cup ...
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0answers
37 views
Partitioning the columns of a matrix
I thought about this problem for a while now and am not able to find a solution for it, be it a direct algorithm or a reduction to a known problem, so I'm asking here:
Suppose you have a matrix $A\in\...
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2answers
113 views
How to generate an instance for an NP_hard proof, where each element has two inputs?
I want to prove the NP-hardness of an scheduling problem. The problem seems to be NP-hard in the ordinary sense, so I am trying with the Partition Problem, precisely the Equal Cardinality Partition (...
2
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1answer
62 views
finding separating words (Nerode)
i have found the equivalence classes of given $R_L$ and i need to find the separating words between the equivalence classes(which i don't know how to do). would appreciate if you could explain to me ...
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1answer
828 views
Array subset division with equal sums
Today I stumbled upon a problem which looked like the partition problem but certainly is different.
Given array of positive integers, guaranteed to not be divisible into two continious subsets of ...
0
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1answer
261 views
Complexity of a set partition derived problem
I am stuck at the complexity of the following problem:
Given a multiset $S = \{x_1,..., x_n\}$ of $n$ integers and a natural number $k$. Can $S$ be partitioned into multisets $S_1,... S_j$ such that ...
