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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931 views

Fastest known algorithm for $3$-$\mathrm{Partition}$ problem

$3$-$\mathrm{Partition}$ problem is $\mathsf{NP}$-Complete in a strong sense meaning there is no pseudo-polynomial time algorithm for it unless $\mathsf{P=NP}$. I am looking for the fastest known ...
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661 views

How to distribute items of varying sizes into bins of varying sizes, such that percent utilization across all bins is minimized?

I have a bunch of databases, each having different access patterns, such that each puts a different amount of load on its database cluster. I would like to distribute them around my set of database ...
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0answers
48 views

How to sample a perfect partition uniformly at random?

I would like to sample $n$ integers (of some fixed length, say $k$ bits) uniformly at random, and have them partitioned into two sets of equal sum. Since finding such a perfect partition (even if it ...
4
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0answers
720 views

Even distribution algorithm

Right now I am working on a distributed system that sends messages from single source to many nodes. It is necessary that certain messages are sent to the same node to ensure order of processing. ...
4
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0answers
265 views

Graph optimization problem with multiple objectives/constraints

Let's assume that we have a directed acyclic graph $G = (V, E)$, non-negative vertex weight functions $w_a(v)$ and $w_b(v)$, and a non-negative edge weight function $t(u,v)$. We want to divide ...
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120 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 ...
3
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0answers
201 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
31 views

Partition columns into m groups to maximize absolute value sums

The Task You are given $n$ columns each of length $m$. All values are either $-1$ or $1$. Find an assignment $s$ of each of columns to 1 of $m$ groups in order to maximize the sum of all the absolute ...
2
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1answer
96 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 ...
2
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0answers
32 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 ...
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 ...
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0answers
1k views

k-way set partition problem--dynamic programming solution

I was reading up on the set partition problem on this site of Wikipedia: https://en.m.wikipedia.org/wiki/Partition_problem Among other things, they present a DP approach to solving the equal subset ...
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183 views

Reducing partition to a partition where sum(partition1) = 3 times sum(partition2)

Given the following NP-complete problem: PARTITION Input: A list of positive integers $a_1, a_2, \dots, a_n$. Question: Can the list be partitioned into $2$ parts, $A_1$ and $A_2$, such ...
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0answers
155 views

min cut for multiple partitions

So I am familiar with the standard minimum cut problem in which the goal is to find the smallest possible set of edges in a graph such that, upon their removal, we have two nonempty, disjoint ...
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50 views

Does an FPTAS exist for the multiple subset sum problem when m is fixed and c is not a variable?

From Wikipedia Multiple subset sum: The multiple subset sum problem (MSSP) is a generalization of the subset sum problem (SSP): given a multiset $S$ of $n$ integers, and an integer $m$, the goal is to ...
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154 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 ...
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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 ...
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45 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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96 views

Polygon decomposition into minimum star-shaped polygons

As the title suggests, I'm trying to implement an algorithm to decompose a polygon into the minimum number of star-shaped polygons. I've been searching for quite some time but I can't find any ...
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0answers
103 views

How to enumerate all partitioning of a set to k-subsets of size at most b

I'm looking for an algorithm to generate/enumerate all possibilities for partitioning a set of size $n$ to $k$ non-empty subsets, each with size at most $b$. More specifically, given a set $V$ where $...
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0answers
585 views

Exact algorithm for the partition problem

The partition problem is: given a set of numbers, find a partition to two subsets in which the difference between the sums in each subset is minimized. This optimization problem is NP-hard. The simple ...
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187 views

Computational Geometry: what is the key of the BST in the algorithm " Partitioning a polygon in y-monotone pieces"

The algorithm to partition a polygon into y-monotone pieces is as follows: ...
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187 views

Non-standard partition problem

I've been writing codes to solve a standard partition problem. I've investigated brute force, greedy, Karmarkar-Karp and complete Karmarkar-Karp algorithms. Standard partition problem: divding a ...
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96 views

DISTINCT 3-PARTITION with all integers between $B/4$ and $B/2$

In the definition of 3-PARTITION of Garey&Johnson, the instance is a set of $3m$ integers such that the sum of all these integers is $mB$ and such that each integer is strictly between $B/4$ and $...
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200 views

Enumerate subtrees of a given size in a graph

Given a graph $G$ with $n$ nodes, is there an algorithm to find $m$ subtrees, each with $\lfloor n/m\rfloor$ or $\lceil n/m\rceil$ nodes, such that every node of $G$ is in exactly one tree? Other ...
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48 views

Graph Partition Across Cluster - Minimize Largest Matrix Size

I am writing some code for modeling semi-biologically realistic neural networks, which is to be run/distributed across nodes in a computer cluster. I begin with a very large adjacency matrix (...
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18 views

Partition a graph into subgraphs such that a partition contains up to X number of a particular node type

I have a DAG graph which contains two types of nodes, A and B. I am looking for a graph partitioning algorithm that can partition a graph in sub-graphs such that each sub-graph contains up to X number ...
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22 views

Finding all k-partitions with additional constraints

The partition problem is a very well known one. To partition an integer array into k equal sum partitions. My problem is I want to partition them in such a way that the sum of their partitions equals ...
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31 views

A Variant to "Boats to Save People"

This question is a variant of LeetCode 881. Boats to Save People by removing the restriction of "each boat carries at most two people at the same time" from the original question. Problem ...
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17 views

Partition in a tree shaped distributed network

We are given a synchronic undirected tree shaped network, with $n$ indexed nodes. We know that there is at least one node with at least $\log_k n$ neighbors, $k>1000$, and $k$ is given. We need to ...
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39 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
26 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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1answer
67 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 = {...
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0answers
23 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 ...
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0answers
85 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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20 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 ...
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105 views

$k$-partitioning problem via brute force

Given a set of non-negative integers $A = \{a_1,a_2,\dots,a_N \}$, the $k$-partitioning problem is to partition the numbers into $k$ sets $\{A_1,A_2,\dots, A_k\} $ such that the deviation: $$ \...
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77 views

Is a minimal deviation version of set partition problem NP-Hard?

I want to know if a specific version (and what´s its name?) of the Set Partitioning Problem is NP-Complete, and if, is NP-Hard? Problem: Given a set of elements $e_i$ where $i={1,...,n}$, each with ...
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168 views

Reduction from 3-Partition to a cutting problem

My problem is the following: Input: a set of $m$ non-negative integers $\{b_1,...,b_m\}$ and a parameter $n$ with $n<m$. Output: $n$ sets of 3 numbers Task: Cut the $b_i$'s into $3n$ integers such ...
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1answer
553 views

How to prove this "np-complete" problem?

I have a problem that I need to prove its np-completness: My original question I need to reduce it from some problem, i´m trying to do it from some other than knapsack, that´s why I ask again. I need ...