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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1answer
38 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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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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1answer
60 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
38 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
41 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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20 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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1answer
47 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 ...
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1answer
22 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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87 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 ...
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27 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 ...
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78 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" ...
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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 ...
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50 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 ...
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1answer
49 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
64 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 ...
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1answer
69 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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60 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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53 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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14 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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1answer
63 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 $\sum_{i\in[n]}a_i = \sum_{i\in[n]}b_i = 2 S$,...
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35 views

Forming groups of people such that no group has two people that dislike each other

I had an assignment for the graph theory unit of my data structures course. The problem was given as follows: Every person in a class has at least one other person that they dislike and will not ...
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1answer
48 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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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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1answer
30 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 ...
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322 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 ...
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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 ...
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17 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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1answer
135 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 ...
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1answer
653 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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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
107 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 (...
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1answer
56 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
364 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 ...
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1answer
150 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 ...
1
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1answer
72 views

Karp hardness of directed monochromatic triangle problem

Monochromatic problem is a classic NP-complete problem. Does the complexity stay NP-complete if we use directed graph? DIRECTED MONOCHROMATIC TRIANGLE problem: Input: A digraph $G(V,A)$ ...
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1answer
137 views

Is this an $NP$-complete problem: Product-2-Partition

I want to prove the NP-hardness of my problem P in scheduling. I was trying with Partition, 3-Partition and Subset product, But neither was successful. Now, I can reduce a problem, say PRODUCT-2-...
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1answer
138 views

Can interval partition solve by sort by different approach

The interval partitioning problem is described as follows: Given a set {1, 2, …, n} of n requests, where ith request starts at time s(i) and finishes at time f(i), find the minimum number of ...
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1answer
68 views

Dijkstra Partitioning Algorithm : Special Case

I have been exploring Dikstra partitioning Algorithm. Below are my given: R = Red W = White B = Blue I have this unpartitioned array. ...
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1answer
772 views

Divide an array into two sub arrays such that their sums are equal and possibly maximum

Given an array A, we should partition A into two subarrays whose sums are equal, and that maximizes this sum. We are free to omit items from the subarrays. For example, ...
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1answer
64 views

Proving that an equal partition does not exist

We are given a set of $n$ numbers and want to know whether it can be partitioned to two sets with an equal sum. To prove that an equal partition exists, it is sufficient to show a partition. What is ...
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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 ...
3
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1answer
41 views

Is there an efficient algorithm for WEAK-PARTITION?

Suppose that we are given a list of non-zero integers $(a_1,...,a_n)$ and we want to decide whether there exist $(x_1,...,x_n)$ such that $x_1a_1 + x_2a_2 + ... + x_na_n = 0$, $x_i$ $\in$ $\{-1,0,1\}...
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426 views

Question regarding Hoare's partitioning scheme and a slight modification to it

This is the pseudocode on wikipedia for Hoare's partitioning scheme: ...
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111 views

Determine if an array is divisible in pairs of equal sum

I'm currently studying the time complexity of the solution provided by my teacher to this problem, and I can't understand the logic: Let A be an unordered array of positive natural numbers. Write ...
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80 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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443 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. ...
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1answer
67 views

sub-optimal but fast partition generation

I have a set of N integers that I want to partition into m subsets. I want these subsets to be well-balanced wrt some criterion say that minimize the max difference between the size of all subsets. ...
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94 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 $...