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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Set partitions and integer partitions

Consider an algorithm that takes the input a finite set $X$ and an integer partition $\sum_{i=1}^k n_i=|X|$ and gives output all the set partitions $\left(S_1,\ldots, S_k\right)$ of $S$ satisfying $|...
2 votes
1 answer
37 views

Optimally partitioning the reciprocals of the first n integers

Do we know anything about the hardness of the optimization version of the multiway number partitioning problem when the set of numbers to partition consists of the reciprocals of the first $n$ ...
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Profitable sequence in a $k$-partite DAG

This question is an extension of this one. Let $D(V, A)$ be a $k$-partite DAG; $P = \{ p_k : 1 \leqslant k \leqslant |P| \}$ such that $p_k \cap p_l = \emptyset$, $\forall k,l : k \neq l$, and $\...
1 vote
1 answer
38 views

Shortest paths in $k$-partite DAG

Let $D(V, A)$ be a $k$-partite DAG; $P = \{ p_k : 1 \leqslant k \leqslant |P| \}$ such that $p_k \cap p_l = \emptyset$, $\forall k,l : k \neq l$, and $\bigcup_{1 \leqslant k \leqslant |P|} p_k = V$ ...
2 votes
1 answer
191 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 ...
3 votes
2 answers
116 views

Does there exist an algorithm / software that finds optimal graph partition while enforcing contiguity on a subgraph?

I am interested in the traditional graph partitioning problem, which roughly speaking seeks to obtain a partition of a graph into a number of components, in which each component has about the same ...
9 votes
2 answers
363 views

Is the ABC-partition problem NP-hard?

In the ABC-partition problem, there are three sets $A, B, C$ with $m$ positive integers in each. The sum of all integers is $m T$. The goal is to construct $m$ triplets with the same sum $T$, each of ...
2 votes
1 answer
65 views

Covering a graph with M cliques maximizing total edges weight

I am working on a problem that involves distributing a set of N supplements across a predefined number of meals (M) in a way that maximizes the total number of positive interactions and minimizes ...
2 votes
1 answer
47 views

NP-hardness of partitioning into k sets

Given a multiset $S$ of positive integers and a fixed positive integer $k$, can $S$ be partitioned into $k$ parts with equal sum? For $k = 2$, this is the well-known Partition problem. For general $k$,...
0 votes
1 answer
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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
0 answers
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How does this partitioning problem map to studied problems?

I have a real-world (see background below) problem wherein a set $S = \{A,B,C,D...\}$ needs to be partitioned into set $P = \{ \{A,B\},\{D\},\{E,F\},\{C,G,H\},...\}$ where $P$ is required to have the ...
0 votes
1 answer
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$k$-way number contiguous partitioning

Given a set $S$ of $n$ positive integers $S=\{a_1,\ldots,a_n\}$, can we partition $S$ into $k$ subsets of equal sum such that each subset has contiguous elements from $S$? Here, a contiguous subset is ...
4 votes
1 answer
344 views

Partitioning a graph into connected pairs and triplets

We need to partition an undirected graph into connected subgraphs of size between $2$ and $k$, where $k$ is an integer. When $k=2$, the problem is equivalent to the perfect matching problem which is ...
1 vote
0 answers
38 views

Algorithm for grouping set items into ordered buckets without crossing boundaries between same set items

I'm trying to order some data in real-time (in an API call) where my item count is on the order of a few million. I'm using Go, so my pseudo code may resemble that. My input items look like this: <...
1 vote
0 answers
41 views

Graph partitioning in 2 clusters minimizing between cluster edges

I've been looking for an algorithm which divides an undirected graph into 2 subgraphs. However, unlike most existing work on graph partitioning out there (like METIS), I don't intend to obtain ...
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List of weakly NP-HARD problems

I need a list of at least 10 weakly NP-HARD problems. I already know the Knapsack problem, partition problem and subset sum problem. Please introduce other weakly NP-hard problems to me.
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Hoare's partition original method

So I was reading the Hoare's partition part of the Quicksort wiki and it says: "With respect to this original description, implementations often make minor but important variations. Notably, the ...
2 votes
1 answer
161 views

Partition of two sets for multi-line fitting, NP-hard?

Given two sets of nonnegative numbers $X=\{x_1,...,x_n\}$ and $Y=\{y_1,...,y_n\}$, my problems consists in finding the partition $S \subseteq \{1,...,n\}$ and $\bar{S}=\{1,...,n\}\backslash S$ ...
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reduction from partition to N3DM or balanced 3 partition problem

I want to know how can I reduce Subset Sum or Partition problem to N3DM problem in which each set has exactly 3 elements and same sum. N3DM Problem: https://en.wikipedia.org/wiki/Numerical_3-...
3 votes
1 answer
71 views

Partition a graph into connected subgraphs of 3 vertices each

We need to partition a graph into subgraphs of 3 vertices each, such that every subgraph has at least 2 edges. The problem is similar to the partition into triangles problem (which is NP-complete) but ...
0 votes
1 answer
46 views

Partitioning a set based on a non-equivalence relation

I have a set of $n$ elements, and a binary relation between these elements. However, this is not guaranteed to be an equivalence relation. (Specifically, the elements are line segments in a plane, and ...
1 vote
1 answer
43 views

Is deciding whether a graph admits two vertex-disjoint spanning trees of bounded size difference NP-hard?

I'd like to decide whether, given a connected graph $G = (V, E)$ and an integer $k$ as input, $G$ admits two vertex-disjoint subgraphs $T_1 = (V_1, E_1)$ and $T_2 = (V_2, E_2)$ such that $T_1$ and $...
0 votes
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Is there any upper bound for the number of ways we can partition a multiset, where each part/segment in the partition has distinct elements?

A question is asked in the below link, which asks for the number of cases we can partition a multiset, where each part/segment in the partition has distinct elements. https://math.stackexchange.com/...
1 vote
1 answer
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Reduction from $2$-Partitioning to (simple) pairwise $2$-Partitioning

I'm currently stuck showing $NP$-hardness of a problem of mine. An instance of my problem (I call it (simple) pairwise $2$-Partitioning) is given by the following: Given a set of tupels $B=\{(b_1,1),\...
1 vote
1 answer
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Minimum number of bits to codify arrays with bounded sums

I have a very big set of non-empty arrays of possibly repeated numbers sorted in a weakly-decreasing order where each number belongs to the interval $[1, 55]$ and the sum of the elements of each array ...
0 votes
1 answer
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Maximal Profit of 'legal' cutting of a board

I'm facing this problem for some time now, I've tried a greedy approach yet I result to trying a DP-ish approach, only to get stuck at a standstill. Given a board of length $n$, and an increasing ...
3 votes
1 answer
754 views

Is it possible to randomly allocate items to bins such that each distinct allocation has equal probability?

I'm trying to randomly allocate N indistinguishable items over B indistinguishable bins with unlimited capacity. Each allocation should occur with equal probability. An allocation identifies the ...
1 vote
1 answer
156 views

The modularity difference in Louvain algorithm for community detection in graphs

I am interested in the formula of modularity difference between two partitions considered in the Louvain algorithm for community detection in graphs. I give a formal expression of this question below. ...
1 vote
0 answers
46 views

Minimize sum of products of partition [closed]

I have a set of positive integer numbers $A = \{a_1,...,a_N\}$ and I need to find a partition of $A$ into two sets, such that the sum of their products is minimal, i.e., $$ \min_{X,Y : X \cup Y = A} \...
4 votes
1 answer
161 views

Can almost equal partition problem be solved in polynomial time?

Given a list of positive integers with sum $s$, decide if there is a subset with sum $0.5s$. This is the well-known PARTITION problem, which is NP-hard. What about the following: Given a list of ...
3 votes
1 answer
273 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 ...
0 votes
1 answer
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Complexity of the partition problem with additional constraint

The "classical" partition problem asks whether a given multiset $S$ of positive integers can be partitioned into two subsets $S_1$ and $S_2$ such that the sum of the numbers in $S_1$ equals ...
1 vote
1 answer
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Set partitions into singletons or pairs

I am trying to build an algorithm to compute the partition of a set into singletons or pairs for a set of cardinality 16. I am doing it in MATLAB and came up with codes that either run forever or ...
1 vote
0 answers
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Split Bipartite Graph

I have a bipartite graph $G=(U=\{U_1, U_2,\cdots\}, V=\{V_1, V_2,\cdots\} , E)$ such that edges don't "skip" the $V$ vertices. Meaning, if edge $(U_i, V_j)$ doesn't exist, neither will edges ...
2 votes
1 answer
261 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 $n$? ...
1 vote
0 answers
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"Icebreaker problem": Dividing set into partitions that maximize pairwise shared memberships?

Suppose I have a set of N*L elements. I wish to generate a sequence of k partitions of size L such that I maximize the number of pairs of elements that share membership of the same partition. What ...
19 votes
5 answers
4k views

"partial sorting" algorithms (aka "partitioning")

Context: When trying to tame real-world datasets that contain outliers and noise, the interquartile mean is a handy tool: you sort the data, throw away the top and bottom 25% of the data and take the ...
1 vote
1 answer
481 views

How to partition N numbers into K groups with constraints on the size and sum of each group?

Suppose we are asked to assign $N$ numbers into $K$ mutually exclusive groups. The partition has to satisfy two constraints for each group. The first one is that group $k$ must contain exactly $n_k$ ...
0 votes
1 answer
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Partitions with subsets of limited cardinality

Is there a way to compute the number of partitions such that each set in the partition has a cardinality lower or equal to two? If yes, is there also an efficient algorithm to compute these partitions?...
1 vote
1 answer
54 views

Partition data into two sets of the same size such that the sum of the average distances is maximized

Say I have a set of strings $S=\{s_1, s_2, ..., s_N\}$, and I want to partition $S$ into two sets $S_1$ and $S_2$ equally, i.e., $||S_1|-|S_2||\leq1$. Define the difference of a set as $$Diff(S_k)=\...
3 votes
2 answers
582 views

Proof that Balanced Partition is NP-complete

In the Balanced Partition problem, there are $2m$ nonnegative integers with sum $2s$. The goal is to decide whether they can be partitioned into two subsets of $m$ integers and sum $s$. The problem is ...
1 vote
0 answers
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Proof that there isn't a $c$-additive approximation to Partition Problem

Define Partition to consist of all tuples $x_1,\ldots,x_n$ which can be divided to two groups in which the sums of the two groups are equal. Is there a proof that there isn't an additive approximation ...
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Is there an algorithm to efficiently generate all partitions of a set such that no cell contains fewer than k elements of the set?

I am trying to generate partitions of networks to evaluate clustering algorithms. I know that generating all partitions is infeasible (since they grow with Stirling number of the second kind which get ...
4 votes
1 answer
301 views

Parition a multiset of numbers into two subsets, how to maximize the sum of their medians?

Given a multiset $S$ of numbers, partition it into two subsets $S_1 $ and $S_2$. How to maximize the sum of their medians? For example, the median of {1,2} is ...
1 vote
1 answer
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Similar problem to Subset Sum?

I've been trying to search for a problem which I think could be similar to Subset Sum. The definition of the problem would be as follows: Given k $\in$ $\mathbb{Z}$ and S = {$s_1$,...,$s_n$} s.t. $s_i ...
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0 answers
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Efficient way of partitionning a set into a fixed number of parts

I am looking for a way of efficiently partitionning a set of consecutive integers in to a set where the number $K$ of parts is given. For instance, for the set {1, 2, 3, 4}, I would like the best way ...
2 votes
0 answers
170 views

Is it known whether PARTITION is NP-complete via first order reductions?

The PARTITION decision problem is defined as follows (taken from COMPUTERS AND INTRACTABILITY from Garey and Johnson): Instance: A finite set $A$ and a size $s(a) \in \mathbb{Z}^{+}$ for each $a \in A$...
1 vote
0 answers
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Split the given array into K subsets such that maximum sum of all subsets is minimum

Given an array of $N$ elements, $A$, and a number $K$. ($1 \leq K \leq N$) . Partition the given array into $K$ subsets (they must cover all the elements but can be noncontiguous too). The maximum ...
1 vote
0 answers
36 views

Computational Hardness of the $k$-Partition Problem with identical numbers/objects?

The $k$-Partition Problem is NP hard. I want to know if some slight modification of this problem makes it polynomially solvable. Now consider the set $S=\{a_1,\ldots,a_n\}$ of IDENTICAL numbers/...
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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 ...