# Questions tagged [partition-problem]

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### 3-partition problem is NP-complete. How about “N-partition problem”?

It is well known that 3-partition problem is strongly NP-complete. (https://en.wikipedia.org/wiki/3-partition_problem) Then, My question is, how about "N-partition problem"? Is it NP-...
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### Partitioning a connected polygon into connected pieces of equal area

Armaselu and Daescu (TCS, 2015) present algorithms that, given a convex polygon $P$ and an integer $m$ (which must be a power of $2$), return a partition of $P$ into $m$ convex polygons with the same ...
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### A k-approximation to k-way number partitioning

The $k$-way number partitioning problem accepts as input a multiset $S$ of positive numbers, and returns a partition of $S$ into $k$ subsets such that the subset sums are as nearly-equal as possible, ...
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### 3-partition problem without the restriction to triplets

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 ...
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### A partition problem with order constraints

In the OrderedPartition problem, the input is two sequences of $n$ positive integers, $(a_i)_{i\in [n]}$ and $(b_i)_{i\in [n]}$. The output is a partition of the ...
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### Partition refinement in transition state systems (bisimulation contraction)

I am trying to understand bisimulation contraction of Kripke models. I have read these lecture slides and this Wikipedia page, but I still don't fully understand it. I can understand that the two ...
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### Maximal Clique partition of vertices with smallest number of cut edges

I am given a simple undirected graph $G(V, E)$. I want to partition $V$ into $b$ Maximal cliques: $\{C_1, C_2, ..., C_b\}$ such that the number of edges that cut across two cliques is the minimum. $b$ ...
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### Max-sum graph-partition for maximizing intra-edge weights?

I would like to know if the following problem has already been studied, and if so how is it called. In particular I'm interested in approximability results. Input: A graph G with negative or non-...
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### Variant of Subset Sum Problem with Changing Bound

Given a sequence of decreasing integers, i.e., $a_1 \geq a_2 \geq \cdots \geq a_T$ and a positive real $k\geq 1$, find a subset $S$ such that $$\max_{S\subseteq \{1,\ldots,T\}} \sum_{i\in S} a_i$$ ...
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### Graph partition with weighted vertices and edges

I am searching for an algorithm to apply to a specific graph partition problem that I am interested in. It feels like a topic that people from CS may have worked on but it is also different from ...
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### Distributing bags of apples equally

Assume you have $N$ bags of apples that you want to equally distribute to $K$ people. Each bag contains $n_i$ apples and you are not allowed to open and divide the bags; you must distribute the bags ...
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### NP-hardness of minimizing sum of complicated objective function

In our research, we faced the following problem optimization problem: Input: a list of $k$ pairs of positive integers $(n_1,d_1), \ldots, (n_k,d_k)$; an integer $m$. Output: $P$, a partition of the ...
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### How many edges are cut in a balanced partition of a graph?

Consider a graph on n nodes and e edges that is partitioned into k "balanced" subgraphs in the sense that each block has an equal number of nodes and the number of cut edges is minimized. Is there a ...
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### Array partitioning with limitations on partition size

Consider an array of bytes. I want to partition the array, such that the following two conditions hold: The number of bytes within each partition (except perhaps the last one) is between L and U, ...
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### NP-hardness of minimizing sum of weighted product

Consider a total of $d$ items, $\{I_1,I_2,\cdots,I_d\}$, each having a weight $w_i$ (a positive integer), and a total of $m$ bins, $\{B_1,B_2,\cdots,B_m\}$. We would like to distribute the items into ...
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### Partition planar graph into connected subgraphs of equal size

Work Jünger, Michael, Gerhard Reinelt, and William R. Pulleyblank. "On partitioning the edges of graphs into connected subgraphs." Journal of graph theory 9.4 (1985): 539-549. states that for 4-...
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### What is a minimum vertex separator as in this definition?

In a research paper the following definition appears that I'm not able to understand completely. Let $G=(V,E)$ be an undirected unweighted graph with vertex set $V$ and edge set $E$, no self-loops, ...
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### Another variant of PARTITION

I've got a reduction of the following partition problem to a certain scheduling problem: Input: A list $a_1\leqslant\cdots\leqslant a_n$ of positive integers in non-decreasing order. Question: Does ...
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### Graph partition with objective over intra-partition weights

I have a problem in which I need to find an optimal graph cut that maximizes an objective over weights not on the cut. I have looked at the literature but have not been able to find any similar ...
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### Repartitioning a binary tree

Suppose I have a binary tree $G = (V, E)$ (with undirected edges) that is partitioned into sets of k vertices, where each set of vertices is a connected subgraph of $G$. Additionally, if there are ...
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### Partitioning graphs while minimizing inter-partition edges

I'm working on trying to partition a triangulated graph into connected subgraphs with some guarantees on the number of inter-partition edges. Here's an example of a triangulated graph that has been ...
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### Partition planar graph of vertices with at most degree 3 into connected subgraphs

I'm currently working on my thesis which deals with pathfinding over a Delaunay triangulated graph. I want to be able to partition my Delaunay triangulation into disjoint (regarding vertices) ...
Suppose there is a graph $G=(V,E)$. I want to test if $V$ can be partitioned into two disjoint sets $V_1$ and $V_2$ such that the subgraphs induced by $V_1$ and $V_2$ are unit interval graphs. I know ...