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1
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1answer
121 views

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 ...
6
votes
2answers
196 views

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 ...
0
votes
1answer
89 views

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, ...
10
votes
1answer
257 views

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 ...
2
votes
0answers
117 views

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 ...
1
vote
0answers
54 views

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 ...
7
votes
4answers
396 views

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 ...
1
vote
0answers
44 views

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) ...
0
votes
0answers
49 views

partition into unit-interval graphs [duplicate]

I am re-opening this question as i have the following question. I was going through the paper by Farrugia which was mentioned in an answer in that post. Initially i beleived that the follwoing problem ...
12
votes
1answer
423 views

Partition into interval graphs

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 ...
6
votes
2answers
285 views

Partition a graph into 2 connected subgraphs

I'm stumped on a sub problem that I'm working on for my thesis. I need to be able to partition a graph into 2 connected subgraphs of almost equal size. So if there are $m$ vertices in $G$, subgraphs ...
3
votes
0answers
180 views

DAG partitioning for parallel computing

Consider a set of processes ($P=\{p_1, p_2,\dots, p_n \}$) and their data dependencies. Each process $p_i$ has an execution runtime which is denoted by $d_i$. We are interested to parallelize these ...
2
votes
1answer
149 views

Partitioning a matrix into equal-sized regions: finding the maximum

I am facing the following research problem. We are given a matrix $M[1..p,1..p]$ of elements such that: each element has value in the range $[0, \frac 1 j]$, $j <= p$, $j$ is given, the sum of ...
0
votes
1answer
96 views

Planning jobs as partition problem

I think this should be a famous problem but I could not find its name. I have $n$ jobs with size $s_i$ that are offered at time $t_i$ and $k$ machines. How can I assign jobs to machines to minimize ...
-3
votes
1answer
92 views

what problem is this? [closed]

I have this instance: Let's say I have two (could be more) friends, one weighing 200 pounds and another weighing 100 pounds; I won a box with 30 chocolates in a contest and I want to divide among ...
6
votes
0answers
313 views

Partitioning DAG into Paths

What bounds (lower or upper) are known about the complexity of partitioning a Directly Acyclic Graph (DAG) into paths of respective sizes $n_1,\ldots,n_w$, such that to minimize their entropy $n{\cal ...
5
votes
0answers
100 views

Variation on block design/set cover

Given 3 parameters $s, r$ and $t$, where $r \leq t$, I want to construct $t$ sets such that each integer $\{1, \ldots, s\}$ appears in exactly $r$ of these sets. The question is: Is it possible ...
0
votes
1answer
117 views

Expansion vs Sparsest cut

let $G=(V,E)$ and $S\subsetneq V$ then expansion of set $S$ is $$\alpha(S)=\frac{|E(S,\overline{S})|}{\min\{|S|,|\overline{S}|)\}}$$ where $\bar{S}=V\setminus{S}$ and $E(S,\bar{S})$ are edges ...
2
votes
1answer
63 views

partition to min the max number of intersections

Given $n$ items and $m$ customers, each of whom is interested in some subset of the items, partition the set of items among $k$ different stores so that the maximum number of customers visiting any ...
0
votes
1answer
190 views

FPTAS for Number Partition Problem

I've been given a task to implement two algorithms (an exact algorithm and fully polynomial approximation scheme) for number partitioning problem. I found out that I can use some modification of ...
11
votes
0answers
462 views

Intermediate $\mathsf{NP}$-complete problems?

Partition problem is weakly NP-complete since it has polynomial (pseudo-polynomial) time algorithm if input integers are bounded by some polynomial. However, 3-Partition is strongly NP-complete ...
6
votes
1answer
341 views

K-Clustering of a Graph maximizing intra-cluster 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 complete graph G with ...
0
votes
1answer
302 views

Graph building with weighted nodes

I have a set of nodes which can be connected together through arcs. Every node has an associated value, reflecting the "fitness" that this particular node has in the graph. I have to find the best ...
0
votes
0answers
76 views

Partitioning based on distribution

Having a set of numbers $S={s_i}$, I want to assign them to bins, $b_i$, such that the sum of items on bins follow a specific distribution. For two bins and uniform distribution, this problem is ...
4
votes
0answers
400 views

Approximation results for 3-partition

The 3-partition as defined here is a strongly NP-complete decision problem. Consider one optimization problem of 3-partition where the $m$ subsets each have at most three elements and a sum of not ...
3
votes
1answer
156 views

Set partitioning algorithm

I'm a working software engineer and I'm trying to develop some planning software. I have faced the following problem. I have some finite set $ U $ of some distinct elements $ e_i \in U $. I have ...
8
votes
0answers
452 views

Is this minimization problem NP-Complete?

We are given an $n \times (n + k)$ matrix $A$, with entries in GF(2), of the form $A =[I_n\ B]$, where $I_n$ is the $n \times n$ identity matrix, and $B$ has no "zero" rows or columns. The problem is ...
2
votes
0answers
301 views

Post Correspondence Problem “binary” variant

Bounded Post Correspondence Problem is defined as follows: given list of pairs of words $ (x_1,y_1), \ldots, (x_n, y_n) $ and $K$ find sequence of indexes $i_1, \ldots, i_k$, $k \leq K$ so that ...
5
votes
0answers
718 views

Faster pseudo-polynomial time algorithms for PARTITION

I want to partition N given numbers (may or may not be equal) into 2 subsets such that the 2 subsets have sum as close as possible and also the cardinality of the sets are equal (if n is even) or ...