All Questions
Tagged with co.combinatorics computational-complexity
77 questions with no upvoted or accepted answers
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On an optimization question
Suppose we have a square matrix $M=(1-z)A+zB$ where $A,B$ have integer entries from $\{0,1\}$ with $\det(A)+\det(B)=1$ and $\det(A),\det(B),per(A),per(B)\in\{0,1\}$ and we want to find $z\in[0,1]$ ...
- 13.9k
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185
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Maximum independent set in dense graphs
Let $0 < A < 1$ and $G$ be connected d-regular graph
with degree $d=[A n]$. The density of $G$ is about $A$.
Q1 Are there constraints on $A$ such that finding maximum
independent set of $G$ is ...
- 25.4k
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75
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Subgraph isomorphism problem with linear map
I am working on proving the NP-hardness of a problem by reducing it from the subgraph isomorphism problem. Currently, I can reduce it from the following problem:
Problem 1: Given two graphs $G=(V, E)$ ...
- 101
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209
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Solution to system of linear equations
Input: System of linear equations $$A[x_1,\dots,x_{t}]=b$$ where number of equations is at least number of variables but independence is not guaranteed. However there is atmost one non-negative ...
- 13.9k
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176
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Reduction graph isomorphism to maximum independent set in very dense graph
We got a reduction graph isomorphism to MIS in a very dense graph,
or alternatively negative monotone 2-CNF to MAX-ONEs with a formula
with many clauses.
Let $G,H$ be graphs of order $n$ and adjacency ...
- 25.4k
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147
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The chromatic polynomial of a line graph
Is there a way to obtain the chromatic polynomial of the line graph of a regular simple graph, having known the chromatic polynomial of the graph?
There already exist characterizations of line graph ...
- 2,089
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29
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Existence of Costas array with specified displacment vectors?
Costas array is a set of $n$ points lying on the square of a $n×n$ checkerboard, such that each row or column contains only one point, and that all of the $n(n − 1)/2$ displacement vectors between ...
- 2,993
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78
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Bipartite clustering is NP-hard?
Let $G = (A\cup B, E)$ be a bipartite graph with edge weights $w: E\to \mathbb{R}$. Find a partition $B_1, B_2$ of $B$ and a nonempty disjoint subsets $A_1, A_2$ of $A$ such that $w(A_1,B_1) + w(A_2, ...
- 221
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139
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bounded degree graph colouring.
I was wondering if anyone could provide references on the following:
Is determining the chromatic number of a bounded degree graph APX-complete?
2.I've seen the result that states it is NP-hard to ...
- 903
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28
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The complexity of Max-K interval selection
I came up with the following problem, but do not know how to analyze it.
Let $S$ be an ordered set of integers with size $n$ (i.e., $S=\{1,2,...,n\}$). An interval $INV(a,b)$ covers the elements in $...
- 11
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Some confusion regarding the definition of NPO reduction
I've seen the following definition in a paper on approximation preserving reductions.
Definition:Let $\pi_{1}$ and $\pi_{2}$ be two NPO maximization problems. Then we say that $\pi_{1} \leq_{R} \pi_{...
- 213
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128
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Reducing enumeration of reduced words in general Coxeter groups to another #P-complete problem
There is a polynomial time algorithm that takes as inputs a Coxeter system $(W,S)$ with $S$ finite (but with $W$ not necessarily finite), say encoded as a Coxeter matrix, as well as two reduced words ...
- 2,168
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68
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Polynomial problems in graph classes defined by forbidden induced cyclic subgraphs
Let $C$ be a graph class defined by a finite
number of forbidden induced subgraphs, all
of which are cyclic (contain at least one cycle).
Are there graph problems that can be solved in
polynomial ...
- 25.4k
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74
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TSP: Approximation Ratio of the Double Tree Heuristic after Diagonals have been Removed
In their article "Double-Tree Approximations for the Metric TSP: Is the Best One Good Enough?", Vladimir Deineko and Alexander Tiskin derive a lower bound for the approximation ratio of the double-...
- 13.2k
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157
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Indecomposability of image transformations (pure algebra). Open questions
W-transformations -- definitions
We will consider a class called finite window transformations $\ T:C^\mathbb Z\rightarrow C^\mathbb Z\ $ defined a paragraph below; $\ \mathbb Z\ $ is the ring of ...
- 7,500
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112
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Did anybody come across a computational problem which is related to the notion of cartesian product and is at least NP-hard?
Did anybody come across a computational problem which is related to the notion of cartesian product and is at least NP-hard?
Equally interesting would be to learn about such problems with a non-...
- 41
1
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266
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Multiobjective semidefinite programming
Let $C$ be size $n \times n^{2}$.
Let $B$ be size $2^{g(n)} \times n^{2}$ where $g(n) > n$.
There is only one $\mathcal{1}$ per row of $C$ and remaining entries of $C$ are $\mathcal{0}$.
$B$ is ...
- 800
1
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576
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Minimizing quadratic form over permutations
Let $Q$ be an $n \times n$ real symmetric matrix and $x$ an $n \times 1$ real vector. Consider the following minimization problem:
$\min_{\pi \in S_n} ~(\pi x)^{\rm T} Q (\pi x)$,
where $S_n$ ...
- 1,839
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92
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Algorithm that can solve or approximate the solution to a combination problem
I have a computational problem on my hands and I would like your help.
Here is my problem (simplified)
Let $X = \{x_1, x_2, \ldots, x_n\}$ represent a set of $n$ values.
Each value $x_i$ has a ...
- 1
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59
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NC0 randomness vs. non-uniformity
In
Ajtai and Ben-Or. A theorem on probabilistic constant depth
Computations. STOC '84, 1984
Ajtai and Ben-Or show a non-uniform derandomization of BPAC0.
Is there a similar relation known for ...
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84
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Shattering of a set of binary classifiers
Let $S$ be a set, and let $\mathcal{F}_{S}=\{f:S\to\{-1,+1\}\}$ be a set of different label assignments. Show that $\mathcal{F}_{S}$ shatters at least $|\mathcal{F}_{S}|$ subsets of $S$.
Here is what ...
- 143
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82
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Proving Vizing's and Brooks' theorem using the polynomial approach
It is known that the graph polynomial defined by $\prod_{i<j}(x_i-x_j)$ where the vertices $x_k\ \ , \ \ k=\{1,2\ldots,n\}$ are ordered with respect to some order; can be used to verify the proper ...
- 2,089
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149
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Is 3-coloring bounded degree graphs subexponential: $O(\exp{(\sqrt{n}\log^2{n})})$?
We got an argument that 3-coloring bounded degree graphs is subexponential
with complexity $O(\exp{(\sqrt{n}\log^2{n})})$.
The treewidth of a planar graphs on $n$ vertices is $O(\sqrt{n})$
and 3-...
- 25.4k
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165
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Resources about integral maximization problem
I am looking at the following problem. Given an interval I, and a function f over that interval, find sub-intervals for which:
The sum of the length of the sub-intervals is < k;
The sub-intervals ...
- 109
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87
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maximum weight k-edge problem
Given positive integer $k$ and an undirected graph $(V,E)$, with nonnegative (non-uniform) weights on the nodes. Find $k$ edges whose spanning nodes have the maximum weight.
Is this in P or NP? I ...
- 101
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555
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VC dimension and boolean hypercube subgraphs
Are there any well studied graph theoretic properties that are common to all subgraphs of the boolean hypercubes that have a given VC dimension d.
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220
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What is the computationally simplest way to universally index the set of simple graphs?
If given a simple, integer-labeled, but not necessarily connected, graph $G := (V,E)$ consisting of at least one vertex, i.e. $\lvert \rvert V \lvert \rvert \geq 1$, then is there a function to ...
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