All Questions
Tagged with co.combinatorics computational-complexity
216 questions
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Proof for non-existence of short integer program for squares
We do not know if $P=NP$ or not or if there is a superfast integer mutiplication algorithm. But I do not think either assumption is necessary to answer this question.
Is there a way to show within an ...
- 13.9k
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0
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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
4
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0
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155
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Permutation generation problem using swaps
This is motivated by Aaronson's post, Probability of generating a desired permutation by random swaps. I am interested in a related problem where the swaps are given in the input.
We're given as input ...
- 2,993
2
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81
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Degeneracy and the "Linear Degeneracy Testing" problem
The Affine Degeneracy problem is about deciding whether $n$ given points in $\mathbb{R}^d$ (or $\mathbb{Q}^d$) are "in general position". i.e. there is no $d+1$ tuple of points which lies in ...
- 153
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101
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On determinant and permanent of certain homotopy defined simple matrices
Let $A_1,A_2,B_1,B_2$ be four $n\times n$ $0/1$ square matrices where $$\det(A_1)=\det(A_2)=per(A_1)=per(A_2)=1$$
$$\det(B_1)=\det(B_2)=per(B_1)=per(B_2)=0$$
hold ($per$ refers to permanent).
I. What ...
- 13.9k
1
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161
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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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245
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Pancake sorting problem – Is computing f(n) NP-hard?
The so-called Pancake flipping problem first discussed by Jacob E. Goodman here yields two entangled problems:
MIN-SBPR (Sorting By Prefix Reversals) - Given a permutation, find the smallest sequence ...
- 51
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0
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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 ...
2
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1
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209
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Computational complexity and commuting functions, examples and conjectures
History of the question. I was proposing a conjecture here, called Prop. 1. Fedor Pakhomov showed a counter-example. Here I am proposing a slightly weaker version of the conjecture, Prop. 2, that ...
- 469
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225
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Computational complexity and commuting functions
EDIT: in this question, I was proposing a conjecture, Prop. 1. Fedor Pakhomov showed a counter-example. In this new question I propose a slightly weaker conjecture that holds even for that example and ...
- 469
1
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1
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187
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A combinatorial matrix reconstruction problem II
For a positive integer $n$, let an $n$-shuffle be a multiset
$S=[(S_i,d_i)|i=1,\ldots,n]$ of pairs $(S_i,d_i)$, where each
$S_i$ is a multiset of $n$ numbers containing the number $d_i$.
A realization ...
- 3,873
1
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1
answer
209
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Deciding if given number is a permanent of matrix
The permanent of an $n$-by- $n$ matrix $A=\left(a_{i j}\right)$ is defined as
$$
\operatorname{perm}(A)=\sum_{\sigma \in S_{n}} \prod_{i=1}^{n} a_{i, \sigma(i)}
$$
The sum here extends over all ...
3
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1
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271
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The number of $3$-CNF formulas in $n$-variables and the fraction of satisfiable ones
What is the number of $3$-CNF (conjunctive normal form) formulas with $n$ sentential variables and what is the fraction of satisfiable ones? I consider two formulas the same if they are syntactically ...
- 33
11
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1
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410
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Complexity of counting regions in hyperplane arrangements
Let $H_1,\ldots,H_n$ be hyperplanes in $\Bbb R^d$. Denote $\mathcal{H} :=\{H_1,\ldots,H_n\}$ and let $c(\mathcal{H})$ be the number of regions in the complement: $\Bbb R^d\setminus \bigcup H_i$.
...
- 17k
3
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130
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Is counting Latin squares #P-complete?
I feel like I should know the answer to this. I did some Googling and didn't easily find the answer...
Question: Is counting Latin squares #P-complete?
Obviously the corresponding decision problem &...
- 1,327
1
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1
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119
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Problem NP-completeness on a specific graph class
Consider the class of simple connected n/2-regular graphs, n even. Are the maximum clique problem and/or maximum independent set problem NP-complete on such graphs? Is there any known result which ...
- 105
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1
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890
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How hard is it to compute the Davenport constant?
The Davenport constant $D(G)$ of a finite abelian group $(G,+)$ is the least positive integer $k$ such that every sequence in $G$ of length $k$ has a zero-sum (nonempty) subsequence.
It seems that the ...
- 1,446
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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
22
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2
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6k
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$\mathbf{P} = \mathbf{NP}$, what's the problem?
Let's take the problem of the backpack: $A_1,\ldots ,A_n$ the weights that are integers, and we want to know if we can achieve a total weight of $V$.
We take $$I=\dfrac{1}{2\pi}\int_0^{2\pi} \exp(-iVt)...
- 4,074
10
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0
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454
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Fast method to verify if a point belongs to a given convex $d$-polytope
We are given a $d$-dimensional convex polytope $P\in\mathbb{R}^d$. Assume we have all the supporting hyperplanes describing $P$ and its vertices. Let $S$ be a sequence of $n\gg 1$ points $\mathbb{R}^d$...
- 1,677
4
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1
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362
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Lower bound on the number of solutions of 2SAT
To compute the number of solutions of a 2SAT is a hard problem. Is there some nontrivial lower or upper bound on this number in terms of a “coarse-grained” description of the Boolean formula, for ...
- 1,207
2
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91
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Blind construction of planar graph with additive spanning tree count
Suppose we have two planar graphs $G_1$ and $G_2$ with number of spanning tree count $P_1$ and $P_2$ respectively then there is an easy construction which gives a planar graph with spanning tree count ...
- 13.9k
7
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203
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Upper bound on the number of perfect matchings in $K_{3,3}$-free graphs
Let $G=(V,E)$ be a graph with an even number of vertices $|V|=2n$. Assume that $G$ is $K_{3,3}$-free i.e. it does not contain a graph isomorphic to biclique $K_{3,3}$. A perfect matching of $G$ is a ...
- 965
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237
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Size of 3-SAT assignments
Let $F(N,M)$ be the set of 3-SAT formula with $N$ variables and $M$ clauses. For a given formula $f\in F(N,M)$, we can ask for the set $s_f$ of truth assignments that satisfy $f$. (If $f$ is ...
- 3,979
4
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182
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Determine the minimal elements of a Dynkin system generated by a finite set of finite sets
(This is a refined version of https://cs.stackexchange.com/q/144371)
Let $\Omega$ be a finite set. A Dynkin system on $\Omega$ is a subset of the power set of $\Omega$ containing $\Omega$, which is ...
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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
2
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0
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64
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Polynomial-time algorithm for uniformly sampling the $n$-slice of a context-free language
Let $L\subset \Sigma^*$ be a context-free language. The $n$-slice is the intersection $L\cap \Sigma^n$ for a non-negative integer $n$.
Is there a polynomial-time algorithm for uniformly sampling from ...
- 21
1
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1
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218
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What is this invariant graph?
Let $G$ be a simple graph (finite or infinite), $[n]\mathrel{:=}\{1,...,n\}$. Define the function:
$$\varepsilon_n(G)\mathrel{:=}\min_\phi{\lvert{\operatorname{dom} (\phi)}\rvert},$$
where $\phi$ is ...
- 107
1
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0
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75
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Subgraph isomorphism problem with linear map
I am working on proving the NP-hardness of a problem by reducing it from the subgraph isomorphism problem. Currently, I can reduce it from the following problem:
Problem 1: Given two graphs $G=(V, E)$ ...
- 101
4
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1
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209
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Finding a binary variable assignment to make a matrix with variables singular (over F_p)
Consider a square matrix defined over a finite field $M\in\mathbb{F}_p^{n\times n}$ having the following form
$$M=\begin{bmatrix}a_{11}+b_{11}x_1&a_{12}+b_{12}x_1&\dots&a_{1n}+b_{1n}x_1\\...
- 153
1
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0
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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
1
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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
3
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132
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Can equality of chromatic symmetric functions of two trees be checked in polynomial time?
Stanley defined chromatic symmetric functions (CSF) in 1995 (Advances in Math) where he conjectured that trees can be distinguished by their CSF. However, tree isomorphism is decidable in P (...
- 31
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70
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Isomorphism preserving transformation graph to graph of logarithmic boolean width and bounded degeneracy
The paper On graph classes with logarithmic boolean-width
claims that some graph problems are fixed parameter tractable with parameter
the boolean width.
In particular, boolean-width of the complement ...
- 25.4k
5
votes
1
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276
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NP-hardness of a sequence problem
Given $n$ binary sequences $s_i$ ($1\le i\le n$) with common period $T$. Let $s_i^{t_i}$ denote the sequence obtained by cyclically shifting $s_i$ for $t_i$ bits. The $n$ sequences form a good system ...
- 367
1
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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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147
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Complexity of edge coloring of class 1 graphs
We know that the decision problem of classifying the graphs as class $1$ or class $2$ (with respect to edge coloring) is NP-complete. But, suppose we have to prove a graph to be in class $1$. Does it ...
- 2,089
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81
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Number of solutions to linear diophantine equations, with natural coefficients in a box
Let c, k, d $ \in \mathbb{N} $, let a, x $ \in \mathbb{N}^k $ suppose for all i $ \leq $ k, $ x_i \leq d $, $ a_i \in \mathcal{O}(d2^i) $ and $ \sum{a_ix_i} = c $ my question is for the value of c ...
3
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1
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96
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What is known about computing all binary error correcting codes of given parameters?
Define a binary $(n, M, 2e + 1)$ code to be a code $C$ having $M$ code words in $\mathbb{F}_2^n$ whose minimum distance is $2e + 1$.
Are there any sources about using algorithms to find all given ...
- 143
1
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1
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157
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Is the graph minicut with the node cardinality constraint NP-hard?
I wonder whether the following problem is a well-studied NP-hard problem?
Get a graph $G$ and a number $k$, we partition the graph $G$ into two components where each component should have at most $k$ ...
- 111
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301
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The expressiveness of functions computable on trees
Motivation:
Let's define a function computable on a $k$-ary tree as a function composed with simpler computable functions defined at each node such that a function of this kind defined on a binary ...
- 3,871
1
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1
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137
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Complexity of edge coloring graphs with $\Delta(G) \ge n/3$ assuming the overfull conjecture
Closely related to this on cstheory.
Let $G$ be graph of order $n$ with $\Delta(G) \ge n/3$.
Assume the overfull conjecture.
Can we edge color $G$ with minimal number of colors in polynomial time?
...
- 25.4k
2
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62
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A combinatorial question about encoding the subsets of logarithmic-bounded cardinality
Let $k \in \mathbb N - \{0\}$ and $f(n) = \binom n 0 + \binom n 1 + \dotsc + \binom n {\log^k n}$.
Our question is:
$f(n) = o(2^{\log^{k+1} \ (n)})$ or $f(n) = \Theta(2^{\log^{k+1} \ (n)})$, which ...
- 193
1
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0
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45
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Coloration of an interval graph with constraints [closed]
Given an interval graph that represents a set of tasks, in a given period of time, to be assigned to a set of employees, the objective is to find a minimum coloring of this graph such that the total ...
- 11
3
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1
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182
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Edge coloring graphs is in P?
It is known that there exist polynomial time algorithm to approximate the Lovasz number or the supremum of Shannon capacity of graphs.
By Vizing's theorem, the graph $G$ has only two chromatic ...
- 2,089
4
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1
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166
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Complementing the red and blue boolean cube?
Given a boolean $0/1$ cube in $n$ dimensions with $2^{n-1}$ red and $2^{n-1}$ blue points can we complement the cube (blue becomes red and vice versa) in $\operatorname{poly}(n)$ transformations?
...
- 13.9k
0
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0
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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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1
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214
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Graphs with Hermitian Unitary Edge Weights
Very recently, Hao Huang proved the Sensitivity Conjecture, which had been open for 30 years or so. Huang's proof is surprisingly short and easy. Here is Huang's preprint, a discussion on Scott ...
3
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2
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235
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Strong chromatic index of some cubic graphs
Edit 2019 June 26 New computer evidence forces us to revise our guesses relating strong chromatic index and girth
Edit 2019 June 25 Some mistakes have been corrected. Question 2 has changed.
...
- 1,018
2
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1
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140
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Matrix completion problem with determinant condition?
Given two $\{0,1\}^{n\times n}$ matrices $L$ and $M$ and an integer $m$ is there a polynomial in $n$ algorithm to find a $\{-1,0,+1\}$ matrix $T$ such that $$\mathsf{det}(L+T\odot M)=m$$ where $\odot$ ...
- 13.9k