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2
votes
1answer
151 views

Is there any efficient algorithm for computing all semigroups of order n? [closed]

Is there any efficient algorithm for computing all semigroups of order n? I found the following paper which solves a bit different problem. Veronique Froidure and Jean-Eric Pin, "Algorithms for ...
8
votes
0answers
103 views

Recognition of a primitive root

Adleman and McCurley published a paper in 1994 called "Open problems in number theoretic complexity, II" (http://ww.cstheory.com/papers/open.ps.gz) Problem 18 of this list of open problems is about ...
4
votes
0answers
265 views

Construction of a Global Isomorphism(permutation) for Graph Isomorphism using Local Isomorphism

Given two graphs $G, H$ (each has $n$ vertices). We, split $G$ into subgraphs $G_1, G_2... G_x$ (total $x$ vertex set). Similarly,assume $H$ has subgraphs $H_1, H_2... H_x$ (total $x$ vertex set). ...
1
vote
1answer
47 views

How the hardness of hidden subgroup problem in $S_n$ changes as the order of the subgroup grows?

In Normal Subgroup Reconstruction and Quantum Computation Using Group Representations by Hallgren et al. In this paper it is showed that no hidden subgroup algorithm can distinguish the trivial ...
0
votes
0answers
40 views

Measurements in non-Abelian hidden subgroup algorithms

More than one measurement take place in a non-Abelian hidden subgroup algorithm. In this question, I would like to verify whether my understandings about them are right. I am using Andrew Childs' ...
0
votes
0answers
18 views

Choice of basis for non-Abelian hidden subgroup problems

I am going through QUANTUM MECHANICAL ALGORITHMS FOR THE NONABELIAN HIDDEN SUBGROUP PROBLEM by Grigni et al. In section 1.1 it is said that, It is still possible that a clever choice of basis ...
3
votes
2answers
94 views

Is there any hidden subgroup of a symmetric group which can be efficiently determined?

There have been a number of cases where efficient hidden subgroup algorithms have been found for specific non-Abelian groups with very specific structures. Why haven't we found any efficient quantum ...
3
votes
0answers
68 views

Why hidden subgroup problem is easy for very large subgroup?

I am going through QUANTUM MECHANICAL ALGORITHMS FOR THE NONABELIAN HIDDEN SUBGROUP PROBLEM by Grigni et al. On page 2, it is said that solving the hidden subgroup problem becomes very easy when the ...
0
votes
1answer
52 views

Question about discarding the second register in the standard approach of hidden subgroup algorithm

My questions: What does discarding the second register mean for the standard approach of hidden subgroup algorithm? Why does discarding let the first register end up in a mixed state? My ...
11
votes
6answers
2k views

Book for self study of algorithms in group theory

I am a math major interested on TCS. I want to self-study the algorithms, and complexity of them for solving the group theoretical problems like find order of elements, coset enumeration, find ...
-3
votes
1answer
125 views

Dimension of the Fourier transform for $S_5$ [closed]

My question: What is the dimension of the Fourier transform for $S_5$? My effort: The dimensions of the seven irreps of $S_5$ are $1,1,4,4,5,5,6$. According to the notes of Andrew Childs, the ...
-1
votes
1answer
156 views

Are Graph and Group Isomorphism problems random self-reducible?

Are Graph and Group Isomorphism problems known to be random self-reducible? If so is there a good proof? Are there other non-trivial examples of random self-reducibility? Is there a good reference?
0
votes
0answers
72 views

Fourier transformation of the automorphism group of a graph

Following is an example of permutation cycle graph $\Gamma$ for a given permutation $\pi = \left(1\text{ } 2\right) \left(3\text{ } 4\text{ } 5\text{ } 6\right)$. The adjacency matrix $A$ is given ...
3
votes
0answers
78 views

Embedding distortion under group quotient

The high level question is as follows: Suppose some group (here assumed to be a vector space of $\mathbf{F}_2^n$) has a low-distortion embedding into $l_1$. Under what condition does the quotient of ...
3
votes
1answer
105 views

Complexity class for some group and graph homomorphism problems

Given two groups $G_1$ and $G_2$ what is the complexity class in which the following problem belongs? $$\mathsf{Is }|Hom(G_1,G_2)|>0$$ Given two graphs $H_1$ and $H_2$ what is the complexity ...
2
votes
1answer
89 views

Some nuances on Group and Subgroup Isomorphism?

(1) Is it known Group Isomorphism is in $\mathsf{coNP}$ and is the conjecture so? Is there a good reference for $\mathsf{coNP}$-ness in similar situations? (2) Is subgroup isomorphism ...
1
vote
0answers
47 views

First register in the hidden subgroup representations of Simon's and graph isomorphism problems

The Simon's problem involves a function which takes binary strings as inputs. One seeks to find the period of the function which acts on those inputs. In the standard method, the first register has ...
0
votes
0answers
48 views

Complexity of quantum shuffle transform

I define the shuffle permutation matrix by quoting from Hoyer first. Shuffle permutation matrix: The shuffle permutation matrix of dimension $mn \times mn$, denoted by $\Pi_{mn}$ as shorthand for ...
5
votes
1answer
325 views

Quasi-polynomial time algorithm for permutation group isomorphism

Is there a known $n^{\alpha \log n+O(1)}$ algorithm for permutation group isomorphism? Here $n$ is the size of the group, and the isomorphism must be a permutational isomorphism. My hope for such an ...
8
votes
1answer
95 views

Determining what can be achieved by a permutation of elements of a noncommutative group

Fix a finite group $G$. I am interested in the following decision problem: the input is some elements of $G$ with a partial order on them, and the question is whether there is a permutation of the ...
9
votes
2answers
286 views

Number of Automorphisms of a graph for graph isomorphism

Let $G$ and $H$ be two $r$-regular connected graphs of size $n$. Let $A$ be the set of permutations $P$ such that $PGP^{-1}=H$. If $G=H$ then $A$ is the set of automorphisms of $G$. What is the ...
0
votes
2answers
301 views

Complexity of simple undirected graph isomorphism problem

We define a simple undirected graph as a graph where no vertex has a loop and there is only zero or one undirected, unweighted edge between any pair of vertices. My question: What is the complexity ...
10
votes
2answers
369 views

Complexity of computing the order of a permutation group

Given two permutations $g$ and $h$ over $n$ elements (i.e., members of $S_n$), what is the complexity of computing the order of the subgroup generated by $g,h$? Or just of deciding whether the ...
1
vote
0answers
32 views

Coset state of $3$-node graph isomorphism problem

The hidden subgroup representation of a $3$-node graph isomorphism problem is defined over the symmetric group, $G = S_6$. So, any hidden subgroup algorithm that wishes to solve the problem should ...
7
votes
1answer
85 views

Problem with a group as complexity parameter?

I am currently studying a complexity problem related to symmetries, and am considering a study of the parameterized complexity of the problem. In theory, any part of the input can be fixed as a ...
5
votes
1answer
193 views

possible bridge between group growth theory and complexity theory?

RJ Lipton conjectures a link between group growth theory and complexity theory. Group growth theory has undergone rapid advance in the last decade and has many surface similarities/ parallels with ...
3
votes
1answer
163 views

Efficient generation of permutational invariant quantum states

Starting from $|00\cdots 0\rangle$, can permutational invariant quantum states, i.e. the following one: $$ |\psi_n\rangle = \frac1{n!} \sum \prod_{\pi\in S_n} ...
10
votes
1answer
110 views

Diameter of Cayley graphs of subgroups of $S_n$ without inverses

Babai and Seress proved that given a subgroup $G \leq S_n$ and a generating set $S$ of $G$, any permutation in $G$ can be written as a product of generators and their inverses of length ...
17
votes
2answers
437 views

Complexity of the coset intersection problem

Given the symmetry group $S_n$ and two subgroups $G, H\leq S_n$, and $\pi\in S_n$, does $G\pi\cap H=\emptyset$ hold? As far as I know, the problem is known as the coset intersection problem. I am ...
7
votes
0answers
52 views

Explicit error bounds on the abelian hidden subgroup problem

What are some explicit forms for the error probability in the typical quantum abelian hidden subgroup algorithm as a function of oracles queries? Ettinger, Hoyer, and Knill give a result that the ...
4
votes
1answer
156 views

$\ell_1$ norm of Fourier coefficients vector for the hypercube

Let $G$ be the normzlied hypercube graph on $2^d$. It is a Cayley graph and it is well known that its eigenvalues are given by $\lambda_r = 1-2\frac{|r|}{d}$ for every $r \in \{0,1\}^d$. Given a ...
8
votes
1answer
200 views

Recent progress in permutation groups algorithms?

I am interested in algorithms for finite groups as implemented in the GAP package. It seems that all known algorithms in this field deal with permutation groups/matrix groups; two fundamental ones are ...
11
votes
0answers
235 views

Generalizations of the determinant/permanent problem?

A tantalizing open question in computational complexity is to understand the 'behavioral differences' between the determinant and the permanent. While the former is computable in polynomial time with ...
4
votes
2answers
320 views

TCS oriented refs/survey on group theoretic word problem

The word problem for groups was shown to be Turing-complete in 1955 but has many decidable subcases. This problem arose more in mathematical group theory than in theoretical computer science, but now ...
-5
votes
1answer
206 views

Some questions (still) unresolved about braids?

I was looking for interesting questions pertaining to braid theory. I don't know if the following are considered important, but I'd like to ask: (1) in relation with the following link, is it true ...
0
votes
1answer
193 views

Classifying noetherian simple groups by order type?

A (possibly infinite) group $G$ is noetherian if it satisfies the following equivalent conditions: (1) every subgroup of $G$ is finitely generated, (2) there is no infinite strict ascending chain of ...
2
votes
0answers
120 views

Extending the notion of independence

Background I was looking for a formulation of 'free sets' and 'independent sets' from linear algebra that would extend to groups. This question was considered here but I couldn't find a satisfactory ...
3
votes
3answers
126 views

On proving it is hard to compute $g^{rb}$ with knowledge of $r$, given $g, g^a, g^{ab}$

I am trying to prove the following Given $g, g^a, g^{ab}$ it is hard to compute $r, g^r, g^{rb}$, for some arbitrarily chosen value of $r$ where $g ∈ \mathbb{G}, \mathbb{G}$ is a cyclic group ...
10
votes
1answer
95 views

Encoding sets of permutations with a generating set and a set of excluded elements

Polynomial-time algorithms are known for finding generating sets of permutation groups, which is interesting since we can then represent those groups succinctly without giving up on polynomial-time ...
1
vote
0answers
73 views

Upper bound on the number of vertex-transitive graphs

Is there a known upper bound on the number of vertex transitive graphs on $n$ vertices?
0
votes
1answer
110 views

How to go about finding the most “complex” function?

Intro Hi, I'm a hobbyist, with no formal education, tinkering with SAT solving and boolean algebra minimization. So expect bad terminology. I hope you will forgive me I'm asking a wrong question in a ...
14
votes
0answers
249 views

Are there other proofs for Barrington's theorem?

I know that you can use other non-solvable groups, but is there a proof that uses a completely different approach? In case someone would not know the theorem: ...
11
votes
2answers
262 views

Difficulty in understanding the quantum algorithm for the abelian hidden subgroup problem

I've difficulty in understanding the last steps of the AHSP algorithm. Let $G$ be an abelian group and $f$ be the function which hides the subgroup $H$. Let $G^*$ represent the dual group of $G$. ...
8
votes
0answers
134 views

Complexity of checking if AB intersects C

Let $A,B,C$ be subsets of a nonabelian group $G$, and assume we know the structure of $G$ "fairly well" (e.g., $G = S_n$ or $A_n$). Assume that group operations take $O(1)$ time. Is it ...
8
votes
0answers
166 views

Is there a candidate for a post-quantum one-way group action?

Is there a known family of group actions with a designated element in the set that is being acted on, where it is known how to efficiently $\:$ sample (essentially uniformly) from the groups, ...
1
vote
2answers
324 views

Primitive Recursive Isomorphisms

What is the relationship between invertible primitive recursive functions (that is, a primitive recursive function that is an isomorphism) and all primitive recursive functions? Can every primitive ...
15
votes
1answer
451 views

Complexity of recognizing vertex-transitive graphs

I am not knowledgeable in the area of complexity theory involving groups so I apologize if this is a well known result. Question 1. Let $G$ be a simple undirected graph of order $n$. What is the ...
33
votes
12answers
2k views

Applications of representation theory of the symmetric group

Inspired by this question and in particular the final paragraph of Or's answer, I have the following question: Do you know of any applications of the representation theory of the symmetric group ...
13
votes
2answers
630 views

Gaussian Elimination in terms of Group Action

Gaussian elimination makes determinant of a matrix polynomial-time computable. The reduction of complexity in computing the determinant, which is otherwise sum of exponential terms, is due to ...
3
votes
1answer
112 views

Choices for the group in Public Key Cryptography

I am only aware of two alternatives for the group used in Diffie-Hellman scheme (and similar ones) where logarithms are conjectured to be hard. Those are $\mathbb{F}_p$ and Elliptic Curves. Are there ...