computational complexity theory; complexity classes, such as P, NP, PSPACE, and so on; resource-limited computation; NP-completeness and other completeness concepts; oracle analogues of complexity classes; complexity-theoretic computational models such as automata, circuits; regular languages; ...

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8
votes
1answer
188 views

Groups whose word problem can be solved in constant time

Given a finitely generated group $G$, define an encoding of $G$ to be a one-to-one function $\Phi:G\to \bigcup_n \{0,1\}^n$ that sends each group element to a unique finite word. For $a,b\in G$, ...
-2
votes
0answers
42 views

Analysis of algorithms [closed]

I'm looking at the video http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-00sc-introduction-to-computer-science-and-programming-spring-2011/unit-1/lecture-3-problem-solving/ ...
3
votes
0answers
186 views
+100

On weight enumerators of codes

Are there $[n,k]_q$ constant rate $\frac kn$ and constant alphabet linear code families with automorphism group of size $\Omega((n-n^\beta)!)$ that have minimum distance $d=O(n^\alpha)$ and number of ...
1
vote
2answers
44 views

Finding minimum weight codeword of MDS RS code

For a $[n,k,n-k+1]_q$ Reed Solomon code is there a polynomial time algorithm to find at least one minimum weight $(n-k+1)$ codeword? I searched in literature and I could not find one and hence I am ...
5
votes
1answer
80 views

Complexity for solving linear equations?

What is the best known complexity for finding a vector $x \in \mathbb{R}^n$ to minimize $||Ax - b||^2$ and/or to solve (when possible) the system of linear equations $Ax=b$? I am interested in ...
1
vote
0answers
97 views

Efficient deterministic algorithms of factorizing

My question is about efficient deterministic algorithms of factorizing polynomials of degree $n$ over $\mathbb{F}_q$. Are there such algorithms that use poly$(n, \log q)$ bit operations? I know ...
11
votes
1answer
422 views

An efficient isomorphism between finite fields

Let $p$ be a prime number. Let $f$ and $g$ be irreducible polynomials over $\mathbb{F}_p$, both of degree $n$. We know that factor-rings $\mathbb{F}_p[x]/(f)$ and $\mathbb{F}_p[x]/(g)$ are isomorphic ...
8
votes
1answer
329 views

Computational complexity of computing simplicial homology

Is there any literature regarding the fastest known algorithm to compute the homology groups of a simplicial complex (on n vertices)? What about computing the fundamental group? The context is to tell ...
11
votes
0answers
449 views

Why should Algebraic Geometers and Representation Theorists care about Geometric Complexity Theory?

Geometric Complexity Theory has demonstrated that Complexity Theorists should care about Algebraic Geometry and Representation Theory, but, why should Algebraic Geometers and Representation Theorists ...
2
votes
0answers
71 views

Effective “almost enumeration” of monotone boolean functions

Denote by $\mathcal{M}(n)$ the set of all monotone functions $\{0,1\}^n \to \{0,1\}$. Can $\mathcal{M}(n)$ be represented as $\mathcal{M}(n) = \{ f(t) | t\in \{0,1\}^k \}$ such that: 1) $k = \log |\...
1
vote
0answers
55 views

Recognizing cubic graphs decomposable into 2-factor with given cycle type

Petersen's theorem states that every cubic, bridgeless graph contains a perfect matching. It implies that the edge set $E$ can be partitioned into a perfect matching and a 2-factor. Determining the ...
0
votes
0answers
76 views

Generating set of Graph-Automorphism from Direct Product

Notation: $H$ is the adjacency matrix of graph $\mathcal{H}$ . $$H = \begin{bmatrix} H_{(3)} & R_{(3, 2)} & R_{(3,1)} \\ R_{(3,2)} & H_{(2)} & R_{(2,1)} \\ R_{(3,1)} & R_{(2,1)}...
0
votes
0answers
20 views

Counting contingency tables with unary inputs

Given $2$ sequences $A = (a_1,\dots,a_k)$ and $B = (b_1,\dots,b_l)$ of natural numbers. A contingency table is a matrix in $\mathbb{N}^{k \times l}$ with row sums $A$ and column sums $B$. Counting ...
0
votes
0answers
62 views

GI-hard problems that would benefit from efficient algorithm for GI

The exponential time hypothesis (ETH) states that 3SAT can not be solved in subexponential time. As far as I know, it is not known whether an efficient algorithm for graph isomorphism problem (GI) ...
8
votes
0answers
161 views

Is there an efficient algorithm for testing isomorphism of projective planes?

Isomorphism testing is a core problem in computational complexity. Recently, Babai has shown that Graph Isomorphism problem for general graphs can be solved in quasipolynomial time. Long time before ...
0
votes
0answers
43 views

Analogues of Specker sequences for different complexity classes

Consider the standard definition of computable real numbers: a real number $r$ is computable just in case $r$ is the limit of a sequence $(a_i)_{i \in \mathbb{N}}$ such that (1) the function $i \...
5
votes
1answer
97 views

Complexity of integer programming with added predicates

A classical theorem in Integer Programming by Lenstra says that any integer system $$A x \le b$$ can be solved in polynomial time, where $A \in \mathbb{Z}^{m \times n}, x \in \mathbb{Z}^n, b \in \...
10
votes
2answers
245 views

Finite objects for which isomorphism is NP-hard or harder?

Are there finite objects for which deciding isomorphism is NP-hard or harder? Graphs and groups are not solutions. Searching the web didn't return answer for me. Partial result based on Chow's ...
1
vote
0answers
59 views

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 ...
3
votes
0answers
73 views

Hypergraph edge colouring

I'm interested in knowing if finding the edge-chromatic number of a $k$-uniform $k$-partite hypergraph is NP-hard for $k\geq 3$ Could anyone provide a reference for the result? By edge-chromatic ...
-3
votes
1answer
139 views

Turing and Many one reductions in computability versus complexity

What are some non-trivial (please exclude poly time definitional difference) differences between Turing versus Many-one reductions in computability theory and those that occur in complexity theory?
7
votes
2answers
722 views

Complexity of Turing Machine behavior

If one looks at the code for a Turing Machine (TM) with $q$ states and, let's say, $2$ symbols, they all look pretty much the same: A list of $5$-tuples: $$ < state, symbol{-}read, symbol{-}to{-}...
1
vote
0answers
68 views

Calculation of cardinality of Jacobians

The problem of calculation the number of rational points on curves over finite fields is $\#P$-complete - "Counting curves and their projections". Is it true for calculation of number of rational ...
18
votes
0answers
375 views

Checking Mertens and the like in less than linear time or less than $\sqrt{x}$ space

Say you want to check that $|\sum_{n\leq x} \mu(n)|\leq \sqrt{x}$ for all $x\leq X$. (I am actually interested in checking that $\sum_{n\leq x} \mu(n)/n|\leq c/\sqrt{x}$, where $c$ is a constant, and ...
5
votes
1answer
127 views

Decidability of convex rearrangements of polygons

Triggered by the MO question, "How many convex shapes can be made with the pieces of the Stomachion?," I would like to pose this question: Q. Given $n$ polygons in a set $S$, say each with integer ...
4
votes
1answer
296 views

an algebraic variety for a boolean circuit

There is a polynomial reduction from a $3-CNF$ $SAT$ problem to some system of polynomial equations over $\mathbb{F}_2$. I mean there is polynomial reduction $F$ such that for every boolean ...
1
vote
0answers
32 views

Reference requests for tiling easiness [closed]

For Wang tile problem, is there some general statements in a paper stating that the more tiles (supposed provided by random) available, the easier it is for these tiles to tile the plane? Thank you.
15
votes
4answers
2k views

Languages beyond enumerable

A language is a set of finite-length strings from some finite alphabet $\Sigma$. It is no loss of generality (for my purposes) to take $\Sigma=\{0,1\}$; so a language is a set of bit-strings. ...
3
votes
2answers
85 views

Complexity of solving systems of linear diophantine equations

It is "well known" that a matrix system $Ax=b$ where $A\in \Bbb Z^{m\times n}$, $x\in \Bbb Z^n,b\in\Bbb Z^m$ for some $m,n \in \Bbb N$, can be solved in polynomial time, using Smith/Hermite Normal ...
7
votes
5answers
1k views

Zero knowledge proof of equality

Alice and Bob each secretly chooses an integer between 1 and 10, a and b. They want to know (with high probability) whether or ...
3
votes
0answers
116 views

Average nastiness of a Newton polytope

Given a Newton polytope $P$ inside the $d$-scaled simplex ( i.e $\alpha \in P \implies | \alpha |=d $ ), then we define the following quantity: $$ P(x)= \left( \sum_{\alpha \in P} \binom{d}{\alpha}...
12
votes
2answers
212 views

Permutation search problems with no known $o(n!)$ algorithms

I am looking for problems for whose solution no known subfactorial algorithms are known. I am particularly interested in questions of isomorphism; that is, is there a permutation that converts one ...
6
votes
2answers
188 views

Finding an “optimal” quotient in a free group

Consider the abelian free group $G = \mathbf{Z}^n$ of rank $n$ and a finite subset $A \subset G \setminus \{0\}$. Since $G$ is residually finite, there is a subgroup $H \subset G$ such that $A \cap H =...
2
votes
1answer
123 views

About Renyi entropy

If one is given a joint probability distribution over a finite set of discrete random variables then I guess there a notion of $\alpha-$Renyi entropy defined for it as $S_\alpha (X_1,..,X_n) = \frac{...
3
votes
0answers
86 views

State of the art for univariate complex polynomials factorization with algebraic coefficients

Let $\mathbb{K}:=\overline{\mathbb{Q}}$ be the field of algebraic numbers. We choose to represent an element of $\mathbb{K}$ as its minimal monic polynomial, which is a vector in some $\mathbb{Q}^n$. ...
5
votes
0answers
119 views

Complexity of graph isomorphism

Last year, Laszlo Babai proved that the graph isomorphism problem can be solved in time: $$ \exp(O(\log^c n)) $$ where $n$ is the number of vertices. What is the best bound we have for $c$? (The ...
1
vote
0answers
131 views

How I can prove the equality $P^{P_{\operatorname{space}}}=NP^{P_{\operatorname{space}}}=P_{\operatorname{space}}^{P_{\operatorname{space}}}$ [closed]

I know how to prove that if $A \in P^{P_{\operatorname{space}}}$ then $A \in NP^{P_{\operatorname{space}}}$ and $A \in P_{\operatorname{space}}^{P_{\operatorname{space}}}$. I don't know how to prove ...
3
votes
0answers
257 views

Why is solving polynomial systems NP hard?

Solving polynomial systems is known to be a NP hard problem; however it is not completely clear to me where this complexity comes from. My interest is in the case of systems of multivariate ...
8
votes
1answer
328 views

Is there any real quadratic ring for which the Euclidean algorithm is polynomial?

We know from Rolletschek's work that the Euclidean algorithm of $\mathbb{Z}[i]$ is polynomial. Indeed, let $n$ be the maximum number of steps in the Euclidean algorithm applied to $u,v \in\mathbb{Z}[i]...
7
votes
0answers
195 views

Can primes be (almost) random sequence in von Mises sense?

Random models for primes (such as Cramer's model) have been extensively used for informal justification of various conjectures involving primes. It is crucial to understand in what sense sequence of ...
5
votes
0answers
125 views

Complexity of $\mathbb{Z}^n$ tilings

Let $\mathcal{T} \subset \mathbb{Z}^n$ be a finite set. Let $\Lambda \subset \mathbb{Z}^n$ be a full rank lattice. We say that $\mathcal{T}$ is a $\Lambda$-tile for $\mathbb{Z}^n$ if the following ...
3
votes
0answers
90 views

What is the complexity of finding a third Hamilton Cycle in cubic graph?

According to Smith Theorem: if a cubic graph has a hamilton circuit then it must have a second one. SMITH : Given a Hamilton circuit in a 3-regular graph, find a second Hamilton circuit. It is known ...
5
votes
0answers
116 views

Littlewood-Richardson rule for the complete flag variety: GapP complete?

The cohomology ring of a complete flag variety $X$ has a basis of Schubert classes $S_u$ for permutations $u$. Define the Littlewood-Richardson coefficient $c_{uv}^w$ for permutations $u,v,w$ to be ...
1
vote
1answer
76 views

Graph colouring for bounded degree graphs

I'm fairly new to colourings on bounded degree graphs i'm interested in the following questions, For planar graphs with bounded degree $4$ is finding the colouring number $NP$-hard? So is ...
9
votes
2answers
277 views

Bounded Arithmetic vs Complexity Theory

In this post, when I talk about bounded arithmetic theories, I mean the theories of arithmetic according to "Logical Foundations of Proof Complexity", which capture the complexity classes between $AC^...
3
votes
1answer
45 views

Complexity class of matrix generalization of knapsack problem

Let $n$ be a natural number, $u_+,v_+,u_-,v_-$ be real or complex column vectors of length $n$, and $M_1,M_2,\ldots,M_k$ be a finite collection of $n\times n$ real or complex matrices. Consider the ...
2
votes
1answer
70 views

Directed edge-colouring

I'm interested to know whether the following problem is NP-complete or if there is an algorithm to solve it. Suppose we are given a directed graph $G=(V,E^{\rightarrow})$ and we want to colour the ...
1
vote
1answer
84 views

How to complete the NP-hardness proof of GENERAL-SQUARE-PRODUCT?

I am interested in the complexity of the following problem: GENERAL-SQUARE-PRODUCT INSTANCE: Two sets $A=\{a_1,\ldots,a_n\}$ and $B=\{b_1,\ldots,b_n\}$ of integers, a positive integer $k<n$ and a ...
14
votes
1answer
1k views

Is deciding if one planar graph is dual to another really NP-hard (Wikipedia claim)?

Wikipedia claims (permanent link) without reference: Testing whether one planar graph is dual to another is NP-complete. Another claim with reference: For any plane graph G, the medial graph ...
7
votes
1answer
152 views

Factoring a multiset into a product of two multisets

Given a multiset $S$ of $mn$ numbers, how hard is to find multisets $A$ and $B$, of $m$ and $n$ numbers respectively, such that $$ S = \{ xy \mid x\in A, y\in B \}~~~~\text{(multiset sense)}, $$ or ...