Tagged Questions

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; ...

295 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 ...
89 views

$0/1$ programming multiple quadratic constraints

If we have an $n$-variable rank $n$-linear system it is clear we can find whether there exists a $0/1$ solution in polynomial time. If we have an $n$-variable degree $2$ system how many constraints ...
97 views

154 views

Complexity of $\mathsf{gcd}(a,b)\bmod N$

Given $a,b\in\Bbb N$ where each $a,b$ is $n$-bits, we can compute $\mathsf{gcd}(a,b)$ in $cn^{1+\epsilon}$ bit operations for some fixed $c\geq1$. My query is given $N,a,b$ where $a,b$ is $n$-bits ...
53 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 ...
323 views

Is this minimization problem NP-Complete ?

We are given an $n\times (n+k)$ matrix $A,$ with entries in $\mathrm{GF}(2),$ of the form $A=(I_n|B)$ where $I_n$ is a $n\times n$ identity matrix where the matrix $B$ has no "zero" rows or columns. ...
75 views

64 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 ...
374 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 ...
295 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 ...
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.
125 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 ...
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. ...
80 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 ...
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 ...
116 views

130 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 ...
255 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 ...
775 views

Efficient Hamiltonian cycle algorithms for graph classes

Generally speaking finding a Hamiltonian cycle is NP-Hard and so tough. But if $G=L(H)$ is the line graph of $G$ then we can reduce the finding of a Hamiltonian cycle in $G$ to a Eurler your of $H$ ...
246 views

Can we solve Hamiltonian Path problem for biconnected planar graphs in linear time?

Assume that we have a bi-connected planar graph $G$ with $\Delta(G)>3$, and we want to find a Hamiltonian Path in $G$. As we know the st-order of a bi-connected planar graph can be computed in ...
6k views

How fast can we *really* multiply matrices?

Background: The Strassen Algorithm, described here, has a computational complexity of $\text{O}(n^{2.807})$ for the multiplication of two $n \times n$ matrices (the exponent is $\frac{\log7}{\log2}$). ...
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 $u,v \in\mathbb{Z}[i]... 0answers 194 views Can primes be (almost) random sequence in von Mises sense? Random models for primes (such as Cramer's model) have been extensively used for informal justification of various conjectures involving primes. It is crucial to understand in what sense sequence of ... 0answers 125 views Complexity of$\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$if the following ... 0answers 84 views What is the complexity of finding a third Hamilton Cycle in cubic graph? According to Smith Theorem: if a cubic graph has a hamilton circuit then it must have a second one. SMITH : Given a Hamilton circuit in a 3-regular graph, find a second Hamilton circuit. It is known ... 0answers 110 views Littlewood-Richardson rule for the complete flag variety: GapP complete? The cohomology ring of a complete flag variety$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 ... 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 ... 2answers 271 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^...
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 ...