**24**

votes

**0**answers

587 views

### A combination of two well-known complexity problems

Suppose you are given two graphs $G$ and $H$ and are told that one of the following two situations occurs. Either they are isomorphic, or one of the graphs contains a Hamilton cycle and the other ...

**18**

votes

**0**answers

617 views

### Computational complexity of topological K-theory

I am a novice with K-theory trying to understand what is and what is not possible.
Given a finite simplicial complex $X$, there of course elementary ways to quickly compute the cohomology of $X$ with ...

**17**

votes

**0**answers

494 views

### Reference request: Parallel processor theorem of William Thurston

Sometime in the 1980's or 1990's, Bill Thurston proved a theorem regarding the existence of a universal parallel processing machine, using a certain class for such machines having finite deterministic ...

**12**

votes

**0**answers

332 views

### Splay trees and Thompson's group $F$

( I apologize for only indicating some easy to find references, but new users are not allowed to link more than five). This is very speculative, but:
Question: Is there a reformulation of the Dynamic ...

**12**

votes

**0**answers

2k views

### How can an approach to $P$ vs $NP$ based on descriptive complexity avoid being a natural proof in the sense of Raborov-Rudich?

EDIT: This question has been modified to make it a stand-alone question. Feel free to retract your votes for the previous version.
Here are Vinay Deolalikar's paper, and Richard Lipton's first post ...

**12**

votes

**0**answers

1k views

### Razborov's response to Almost Natural Proofs

This post is about Natural Proofs barrier in computational complexity.
There are two recent papers related to this. They are:
Amplifying lower bounds by means of self-reducibility by Eric Allender ...

**11**

votes

**0**answers

730 views

### Primes and Parity

This problem is motivated by the polymath4 project. There, the aim was to find an efficient deterministic algorithm for finding a prime larger than $N$. The hope was to find a polynomial algorithm in ...

**11**

votes

**0**answers

590 views

### Regular languages of matrices and their generating functions

My question is somewhat related to this question.
Let us fix natural numbers $k$ and $C$. Let $A$ be an automaton whose alphabet consists of $k\times k$ matrices with integer coefficients of ...

**9**

votes

**0**answers

1k views

### Is Witten's new method of quantization useful for geometric complexity theory?

The Kempf-Ness theorem (see e.g. arXiv:0912.1132) - that the algebraic quotient of geometric invariant theory is also a symplectic quotient - suggests (to me) that certain physical constructions used ...

**8**

votes

**0**answers

168 views

### Ricocheting pinball-like shot: Complexity?

Suppose one has $n$ perfect two-sided mirror segments in the plane $\mathbb{R}^2$.
The segments are open, excluding their endpoints.
They are disjoint as closed segments, i.e., no pair shares an ...

**8**

votes

**0**answers

238 views

### Computing van Kampen diagrams

If G is a finitely presented group (with generating set X) and w is a word over X such that
w=1 in G, then the latter can be witnessed by a so called van Kampen diagram for w, which is
a planar ...

**8**

votes

**0**answers

498 views

### Is the dominating set problem restricted to planar bipartite graphs of maximum degree 3 NP-complete?

Does anyone know about an NP-completeness result for the DOMINATING SET problem in graphs, restricted to the class of planar bipartite graphs of maximum degree 3?
I know it is NP-complete for the ...

**8**

votes

**0**answers

1k views

### Weighted Hamming distance

Basically my question is, what kind of geometry do we get if we use a "weighted" Hamming distance. This is combinatorics but similar things come up sometimes in theoretical computer science, for ...

**7**

votes

**0**answers

148 views

### Is there an infinite increasing sequence of naturals for which Landau's function can be efficiently computed?

Landau's function
$g(n)$ is the largest order of an element of the symmetric group $S_n$.
Equivalently, $g(n)$ is the largest least common multiple (lcm) of any partition of $n$.
In general $g(n)$ is ...

**7**

votes

**0**answers

193 views

### When is a reduction not a reduction?

Every mathematician understands the concept of reducing a complicated problem to a simpler problem. "Without loss of generality, we may assume…" However, I've noticed that some kinds of ...

**6**

votes

**0**answers

89 views

### Is there a ``Ladner's Theorem" for the PH-vs-PSPACE scenario?

Like a statement of the kind, ``If the Polynomial Hierarchy (PH) $\neq$ PSPACE then there exists $L \in PSPACE \backslash PH$ which is not PSPACE-complete"?
Or is there something else that states ...

**6**

votes

**0**answers

344 views

### Is simultaneous diophantine approximation (in a weaker sense) NP hard?

The traditional problem of simultaneous diophantine approximation is: Given a set of rational numbers $g_1,\ldots,g_d$, an integer $N$, and a rational $\gamma>0$, is there an integer $W$ with ...

**6**

votes

**0**answers

413 views

### Any reason to believe that $NP \neq P$ is unprovable in ZFC

We know $NP \neq P$ from a lot of point of view like empirical reason,or theoretical reasons such as finite model theory or descriptive complexity.Although we find so many reasons to believe $NP \neq ...

**6**

votes

**0**answers

128 views

### An explicit IP algorithm for chess?

If I have 2 large graphs to be tested for isomorphism, and can communicate with some (powerfull but untrusted) machine, I can choose graph at random, permute vertices, ask machine to guess which one ...

**6**

votes

**0**answers

268 views

### Is integer GCD in NC?

Wikipedia, in the page http://en.wikipedia.org/wiki/Greatest_common_divisor#Complexity mentions integer GCD is in NC by citing http://www.cs.cornell.edu/courses/CS6820/2012sp/Handouts/Sedjelmaci07.pdf
...

**6**

votes

**0**answers

288 views

### Enumerating (generalized) de Bruijn tori

Given a cyclic word $w$ of length $N$ over a $q$-ary alphabet and $k \in \mathbb{Z}_+$, consider the directed multigraph $G_k(w) = (V,E)$ with $V \subset$ {$1,\dots,q$}$^k$ given by the $k$-lets ...

**5**

votes

**0**answers

101 views

### Most computationally efficient Littlewood-Richardson rule

There are many, many different versions of the Littlewood-Richardson rule: the original characterization via Yamanouchi words, Remmel's version, a description via the Poirier-Reutenauer bialgebra, the ...

**5**

votes

**0**answers

117 views

### Is Hankelability NP-hard?

This question was previously asked on cstheory but with no answers or substantive comments.
I am trying to write code to detect if a matrix is a permutation of a Hankel matrix. Here is the spec.
...

**5**

votes

**0**answers

123 views

### Complexity of finding three perfect matchings with no edge in common in a bridgeless cubic graph

According to a conjecture:
Conjecture (Fan & Raspaud, 1994) Every bridgeless cubic graph contains three perfect matchings with no edge in common.
Equivalent statement here
Main question:
...

**5**

votes

**0**answers

89 views

### Can the isomorphism relation for countable models become harder when adding finitely many constants?

I am particularly interesting in the case where $T$ is o-minimal, but I would be interested in any theory $T$ (or even an $L_{\omega_1,\omega}$-sentence) which has this property.
Context: view the ...

**5**

votes

**0**answers

178 views

### Feasible Type Theories

I am looking for references about efficient type theories,
efficiency in the sense of computational complexity,
and type theory in the sense of Martin-Lof's type theories.
Has there been any studies ...

**5**

votes

**0**answers

113 views

### Are there sampNP-intermediate problems?

This questions is approximately cross-posted from theoretical computer science stackexchange
Ladner's theorem establishes that if $\mathsf{P} \ne \mathsf{NP}$ then $\mathsf{NPI} := \mathsf{NP} ...

**4**

votes

**0**answers

40 views

### Digraph weak connectivity in $O(V)$ space and $O(V+E)$ time

A digraph is called weakly connected if its underlying undirected graph is connected.
You are given a digraph $G$ with $V$ vertices and $E$ edges as a read-only data structure consisting of lists of ...

**4**

votes

**0**answers

197 views

### Is there a program for theory of incompleteness in NP?

Motivated by Suresh's post, Techniques for showing that problem is in hardness limbo, it seems that there might be an underlying theory that explains why some of these problems can not be complete for ...

**4**

votes

**0**answers

167 views

### What is the complexity of intersecting two matrix algebras over a finite field?

The following question arose in a joint project with Arkadius Kalka and Adi Ben-Zvi.
Let $\mathbb{F}$ be a finite field, and $M_n(\mathbb{F})$ be the $n\times n$ matrices over $\mathbb{F}$.
For a ...

**4**

votes

**0**answers

143 views

### What is known about the complexity of this covering problem?

Let $G=(V,E)$ be a graph. A vertex set $X\subseteq V$ is called critical if $X\neq\emptyset$ and no vertex in $V\setminus X$ is adjacent to exactly one vertex in $X$. The problem is to find a vertex ...

**4**

votes

**0**answers

307 views

### About “natural proof” of Razborov and Rudich

The famous "Natural Proof" paper ,http://www.cs.umd.edu/~gasarch/BLOGPAPERS/natural.pdf , of Razborov and Rudich gives a barrier for any proof that try to separate P and NP. It mainly shows that if ...

**4**

votes

**0**answers

476 views

### Computational complexity of multiplication in a nilpotent group?

What is the computational complexity of multiplication in a Carnot group ?
Background: A Carnot group is a real nilpotent Lie group $N$ whose Lie algebra $Lie(N)$ admits a direct sum decomposition
...

**4**

votes

**0**answers

161 views

### Best lower bound for proof complexity of graph asymmetry

Graph automorphism problem ( GA) of determining whether a graph has a nontrivial automorphism is a good candidate for a problem in $NP$-intermediate. I'm looking for references that study the ...

**4**

votes

**0**answers

172 views

### Suppose P = BPP; Pseudorandom Generators vs NP Adversary

Suppose P = BPP.
Then we know there exist pseudorandom number generators vs P.
Suppose the adversary is NP.
Now, any pseudorandom number generator that only uses P will fail. (Since NP can invert ...

**4**

votes

**0**answers

329 views

### Complexity of edge coloring in planar graphs?

I asked this on StackExchange TCS but did not get a satisfactory answer:
Four Color Theorem is equivalent to "Every cubic planar bridgeless graphs is 3-edge colorable". However, 3-edge coloring of ...

**4**

votes

**0**answers

218 views

### Domination in Nice Lattices

Let an integer vector be nice when it has only two nonzero components, which sum to zero. So (0, 0, 3, 0, -3) and (-1, 0, 1, 0, 0) are examples of nice vectors in $n=5$ dimensions.
Call a lattice ...

**4**

votes

**0**answers

307 views

### Can the Littlewood-Richardson cone be used for combinatorial optimization?

The Littlewood-Richardson cone $LR_{n, k}$ consists of all $k-$tuples $(a_1, a_2, \dots, a_k)$ of real $n-$vectors with monotonically decreasing entries such that there exist $k$ $n \times ...

**3**

votes

**0**answers

41 views

### $n!$ computation in $\mathsf{BSS}$ model

It is well known that if $n!$ cannot be computed in $\mathsf{polylog}(n)$ ring ($\Bbb Z$) operations, then $\mathsf P\neq\mathsf{NP}$ in $\mathsf{BSS}$ model.
Suppose if we assume $\mathsf ...

**3**

votes

**0**answers

71 views

### Are all $k$th-longest-tour problems equally hard?

It is well known, that determining the shortest and, the longest Hamilton Cycle of a complete graph with real edge weights are algorithmically two sides of the same medal: one transforms to the other ...

**3**

votes

**0**answers

115 views

### On the computational complexity of the Hilbert polynomial of numerical semigroup rings

Let $(R, \mathfrak{m}) = k[[X^a, X^b, X^c]]$, $a<b<c$, $gcd(a, b, c) = 1$, be a semigroup ring. We have $R$ is a Cohen-Macaulay local ring of dimension one. It is well known that ...

**3**

votes

**0**answers

168 views

### Partitioning a cubic graph into two induced cycles of equal order

I am aware that deciding the existence of a partition of the vertices of a connected graph $G(V, E)$ into two induced cycles is $NP$-complete(Theorem 2). Induced cycle is a cycle without any chord ...

**3**

votes

**0**answers

187 views

### Nesting big-O with big-Omega $O(g(\Omega(h(n))))$: is it $O$ for all $\Omega$ or for one $\Omega$?

I want to express the following statement about a function $f(n)$: there exists $f_\Omega\in\Omega(h(n))$ such that $f\in O(g(f_\Omega(n))$. What's the correct notation for this? Is it $f\in ...

**3**

votes

**0**answers

59 views

### Is the $d$-dimensional Arrangement of Trees still $NP$-hard?

The $d$-dimensional Arrangement Problem for general graphs is known to be $NP$-hard since the special case $d=1$ (OLA) already is (Garey et al, [1976]). For Trees however, the one dimensional case can ...

**3**

votes

**0**answers

130 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 presence ...

**3**

votes

**0**answers

184 views

### Kolmogorov complexity with bounded ressources

Thanks to symetry of information (i.e $\forall x,y, K(xy) = K(x) + K(y|x) - O(log(|x| + |y|)$), one can easily show that :
$ \exists N \forall x, (|x| = n^{log(n)} and |x| \geq N), \exists y, (|y| ...

**3**

votes

**0**answers

350 views

### Connected Sum Decomposition of a Knot

Given a composite knot, is it possible to decompose it in prime knots by an algorithm that runs in polynomial time?

**3**

votes

**0**answers

305 views

### Amortized analysis of data structure via potential function

One common method for proving that a data structure supports an operation in $O(f(n))$ amortized time is to construct a potential function $\Phi: \mathcal S \rightarrow \mathbf R^{+}$, which ...

**3**

votes

**0**answers

198 views

### Algorithm for testing satisfiable fraction of linear equations mod 2

Hello
Let $F_{n,p}$ be a random process which generates a system of linear equations over $F_2$. The variables are $\{x_1, ..., x_n\}$ and for each of the $ \binom{n}{2}$ $i,j$ pairs, the equation ...

**3**

votes

**0**answers

637 views

### Why isn't Montgomery modular exponentiation considered for use in quantum factoring?

It is well known that modular exponentiation (the main part of an RSA operation) is computationally expensive, and as far as I understand things the technique of Montgomery modular exponentiation is ...