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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22
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550 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 ...
17
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0answers
475 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 ...
12
votes
0answers
316 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
0answers
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
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0answers
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
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574 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
0answers
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
0answers
202 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
0answers
469 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
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975 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 ...
6
votes
0answers
267 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
0answers
372 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
0answers
178 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
0answers
122 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
0answers
246 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
0answers
277 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
0answers
114 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
0answers
87 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
0answers
164 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
0answers
107 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
0answers
248 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
0answers
162 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 ...
4
votes
0answers
409 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
0answers
164 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
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212 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
0answers
299 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
0answers
133 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
0answers
116 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
0answers
175 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
0answers
155 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 ...
3
votes
0answers
319 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
0answers
288 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
0answers
194 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
0answers
321 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 ...
3
votes
0answers
605 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 ...
3
votes
0answers
201 views

For Ising models on finite graphs, is the gradient of Z (w/r/t coupling and field) easier to compute than Z?

Suppose we have a graph $G$ with $n$ vertices, edgeset $E$, $\mathcal{X}=\{1,-1\}^n$. The partition function of the spin-1/2 Ising model on $G$ is $$Z(J,h)=\sum_{x\in \mathcal{X}} \exp\left(J ...
3
votes
0answers
386 views

Does $EXP\neq ZPP$ implies sub-exponential simulation of BPP or NP?

By simulation I mean in the Impaglazzio-Widgerson [IW98] sense, i.e sub-exponential deterministic simulation which appears correct i.o to every efficient adversary. I think this is a proof: if ...
3
votes
0answers
276 views

To what extent MSO = WS1S, when adding relations?

Let me first clarify my definitions. For a word $w \in \Sigma^*$, with $\Sigma=\{a_1, \ldots, a_n\}$, I define two structures: $${\mathbb{N}}(w) = \langle {\mathbb{N}}, <, Q_{a_1}, \ldots, Q_{a_n} ...
2
votes
0answers
90 views

counting irreducible factors

In How hard is it to compute the number of prime factors of a given integer? a question was asked on computing number of prime factors of an integer. Suppose we have a polynomial $f(X)\in \Bbb Z[X]$ ...
2
votes
0answers
46 views

relationship of max-sat and min-cut in theory and practice

I have been using MAX-SAT solver to obtain the exact ground state of ising spin glass model: For 1D periodic model, for systems with 50 binary variables and interaction range of 15th nearest ...
2
votes
0answers
63 views

What is the computational complexity to compute the integral numerically?

Given $$\int_{\Delta}\frac{P_1(x_1,x_2,\dots,x_n)}{P_2(x_1,x_2,\dots,x_n)}$$ where $P_i$ is polynomial(that is $P_1(x_1,x_2,\dots,x_n), P_2(x_1,x_2,\dots,x_n)$ are polynomial) whose coefficients are ...
2
votes
0answers
49 views

Computing basis of a lower set given basis of complementary upper set

In a poset $P$, $U\subseteq P$ is an upper set when for all $x\in U$, we have $y\ge x$ implies $y\in U$. Any subset of $P$ generates an upper set, and the basis of an upper set $U$ is the smallest ...
2
votes
0answers
89 views

Existence of roots of high order polynomial over finite fields

I want to solve the following question: Consider a polynomial $f(x)=a_0+a_1*x^{e_1}+a_2*x^{e_2}+\cdots+x^{e_m}\in F_p[x]$ where $p$ is a prime such that $\log(p)\sim m$ and $e_m\sim 2^m$, I want to ...
2
votes
0answers
182 views

Computing the chromatic polynomial of graph modulo $x-3$

The chromatic polynomial of graph $P(G,x)$ is univariate polynomial which counts the number of colorings of $G$ with $x$ colors for natural $x$. Graph is not $k$ colorable iff $P(G,k)=0$. The ...
2
votes
0answers
121 views

characterization of all periodic tiling of a simple set of Wang Tile

Consider a set of Wang Tile such that all the edges are either 1 or 0.... there are 16 elements in such a set. Now, I wish to characterize all the periodic tilings of this set (better if they are ...
2
votes
0answers
87 views

Complexity of counting words of given length in regular or context-free language

Let $L$ be a regular or context-free language over alphabet $\{0,1\}$. What is the complexity of counting words of length $n$ in $L$? Is it possible to efficiently find if for given $n$ all words ...
2
votes
0answers
112 views

Reference Request: Properties of the Integer Factorization Polytope

The complexity of Integer Factorization is to my knowledge still an open problem, whereas deciding, whether a given integer is a prime number is known to be in $P$ and a proof is available online ...
2
votes
0answers
219 views

cyclotomic polynomials of given degree

Is there a fast algorithm to generate all cyclotomic polynomials $\Phi_n$ for which the degree of $\Phi_n$ is a fixed constant $d?$ This is obviously related to the "inverse totient" function: compute ...
2
votes
0answers
134 views

Odds of projections of a point not on the hyperplane

Let $\mathcal{L}=\{\Bbb x\in\Bbb R^n:x_1+x_2+\dots+x_n=0\}$ be a specific hyperplane. Let a projection of $c\in\Bbb R^n$ be $p(c)=[p_1,p_2,\dots,p_n]$ where $p_i\neq c_i\implies p_i=0$. Let ...
2
votes
0answers
132 views

Intermediate $\mathsf{NP}$-complete problems?

Partition problem is weakly NP-complete since it has polynomial (pseudo-polynomial) time algorithm if input integers are bounded by some polynomial. However, 3-Partition problem is strongly ...