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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Analogues of P vs. NP in the history of mathematics

Recently I wrote a blog post entitled "The Scientific Case for P≠NP". The argument I tried to articulate there is that there seems to be an "invisible electric fence" separating the problems in P ...
47
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8answers
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The “sensitivity” of 2-colorings of the d-dimensional integer lattice

Consider the $d$-dimensional integer lattice, $Z^d$. Call two points in $Z^d$ "neighbors" if their Euclidean distance is 1 (i.e., if they differ by 1 on exactly one coordinate). Let $C$ be a ...
39
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2answers
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Walsh Fourier Transform of the Möbius function

This question is related to this previous question where I asked about ordinary Fourier coefficients. Special case: is Möbius nearly Orthogonal to Morse ! Harold Calvin Marston Morse (24 March ...
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5answers
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Computational complexity of computing homotopy groups of spheres

At various times I've heard the statement that computing the group structure of $\pi_k S^n$ is algorithmic. But I've never come across a reference claiming this. Is there a precise algorithm ...
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Testing whether an integer is the sum of two squares

Is there a fast (probabilistic or deterministic) algorithm for determining whether an integer $n$ is a sum of two squares? By "fast" here I mean polynomial time (i.e. time $O((log n)^{O(1)})$). Note ...
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Is it easy to produce hard-to-color graphs?

This question arises from my recent visit to my daughter's second-grade class, where I led some discussion and activities on graph coloring (see Math for seven-year-olds). In one such activity, each ...
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What is the shortest program for which halting is unknown?

In short, my question is: What is the shortest computer program for which it is not known whether or not the program halts? Of course, this depends on the description language; I also have the ...
31
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What is the time complexity of computing sin(x) to t bits of precision?

Short version of the question: Presumably, it's poly$(t)$. But what polynomial, and could you provide a reference? Long version of the question: I'm sort of surprised to be asking this, because ...
28
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The NP version of Matiyasevich's theorem

By Matiyasevich, for every recursively enumerable set $A$ of natural numbers there exists a polynomial $f(x_1,...,x_n)$ with integer coefficients such that for every $p\ge 0$, $f(x_1,...,x_n)=p$ has ...
28
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1answer
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An edge partitioning problem on cubic graphs

Hello everyone, I already asked this question on the TCS Stack Exchange, but it has not been resolved yet. Maybe readers of this forum will have other ideas or information, although I suspect that ...
27
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7answers
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Example of a good Zero Knowledge Proof.

I am working on my zero knowledge proofs and I am looking for a good example of a real world proof of this type. An even better answer would be a Zero Knowledge Proof that shows the statement isn't ...
26
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4answers
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Why is “P vs. NP” necessarily relevant?

I want to start out by giving two examples: 1) Graham's problem is to decide whether a given edge-coloring (with two colors) of the complete graph on vertices $\lbrace-1,+1\rbrace^n$ contains a ...
25
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Is P=NP relevant to finding proofs of everyday mathematical propositions?

Disclaimer: I don't know a whole lot about complexity theory beyond, say, a good undergrad class. With increasing frequency I seem to be encountering claims by complexity theorists that, in the ...
25
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3answers
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Why do statistical randomness tests seem so ad hoc?

Wikipedia describes Kendall and Smith's 1938 statistical randomness tests like this: The frequency test, was very basic: checking to make sure that there were roughly the same number of 0s, 1s, ...
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Problems known to be in both NP and coNP, but not known to be in P

One such problem I know is integer factorization. What are other interesting cases?
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Can We Decide Whether Small Computer Programs Halt?

The undecidability of the halting problem states that there is no general procedure for deciding whether an arbitrary sufficiently complex computer program will halt or not. Are there some large $n$ ...
24
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4answers
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Algebraic P vs. NP

I recently attended a lecture where the speaker mentioned that what he was talking about was connected to the algebraic version of the $P$ vs. $NP$ problem. Could someone explain what that means in a ...
23
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What is the easiest randomized algorithm to motivate to the layperson?

When trying to explain complexity theory to laypeople, I often mention randomized algorithms but seemingly lack good examples to motivate their usage. I often want to mention primality testing but ...
23
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Given a polynomial-time algorithm, can we compute an explicit polynomial time bound just from the program?

Question. Given a Turing-machine program $e$, which is guaranteed to run in polynomial time, can we computably find such a polynomial? In other words, is there a computable function $e\mapsto p_e$, ...
23
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Why is proving P != NP so hard?

Does anyone have any insight into why it is so hard to prove that P != NP conjecture? There seems to be so much evidence in its favor, and so many problems and techniques with which to attack it, that ...
23
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1answer
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How strong is this conjecture? $(Z/nZ)^*$ is generated by “small” elements

Conjecture: There are constants $c,k$ such that every $(Z/nZ)^*$ is generated by its elements smaller than $k (\log n)^c$. Where $(Z/nZ)^*$ is the multiplicative group of integers mod $n$. My ...
23
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1answer
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Are sums of sequences decidable?

Suppose that $f,g$ are rational functions with integer coefficients such that $\sum_{n=0}^{\infty}f(n)$ and $\sum_{n=0}^{\infty}g(n)$ both converge. Is it decidable whether ...
23
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Counting subgraphs of bipartite graphs

I'm not a graph theorist or computational complexity specialist, so my apologies if this question is stupid or poorly posed! Given a bipartite graph $G$ of $n$ vertices, how many induced subgraphs of ...
22
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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 ...
21
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4answers
987 views

A programming language that can only create algorithms with polynomial runtime?

Has someone constructed a programming language that can construct all the algorithms in P, and no others? I'm interested in this restriction coming from the syntax naturally, as opposed to just being ...
21
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3answers
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Discrete Fourier Transform of the Möbius Function

Consider the Möbius function $\mu (m)$. (Thus $\mu(m)=0$ unless all prime factors of $m$ appear once and $\mu (m)=(-1)^r$ if $m$ has $r$ distinct prime factors.) Next consider for some natural number ...
21
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Are there any interesting examples of random NP-complete problems?

Here's an example of the kind of thing I mean. Let's consider a random instance of 3-SAT, where you choose enough clauses for the formula to be almost certainly unsatisfiable, but not too many more ...
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Simulating Turing machines with {O,P}DEs.

Qiaochu Yuan in his answer to this question recalls a blog post (specifically, comment 16 therein) by Terry Tao: For instance, one cannot hope to find an algorithm to determine the existence of ...
20
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2answers
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What is the complexity of this problem?

Recently on Dick Lipton and Ken Regan's blog there was a post about problems of intermediate complexity, that is, NP problems that are harder than P but easier than NP-complete. The main message of ...
20
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1answer
786 views

How hard is reconstructing a permutation from its differences sequence?

My interest in combinatorially motivated computational problems led me to search for simple problems that turn out to be computationally hard. In this pursuit, I came up with a problem which I hope is ...
19
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6answers
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Can a problem be simultaneously polynomial time and undecidable?

The Robertson-Seymour theorem on graph minors leads to some interesting conundrums. The theorem states that any minor-closed class of graphs can be described by a finite number of excluded minors. As ...
19
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3answers
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Is the theory of categories decidable?

There are a lot of theorems in basic homological algebra, such as the five lemma or the snake lemma, that seem like they'd be more easily proven by computer than by hand. This led me to consider the ...
19
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Is there a syntactic characterization for BPP, BQP, or QMA?

Background The complexity classes BPP, BQP, and QMA are defined semantically. Let me try to explain a little bit what is the difference between a semantic definition and a syntactic one. The ...
18
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1answer
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How fast can we *really* multiply matrices?

Background: The Strassen Algorithm, described here, has a computational complexity of $\text{O}(n^{2.8})$ for the multiplication of two $n \times n$ matrices. However, the constant is so large that ...
18
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4answers
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Discrete logs vs. factoring

One thing that I've never quite understood is why computing discrete logarithms (in the multiplicative group mod p) and factoring seem to be so closely related. I don't think that there's a reduction ...
18
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3answers
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Satisfiability of general Boolean formulas with at most two occurrences per variable

(If you know basics in theoretical computer science, you may skip immediately to the dark box below. I thought I would try to explain my question very carefully, to maximize the number of people that ...
17
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2answers
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Deep theorems and long proofs

I ran across this discussion by Daniel Shanks, "Is the quadratic reciprocity law a deep theorem?." Solved and Unsolved Problems in Number Theory. Vol. 297. AMS, 2001. p.64ff. which made me ...
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The problem of finding the first digit in Graham's number

Motivation In this BBC video about infinity they mention Graham's number. In the second part, Graham mentions that "maybe no one will ever know what [the first] digit is". This made me think: Could ...
17
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1answer
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Möbius Randomness of the Rudin-Shapiro Sequence

The Rudin-Shapiro sequence (also known as the Golay-Rudin-Shapiro sequence) is defined as follows. Let $a_n = \sum \epsilon_i\epsilon_{i+1}$ where $\epsilon_1,\epsilon_2,\dots$ are the digits in the ...
17
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1answer
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An NP-hard $n$ fold integral

We are given rational numbers $[c_1, c_2, \ldots, c_n]$ and $v$ from the interval $[0,1]$. Consider the $n$-fold integral $$ J = \int_{\theta_1 \in I_1, \theta_2 \in I_2 \ldots, \theta_n \in I_n} ...
17
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0answers
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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 ...
16
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4answers
618 views

Enumeration and random selection

In Peter J. Cameron's book "Permutation Groups" I found the following quote It is a slogan of modern enumeration theory that the ability to count a set is closely related to the ability to pick a ...
16
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1answer
571 views

Polynomial-time algorithm to compare numbers in Conway chained arrow notation

I am looking for a polynomial-time algorithm which, given a character string containing two numbers in Conway's chained arrow notation for large numbers, indicates whether the first number is less ...
15
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12answers
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Lower Bounds in Theoretical Computer Science

Besides the classical: you can't do comparison sort with faster than (n logn); what are other lower bounds we know of for algorithms? I can't seem to dig them up via google scholar, yet they must ...
15
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5answers
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Are there complexity classes with provably no complete problems?

A problem is said to be complete for a complexity class $\mathcal{C}$ if a) it is in $\mathcal{C}$ and b) every problem in $\mathcal{C}$ is log-space reducible to it. There are natural examples of ...
15
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2answers
425 views

Minimum number of variables on which a multivariate polynomial depends?

Let $p:F_2^n\rightarrow F_2$ be a multivariate polynomial, let's say of degree 3. (Both the degree and the order of the field could probably be replaced by other constants without affecting this ...
15
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891 views

Is this strange problem NP-complete?

The following quadratic expression can be simplified: (x+1)(x+2) + (x+1)(x-3) + 2x(2x-1) - (3x+1)(x-3) - 2x(x+2). What is the easiest way of doing the simplification? (It would be good to think ...
14
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What would be some major consequences of the inconsistency of ZFC?

I was happily surfing the arXiv, when I was jolted by the following paper: Inconsistency of the Zermelo-Fraenkel set theory with the axiom of choice and its effects on the computational complexity by ...
14
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2answers
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Is #k-XORSAT #P-complete?

k-XORSAT is the problem of deciding whether a Boolean formula $$\bigwedge_{i \in I} \oplus_{j=1}^k l_{s_{ij}}$$ is satisfiable. Here $\oplus$ denotes the binary XOR operation, $I$ is some index set, ...
14
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symmetric integer matrices

Suppose I have a symmetric positive definite matrix $M$ with integer entries. I want to decide whether $M = A A^t,$ with $A$ likewise integral. I assume that decision problem is NP-complete, as is the ...