Questions tagged [cc.complexity-theory]

P versus NP and other resource-bounded computation.

Filter by
Sorted by
Tagged with
38
votes
3answers
8k views

Is optimally solving the n×n×n Rubik's Cube NP-hard?

Consider the obvious $n\times n\times n$ generalization of the Rubik's Cube. Is it NP-hard to compute the shortest sequence of moves that solves a given scrambled cube, or is there a polynomial-time ...
38
votes
2answers
3k views

Mulmuley's GCT program

It is sometimes claimed that Ketan Mulmuley's Geometric Complexity Theory is the only plausible program for settling the open questions of complexity theory like P vs. NP question. There has been ...
37
votes
6answers
2k views

Geometric problems that are NP-complete in $R^3$ but tractable in $R^2$?

A number of geometric problems are easy when considered in $R^1$, but are NP-complete in $R^d$ for $d\geq2$ (including one of my favourite problems, unit disk cover). Does anyone know of a problem ...
37
votes
4answers
3k views

Is $PH \subseteq PP$?

We know that the first level of the polynomial hierarchy (i.e. NP and co-NP) is in PP, and that $PP \subseteq PSPACE$. We also know from Toda's Theorem that $PH \subseteq P^{PP}$. Do we know whether $...
37
votes
3answers
3k views

Does $VP \neq VNP$ imply $P \neq NP$?

As far as I understand, the geometric complexity theory program attempts to separate $VP \neq VNP$ by proving that the permament of a complex-valued matrix is much harder to compute than the ...
37
votes
3answers
6k views

P and NP classes explanation through lambda-calculus

In the introduction and explanation P and NP complexity classes often given through Turing machine. One of the model of computation is the lambda-calculus. I understand, that all of models of ...
37
votes
4answers
1k views

Examples where the uniqueness of the solution makes it easier to find

The complexity class $\mathsf{UP}$ consists of those $\mathsf{NP}$-problems that can be decided by a polynomial time nondeterministic Turing machine which has at most one accepting computational path. ...
37
votes
8answers
2k views

Formal notion for energy complexity of computational problems

Computational complexity includes the study of time or space complexity of computational problems. From the the perspective of mobile computing, energy is very valuable computational resource. So, Is ...
37
votes
2answers
3k views

Axioms necessary for theoretical computer science

This question is inspired by a similar question about applied mathematics on mathoverflow, and that nagging thought that important questions of TCS such as P vs. NP might be independent of ZFC (or ...
37
votes
3answers
2k views

Complexity of exponential function

We know that the exponential function $\exp(x,y) = x^y$ over natural numbers is not computable in polynomial time, because the size of the output is not polynomially bounded in the size of the inputs. ...
37
votes
3answers
1k views

Parameterized complexity of Hitting Set in finite VC-dimension

I'm interested in the parameterized complexity of what I'll call the d-Dimensional Hitting Set problem: given a range space (i.e. a set system / hypergraph) S = (X,R) having VC-dimension at most d and ...
36
votes
13answers
2k views

Easy decision problem, hard search problem

Deciding whether a Nash equilibrium exists is easy (it always does); however, actually finding one is believed to be difficult (it is PPAD-Complete). What are some other examples of problems where ...
36
votes
3answers
2k views

Techniques for showing that problem is in hardness “limbo”

Given a new problem in $\mathsf{NP}$ whose true complexity is somewhere between $\mathsf{P}$ and being NP-complete, there are two methods that I know of that might be used to prove that resolving this ...
36
votes
4answers
3k views

Hardness of approximation without the PCP theorem

An important application of the PCP theorem is that it yields "hardness of approximation" type results. In some relatively simpler cases one can prove such hardness without PCP. Is there, however, any ...
36
votes
2answers
3k views

Semantic vs. Syntactic Complexity Classes

In his "Computational Complexity" book, Papadimitriou writes: RP is in some sense a new and unusual kind of complexity class. Not any polynomially bounded nondeterministic Turing machine can be the ...
36
votes
3answers
908 views

Why does randomness have stronger effect on reductions than on algorithms?

It is conjectured that randomness does not extend the power of polynomial time algorithms, that is, ${\bf P}={\bf BPP}$ is conjectured to hold. On the other hand, randomness seems to have a quite ...
36
votes
2answers
2k views

If P=NP, could we obtain proofs of Goldbach's Conjecture etc.?

This is a naive question, out of my expertise; apologies in advance. Goldbach's Conjecture and many other unsolved questions in mathematics can be written as short formulas in predicate calculus. For ...
36
votes
5answers
1k views

Complexity of testing for a value versus computing a function

In general we know that the complexity of testing whether a function takes a particular value at a given input is easier than evaluating the function at that input. For example: Evaluating the ...
35
votes
10answers
6k views

Most important new papers in computational complexity

We often hear about classic research and publications in the field of computational complexity (Turing, Cook, Karp, Hartmanis, Razborov etc). I was wondering if there are recently published papers ...
35
votes
9answers
2k views

Surprising Results in Complexity (Not on the Complexity Blog List)

What were the most surprising results in complexity? I think it would be useful to have a list of unexpected/surprising results. This includes both results that were surprising and came out of ...
35
votes
4answers
1k views

Interactive proofs for levels of the polynomial hierarchy

We know that if you have a PSPACE machine, it's powerful enough to give an interactive proof of any level the polynomial hierarchy. (And if I remember right, all you need is #P.) But suppose you want ...
35
votes
5answers
2k views

Integer multiplication when one integer is fixed

Let $A$ be a fixed positive integer of size $n$ bits. One is allowed to pre-process this integer as appropriate. Given another positive integer $B$ of size $m$ bits, what is the complexity of ...
35
votes
3answers
3k views

Consequences of Factoring being in P?

Factoring is not known to be NP-complete. This question asked for consequences of Factoring being NP-complete. Curiously, no one asked for consequences of Factoring being in P (maybe because such a ...
35
votes
3answers
1k views

NC = P consequences?

The Complexity Zoo points out in the entry on EXP that if L = P then PSPACE = EXP. Since NPSPACE = PSPACE by Savitch, as far as I can tell the underlying padding argument extends to show that $$(\...
35
votes
1answer
2k views

Consequences of $\mathsf{NP}$ containing $\mathsf{BPP}$

Many believe that $\mathsf{BPP} = \mathsf{P} \subseteq \mathsf{NP}$. However we only know that $\mathsf{BPP}$ is in the second level of polynomial hierarchy, i.e. $\mathsf{BPP}\subseteq \Sigma^ \...
35
votes
1answer
1k views

$BQP$ vs $QMA$?

The central problem of complexity theory is arguably $P$ vs $NP$. However, since Nature is quantum, it would seem more natural to consider the classes $BQP$ (ie decision problems solvable by a ...
35
votes
1answer
1k views

Efficiently computable function as a counter-example to Sarnak's Mobius conjecture

Recently, Gil Kalai and Dick Lipton both wrote nice articles on an interesting conjecture proposed by Peter Sarnak, an expert in number theory and the Riemann Hypothesis. Conjecture. Let $\mu(k)$ ...
35
votes
1answer
1k views

NP-Completeness of the decision problem for the generalized 15-puzzle

I am interested in the natural generalization of the famous 15-puzzle, where you have to slide blocks until you have sorted all given numbers (usally there is a gap of 1 block). Now the ...
34
votes
17answers
2k views

Hardness jumps in computational complexity?

Minimum bandwidth problem is to a find an ordering of graph nodes on integer line that minimizes the largest distance between any two adjacent nodes. A $k$-caterpillar is a tree formed from main path ...
34
votes
4answers
921 views

Correspondence between complexity classes and logic

I took a class once on Computability and Logic. The material included a correlation between complexity / computability classes (R, RE, co-RE, P, NP, Logspace, ...) and Logics (Predicate calculus, ...
34
votes
2answers
4k views

Status of Impagliazzo's Worlds?

In 1995, Russell Impagliazzo proposed five complexity worlds: 1- Algorithmica: $P=NP$ with all the amazing consequences. 2- Heuristica: $NP$-complete problems are hard in the worst-case ($P \ne NP$) ...
34
votes
2answers
2k views

Consequences of $SAT \in BQP$

As a TCS amateur, I'm reading some popular, very introductory material on quantum computing. Here are the few elementary bits of information I've learned so far: Quantum computers are not known to ...
33
votes
8answers
3k views

Problems with big open complexity gaps

This question is about problems for which there is a big open complexity gap between known lower bound and upper bound, but not because of open problems on complexity classes themselves. To be more ...
33
votes
5answers
4k views

The unreasonable power of non-uniformity

From the common sense point of view, it is easy to believe that adding non-determinism to $\mathsf{P}$ significantly extends its power, i.e., $\mathsf{NP}$ is much larger than $\mathsf{P}$. After all,...
33
votes
5answers
4k views

Fast Reduction from RSA to SAT

Scott Aaronson's blog post today gave a list of interesting open problems/tasks in complexity. One in particular caught my attention: Build a public library of 3SAT instances, with as few variables ...
33
votes
3answers
1k views

Hardest known natural problem in P?

I wonder, what is (currently) the largest number $k$, such that a natural problem is known with the following properties: An $O(n^k)$ algorithm has been already found for the problem. For any fixed $\...
33
votes
3answers
2k views

complexity of greatest common divisor (gcd)

Consider the following counting problem (or the associated decision problem): Given two positive integers encoded in binary, compute their greatest common divisor (gcd). What is the smallest ...
33
votes
2answers
1k views

Cohomological approach to boolean complexity

A few years ago, there was some work by Joel Friedman relating lower circuit bounds to Grothendieck cohomology (see papers: http://arxiv.org/abs/cs/0512008, http://arxiv.org/abs/cs/0604024). Has this ...
33
votes
2answers
2k views

NTIME(n^k) ≠ DTIME(n^k) ?

In "On determinism versus nondeterminism and related problems" (Proc. IEEE FOCS, pages 429–438, 1983), Paul, Pippenger, Szemerédi and Trotter proved that $\mathsf{NTIME}(n)\neq\mathsf{DTIME}(n)$. ...
33
votes
2answers
1k views

“Steve's class”: origin of SC

We "know" that $\mathsf{SC}$ is named for Steve Cook and $\mathsf{NC}$ is named for Nick Pippenger. If I'm not mistaken, Steve Cook named NC in honor of Nick Pippenger, and I was told that the reverse ...
33
votes
2answers
2k views

When does “X is NP-complete” imply “#X is #P-complete”?

Let $X$ denote a (decision) problem in NP and let #$X$ denote its counting version. Under what conditions is it known that "X is NP-complete" $\implies$ "#X is #P-complete"? Of course the existence ...
33
votes
1answer
1k views

Computational complexity of pi

Let $L = \{ n : \text{the }n^{th}\text{ binary digit of }\pi\text{ is }1 \}$ (where $n$ is thought of as encoded in binary). Then what can we say about the computational complexity of $L$? It's ...
33
votes
2answers
1k views

Does LOGLOG = NLOGLOG?

Define LOGLOG as the class of languages which can be computed in space O(loglog n) by a deterministic Turing machine (with two-way access to the input). Similarly define NLOGLOG as the class of ...
32
votes
5answers
6k views

NEXP-complete problems

There are tons of NP-complete problems around and sources collecting them, e.g. see the book by Garey and Johnson. I would be interested to see a list of NEXP-complete problems as well. Is there one ...
32
votes
3answers
2k views

Is this variation of TQBF still PSPACE-complete?

Deciding if a quantified boolean formula such as $\forall x_1 \exists x_2 \forall x_3\cdots \exists x_n \varphi(x_1, x_2,\ldots , x_n),$ always evaluates to true is a classical PSPACE-complete ...
32
votes
3answers
2k views

Is $AC^0/poly \cap NP$ contained in $P$?

I thought I would share this question as it might be interesting for other users here. Assume that a function which is in a uniform class (like $NP$) is also in a small nonuniform class (like $AC^0/...
32
votes
5answers
2k views

Programming languages for efficient computation

It is impossible to write a programming language that allows all machines that halt on all inputs and no others. However, it seems to be easy to define such a programming language for any standard ...
32
votes
5answers
2k views

Evidence that PPAD is hard?

There is often-quoted philosophical justification for believing that P != NP even without proof. Other complexity classes have evidence that they are distinct, because if not, there would be "...
32
votes
2answers
1k views

How difficult is it to use the Mulmuley-Sohoni GCT approach to show *known* complexity separations?

In this guest post by Josh Grochow at the complexity weblog he reports on a recent workshop devoted to GCT that was held at Princeton in July. Several of the attendees argued that we should use GCT ...
32
votes
3answers
928 views

An Anthology of Complexity Assumptions

In the paper The Random Oracle Hypothesis Is False, the authors (Chang, Chor, Goldreich, Hartmanis, Håstad, Ranjan, and Rohatgi) discuss the implications of the random-oracle hypothesis. They argue ...

1
2
3 4 5
55