Questions tagged [cc.complexity-theory]

P versus NP and other resource-bounded computation.

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232
votes
11answers
116k views

Is Norbert Blum's 2017 proof that $P \ne NP$ correct?

Norbert Blum recently posted a 38-page proof that $P \ne NP$. Is it correct? Also on topic: where else (on the internet) is its correctness being discussed? Note: the focus of this question text has ...
128
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28answers
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Problems Between P and NPC

Factoring and graph isomorphism are problems in NP that are not known to be in P nor to be NP-Complete. What are some other (sufficiently different) natural problems that share this property? ...
79
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14answers
16k views

What kind of mathematical background is needed for complexity theory?

I am currently an undergraduate student, bound to graduate this year. After graduation, I am considering to work towards a TCS master/PhD. I have begun wondering what fields of mathematics are ...
67
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7answers
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Which interesting theorems in TCS rely on the Axiom of Choice? (Or alternatively, the Axiom of Determinacy?)

Mathematicians sometimes worry about the Axiom of Choice (AC) and Axiom of Determinancy (AD). Axiom of Choice: Given any collection ${\cal C}$ of nonempty sets, there is a function $f$ that, given a ...
66
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7answers
4k views

Are $PSPACE$-complete problems inherently less tractable than $NP$-complete problems?

Currently, solving either a $NP$-complete problem or a $PSPACE$-complete problem is infeasible in the general case for large inputs. However, both are solvable in exponential time and polynomial space....
64
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11answers
4k views

What are good references to understanding the proof of the PCP theorem?

I'm familiar with a lot of results that use the PCP theorem (mainly in approximating algorithms), but I've never come across a clear explanation of the PCP theorem (ie, that $\mathsf{NP} = \mathsf{PCP}...
64
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7answers
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Are runtime bounds in P decidable? (answer: no)

The question asked is whether the following question is decidable: Problem  Given an integer $k$ and Turing machine $M$ promised to be in P, is the runtime of $M$ ${O}(n^k)$ with respect ...
63
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1answer
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More on PH in PP?

A recent question by Huck Bennett asking whether the class PH was contained in the class PP, received somewhat contradictory answers (all true, it seems). On one hand, several oracle results were ...
60
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12answers
3k views

Parameterized complexity from P to NP-hard and back again

I'm looking for examples of problems parametrized by a number $k \in \mathbb{N}$, where the problem's hardness is non-monotonic in $k$. Most problems (in my experience) have a single phase transition, ...
59
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8answers
3k views

What is the complexity class most closely associated with what the human mind can accomplish quickly?

This question is something I've wondered about for a while. When people describe the P vs. NP problem, they often compare the class NP to creativity. They note that composing a Mozart-quality ...
58
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4answers
2k views

Problems that can be used to show polynomial-time hardness results

When designing an algorithm for a new problem, if I can't find a polynomial time algorithm after a while, I might try to prove it is NP-hard instead. If I succeed, I've explained why I couldn't find ...
55
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4answers
23k views

Why is 2SAT in P?

I've come across the polynomial algorithm that solves 2SAT. I've found it boggling that 2SAT is in P where all (or many others) of the SAT instances are NP-Complete. What makes this problem different? ...
54
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9answers
22k views

Explain P = NP problem to 10 year old

It is my first question on this site. I am taking a master's course on theory of computation. How you would explain P = NP problem to a 10 year old child and why it has such a monetary reward on it? ...
54
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3answers
2k views

Surprising algorithms for counting problems

There are some counting problems which involve counting exponentially many things (relative to the size of the input), and yet have surprising polynomial-time exact, deterministic algorithms. Examples ...
54
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2answers
3k views

Can one amplify P=NP beyond P=PH?

In Descriptive Complexity, Immerman has Corollary 7.23. The following conditions are equivalent: 1. P = NP. 2. Over finite, ordered structures, FO(LFP) = SO. This can be thought of as "...
53
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1answer
2k views

Is there a gap amplification type of result for the Graph Isomorphism Problem?

Suppose $G_1$ and $G_2$ are two undirected graphs on vertex set $\{1, \dotsc, n\}$. The graphs are isomorphic if and only if there is a permutation $\Pi$ such that $G_1 = \Pi(G_2)$, or more formally, ...
52
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7answers
3k views

For which problems in P is it easier to verify the result than to find it?

For (search versions) of NP-complete problems, verifying a solution is clearly easier than finding it, since the verification can be done in polynomial time, while finding a witness takes (probably) ...
49
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4answers
3k views

Why do we consider log-space as a model of efficient computation (instead of polylog-space) ?

This might be a subjective question rather than one with a concrete answer, but anyway. In complexity theory we study the notion of efficient computations. There are classes like $\mathsf{P}$ stands ...
47
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20answers
7k views

NP-hard problems on trees

Several optimization problems that are known to be NP-hard on general graphs are trivially solvable in polynomial time (some even in linear time) when the input graph is a tree. Examples include ...
47
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6answers
2k views

Ways for a mathematician to stay informed of current research in complexity theory

Complexity theory is a strong secondary interest of mine but it's not my primary research interest, so there is no hope for me to attend all the conferences, read all the blogs, and ensure that the "...
47
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4answers
1k views

What are the consequences of $\mathsf{L}^2 \subseteq \mathsf{P}$?

We know that $\mathsf{L} \subseteq \mathsf{NL} \subseteq \mathsf{P}$ and that $\mathsf{L} \subseteq \mathsf{NL} \subseteq \mathsf{L}^2 \subseteq $ $\mathsf{polyL}$, where $\mathsf{L}^2 = \mathsf{...
46
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4answers
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What are the best current lower bounds on 3SAT?

What are the best current lower bounds for time and circuit depth for 3SAT?
46
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5answers
1k views

Are there Conservation Laws in Complexity Theory?

Let me start with some examples. Why is it so trivial to show CVP is in P but so hard to show LP is in P; while both are P-complete problems. Or take primality. It is easier to show composites in NP ...
45
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5answers
6k views

Is the Chomsky-hierarchy outdated?

The Chomsky(–Schützenberger) hierarchy is used in textbooks of theoretical computer science, but it obviously only covers a very small fraction of formal languages (REG, CFL, CSL, RE) compared to the ...
45
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4answers
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Generalized Ladner's Theorem

Ladner's Theorem states that if P ≠ NP, then there is an infinite hierarchy of complexity classes strictly containing P and strictly contained in NP. The proof uses the completeness of SAT under many-...
45
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3answers
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An NP-complete variant of factoring.

Arora and Barak's book presents factoring as the following problem: $\text{FACTORING} = \{\langle L, U, N \rangle \;|\; (\exists \text{ a prime } p \in \{L, \ldots, U\})[p | N]\}$ They add, further ...
45
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0answers
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Monotone complexity of s-t connectivity

In the problem CONN, we obtain a directed $n$-vertex graph (encoded as a boolean string of $n^2$ bits, one for each potential edge), and want to decide whether there is a path between all $n^2$ pairs $...
44
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8answers
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The importance of Integrality Gap

I always had trouble in understanding the importance of the Integrality Gap (IG) and bounds on it. IG is the ratio of (the quality of) an optimal integer answer to (the quality of) an optimal real ...
44
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8answers
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Obituaries of dead conjectures

I am looking for conjectures about algorithms and complexity that were viewed credible by many at some point in time, but later they were either disproved, or at least disbelieved, due to mounting ...
44
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10answers
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Kolmogorov complexity applications in computational complexity

Informally speaking, Kolmogorov complexity of a string $x$ is a length of a shortest program that outputs $x$. We can define a notion of 'random string' using it ($x$ is random if $K(x) \geq 0.99 |x|$)...
44
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4answers
12k views

Approximation algorithms for Metric TSP

It is known that metric TSP can be approximated within $1.5$ and cannot be approximated better than $123\over 122$ in polynomial time. Is anything known about finding approximation solutions in ...
43
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8answers
7k views

Best Upper Bounds on SAT

In another thread, Joe Fitzsimons asked about "the best current lower bounds on 3SAT." I'd like to go the other way: What's the best current upper bounds on 3SAT? In other words, what is the time ...
43
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5answers
2k views

Theoretical explanations for practical success of SAT solvers?

What theoretical explanations are there for the practical success of SAT solvers, and can someone give a "wikipedia-style" overview and explanation tying them all together? By analogy, the smoothed ...
43
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3answers
4k views

Wikipedia-style explanation of Geometric Complexity Theory

Can someone provide a concise explanation of Mulmuley's GCT approach understandable by non-experts? An explanation that would be suitable for a Wikipedia page on the topic (which is stub at the moment)...
43
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0answers
908 views

Problem unsolvable in $2^{o(n)}$ on inputs with $n$ bits, assuming ETH?

If we assume the Exponential-Time Hypothesis, then there is no $2^{o(n)}$ algorithm for $n$-variable 3-SAT, and many other natural problems, such as 3-COLORING on graphs with $n$ vertices. Notice ...
42
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22answers
4k views

What hierarchies and/or hierarchy theorems do you know?

I am currently writing a survey on hierarchy theorems on TCS. Searching for related papers I noticed that hierarchy is a fundamendal concept not only in TCS and mathematics, but in numerous sciences, ...
41
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12answers
2k views

Gröbner bases in TCS?

Does anyone know of interesting applications of Gröbner bases to theoretical computer science? Gröbner bases are used to solve multi-variate polynomial equations, an NP-hard problem in general. I was ...
40
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5answers
1k views

The cozy neighborhoods of “P” and of “NP-hard”

Let $X$ be an algorithmic task. (It can be a decision problem or an optimization problem or any other task.) Let us call $X$ "on the polynomial side" if assuming that $X$ is NP-hard is known to imply ...
40
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3answers
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What are the reasons that researchers in computational geometry prefer the BSS/real-RAM model?

Background The computation over real numbers are more complicated than computation over natural numbers, since real numbers are infinite objects and there are uncountably many real numbers, therefore ...
40
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3answers
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A fixed-depth characterization of $TC^0$? $NC^1$?

This is a question about circuit complexity. (Definitions are at the bottom.) Yao and Beigel-Tarui showed that every $ACC^0$ circuit family of size $s$ has an equivalent circuit family of size $s^{...
40
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3answers
2k views

Consequences of a quasi-polynomial time algorithm for the graph isomorphism problem

The Graph Isomorphism problem (GI) is arguably the best known candidate for an NP-intermediate problem. The best known algorithm is sub-exponential algorithm with run-time $2^{O(\sqrt{n \log n})}$. ...
39
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7answers
5k views

Many-one reductions vs. Turing reductions to define NPC

Why do most people prefer to use many-one reductions to define NP-completeness instead of, for instance, Turing reductions?
39
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2answers
2k views

Are the problems PRIMES, FACTORING known to be P-hard?

Let PRIMES (a.k.a. primality testing) be the problem: Given a natural number $n$, is $n$ a prime number? Let FACTORING be the problem: Given natural numbers $n$, $m$ with $1 \leq m \leq n$, ...
39
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3answers
2k views

Why are mod_m gates interesting?

Ryan Williams just posted his lower bound on ACC, the class of problems that have constant depth circuits with unbounded fan-in and gates AND, OR, NOT and MOD_m for all possible m's. What's so ...
39
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3answers
5k views

Evidence that matrix multiplication is not in $O(n^2\log^kn)$ time

It is commonly believed that for all $\epsilon > 0$, it is possible to multiply two $n \times n$ matrices in $O(n^{2 + \epsilon})$ time. Some discussion is here. I have asked some people who are ...
39
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3answers
3k views

Is the integer factorization problem harder than RSA factorization: $n = pq$?

This is a cross-post from math.stackexchange. Let FACT denote the integer factoring problem: given $n \in \mathbb{N},$ find primes $p_i \in \mathbb{N},$ and integers $e_i \in \mathbb{N},$ such that $...
38
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9answers
3k views

Optimal greedy algorithms for NP-hard problems

Greed, for lack of a better word, is good. One of the first algorithmic paradigms taught in introductory algorithms course is the greedy approach. Greedy approach results in simple and intuitive ...
38
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5answers
1k views

Is there a logic without induction that captures much of P?

The Immerman-Vardi theorem states that PTIME (or P) is precisely the class of languages that can be described by a sentence of First-Order Logic together with a fixed-point operator, over the class of ...
38
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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
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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 ...