Linked Questions

133
votes
29answers
20k views

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? ...
61
votes
13answers
8k views

Open problems on the frontiers of TCS

In the thread Major unsolved problems in theoretical computer science?, Iddo Tzameret made the following excellent comment: I think we should distinguish between major open problems that are viewed ...
39
votes
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
votes
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
votes
7answers
6k 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?
36
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 ...
28
votes
2answers
4k views

What would be the consequences of factoring being NP-complete?

Are there any references covering this?
26
votes
4answers
1k views

What specific evidence is there for P = RP?

RP is the class of problems decidable by a nondeterministic Turing machine that terminates in polynomial time, but that is also allowed one-sided error. P is the usual class of problems decidable by ...
26
votes
4answers
2k views

Best known deterministic time complexity lower bound for a natural problem in NP

This answer to Major unsolved problems in theoretical computer science? question states that it is open if a particular problem in NP requires $\Omega(n^2)$ time. Looking at the comments under answer ...
24
votes
4answers
9k views

Major unsolved problems in distributed systems?

Inspired by this question, what are the major problems and existing solutions which needs improvement in (theoretical) distributed systems domain. Something like membership protocols, data ...
21
votes
1answer
2k views

Is there a proof that addition is faster than multiplication?

The best upper bound known on the time complexity of multiplication is Martin Fürer's bound $n\log n2^{O(\log^* n)}$, which is more than linear time complexity of addition. Do we have a proof that ...
19
votes
5answers
1k views

P with integer factorization oracle

I just read the "Is integer factorization an NP-complete problem?" question ... so I decided to spend some of my reputation :-) asking another question $Q$ having $P(\text{Q is trivial}) \approx 1$: ...
14
votes
1answer
725 views

Is Quasi-polynomial time in PSPACE?

I had done some search on this but I was not able to find an answer either way. Huck answered it fully. Thanks :)
12
votes
3answers
7k views

What does research in theoretical computer science involve?

I am trying to understand what is involved in theoretical computer science research. What do theoretical computer scientists do? I know a significant time is spent on teaching, supervising graduate ...
11
votes
2answers
1k views

Sources of open problems?

I'm wondering if there are some known sources of open TCS problems? I'm a junior studying math/CS and would like to know of some accessible problems that I could start thinking about! Thanks so much!...
8
votes
2answers
570 views

Is there a list of known problems?

Is there a database of known problems with information about their complexity and algorithms, related problems, references etc that is available to us? [If not, can we make one? I know this is off ...
7
votes
3answers
5k views

Polynomial Time Algorithm for Graph Isomorphism Testing [closed]

"Michael I. Trofimov" claims that he has found a poly-time algorithm for graph isomorphism, which works for all graphs. The paper is given in arXiv. The companion website gives a proof-of-concept ...
3
votes
0answers
278 views

Why can't we have superlinear bounds on Boolean circuit size for an explicit function?

I am interested about the minimal size (number of gates) of a family of circuits (with negation) over a complete Boolean basis (with fanin 2) that computes some explicit Boolean function. (In other ...
2
votes
1answer
376 views

Complexity lower bound of finding the factorial of a number

I was wondering about the complexity of the factorial of a number mostly because this problem is not referenced in the complexity books I have read. Two similar problems, Matrix Multiplication and ...
-1
votes
1answer
169 views

Evidence integer multiplication is in linear time?

After millenia of quest we have identified two $n$ bit integers can be multiplied in $O(n\log n)$ time. Please refer details in https://www.quantamagazine.org/mathematicians-discover-the-perfect-way-...
-2
votes
1answer
275 views

Finding research problem for PhD(TCS)? [closed]

I am a theoratical computer science PhD student. I am wanted some suggestion in how to find research problem for PhD research. I have supervisor and he has given me first problem. We had get progress ...