Add X as a hypothesis, where X is not known to be either true or false.

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9
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0answers
65 views

Consequences of bipartite perfect matching not in NL?

Are any significant consequences known of $\text{BPM} \not\in \textsf{NL}$? I'm interested in the status of the following well-studied decision problem, in particular whether it is known to be ...
10
votes
0answers
190 views

Does a polynomial-time algorithm for factoring product of two primes imply a polynomial-time algorithm for factoring in general?

Is it known if the existence of a polynomial-time algorithm for the promise problem of factoring of numbers with two prime factors implies that factoring in general has a polynomial-time algorithm?
10
votes
0answers
116 views

Hardness in P: methods to show optimality of $O(m^2n)$-like time?

In recent years, there has been exciting work in proving lower bounds for polynomial-time problems conditional on conjectures like SETH or "All-Pairs-Shortest-Paths (APSP) cannot be solved in ...
17
votes
0answers
241 views

Quantum Hardness of Finding Nash Equilibria

This question is inspired by the recent, beautiful work On the Cryptographic Hardness of Finding a Nash Equilibrium by Bitansky, Paneth, and Rosen. Their main result is that the existence of ...
11
votes
2answers
262 views

Consequence of PIT over $\Bbb Z[x_1,\dots,x_n]$ not having efficient algorithm

Given $p(x_1,\dots,x_n),q(x_1,\dots,x_n)\in \Bbb Z[x_1,\dots,x_n]$ such that coefficients of $p,q$ are bounded by $B$, does $p\equiv q$ hold? Schwartz-Zippel lemma applies here since it holds for ...
-1
votes
1answer
294 views

What is the status of intermediate problems if P is not NP in worst way imaginable?

Assume $P\neq BPP\neq NP$ with caveat that there is a deterministic algorithm for every $NP$ complete problem with input size $n$ bits in $2^{(\log n)^{1+f(n)}}$ arithmetic operations on $\log n$ ...
11
votes
1answer
205 views

L/P/PSpace vs P/NP

in 1979 Hopcroft/ Ullman wrote that L ⊆ P ⊆ NP ⊆ PSpace is known but L ⊊ PSpace is the only proper (& trivial) containment known although all are conjectured to be proper containments, and "where ...
3
votes
2answers
246 views

What happens to complexity classes if all $\#P$ problems have polynomial-time algorithms?

As title says what happens to other complexity classes if all $\#P$ (Sharp-P) problems have polynomial-time algorithms? What happens to PSPACE?
9
votes
1answer
325 views

Does P/poly $\neq$ NP/poly have any interesting implications?

$P/poly = NP/poly$ implies $NP \subseteq P/poly$, which in turn has interesting consequences like the collapse of the polynomial hierarchy. Are there interesting implications for $P/poly \neq ...
14
votes
1answer
306 views

$\mathsf{EXP}$ vs $\oplus\mathsf{EXP}$

In our recent work, we resolve a computational problem which arose in combinatorial context, under assumption that $\mathsf{EXP} \ne \mathsf{\oplus{}EXP}$, where $\mathsf{\oplus{}EXP}$ is the ...
8
votes
2answers
285 views

ETH: k-SAT vs. SAT?

Let SAT$_v$ be the language of those instances of SAT that contain variables $[v] = \{0,1,\dots,v-1\}$, let $k$-SAT be the language of those instances of SAT in which every clause has at most $k$ ...
8
votes
3answers
641 views

What would be the consequences of PH=PSPACE?

A recent question (see Consequences of NP=PSPACE) asked for the "nasty" consequences of $NP=PSPACE$. The answers list quite a few collapse consequences, including $NP=coNP$ and others, providing ...
22
votes
4answers
2k views

Consequences of NP=PSPACE

What would be the nasty consequences of NP=PSPACE? I am surprised I did not found anything on this, given that these classes are among the most famous ones. In particular, would it have any ...
6
votes
0answers
310 views

Why do we believe $\mathsf{fewP \ne NP}$?

$\mathsf{fewP}$ ($\mathsf{NP}$ with few witnesses, see the zoo) is one of the important ambiguity-bounded sub-classes of $\mathsf{NP}$. There are interesting natural problems in this class that are ...
17
votes
1answer
483 views

What evidence do we have for $\mathsf{UP} \neq \mathsf{NP}$?

Following Josh Grochow's suggestion, I am converting my comment from a previous question into a new question. What evidence do we have for $\mathsf{UP} \neq \mathsf{NP}$? Here $\mathsf{UP}$ is ...
21
votes
4answers
2k views

Mathematical implications of complexity theory conjectures outside TCS

Do you know interesting consequences of (standard) conjectures in complexity theory in other fields of mathematics (i.e. outside of theoretical computer science)? I would prefer answers where: ...
26
votes
2answers
1k views

Reasons to believe $P \ne NP \cap coNP$ (or not)

It seems that many people believe that $P \ne NP \cap coNP$, in part because they believe that factoring is not polytime solvable. (Shiva Kintali has listed a few other candidate problems here). On ...
2
votes
1answer
121 views

Intermediate Problems between FP and #P

Do there exist intermediate problems (in the sense of Ladner's Theorem) for FP vs. #P? I assume that something is known, because I read some papers concerned with FP/#P dichotomies. However, I ...
11
votes
0answers
301 views

Is it known that $NEXP = \Sigma_2 \implies NEXP = MA$?

Is it known whether the implication $\mathsf{NEXP} = \Sigma_2 \implies \mathsf{NEXP} = \mathsf{MA}$ holds? (The question is inspired by well-known $\mathsf{NEXP} \subseteq \mathsf{P/poly} ...
4
votes
1answer
452 views

On the proof of Meyer's Theorem

Meyer's theorem is one of the classical results about collapse of the polynomial hierarchy such as famous Karp Lipton's theorem, and states that $EXP \subseteq P/poly \Rightarrow EXP = \Sigma_{2}^{p} ...
2
votes
0answers
97 views

Conditional Results on Bounded Depth Circuit Hierarchy

$\mathsf{AC,ACC,TC}$-hierarchy are basic bounded depth circuit hierarchies. $AC$-hierarchy is $\bigcup _{i =0}^{\infty} AC^{i} $ , where $AC^{i}$ is the $i$-th level of the hierarchy: a family of ...
4
votes
1answer
170 views

bounded language complete for NSPACE(log n)?

What are the consequences of a sparse language being complete for $\mathsf{NSPACE(\log n)}$ under deterministic $O(\log n)$-space many-one reductions? Is there an analog of Mahaney's Theorem for ...
11
votes
1answer
674 views

Consensus on P = NP in a world where RP = NP

$RP = NP$ is widely conjectured to be false. But imagine for a moment that it is true. In such case, how likely would be that $P = NP$? Put in other words: in a world where $RP = NP$, what might ...
18
votes
1answer
376 views

Consequences of $BQP \subseteq P/poly$?

While Adleman's theorem shows, that $\mathsf{BPP} \subseteq \mathsf{P}/\text{poly}$, I'm not aware of any literature investigating the possible inclusion of $\mathsf{BQP} \subseteq ...
2
votes
1answer
120 views

Consequences of a $p$-optimal proof system for $\operatorname{TAUT}$

I'm reading a paper which shows the result: $(1)$ There is a $p$-optimal proof system for $\operatorname{TAUT}$. $\Leftrightarrow$ $(2)$ $L_{\leq}$ is a $P$-bounded logic for $P$. Both $(1)$ and ...
39
votes
2answers
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 = ...
5
votes
0answers
636 views

Does L=P imply any new complexity class separations?

If L=P then P is not equal to PSPACE. This follows from PSPACE properly containing L. I am wondering if L=P implies any stronger separation between complexity classes? Does it imply P is properly ...
9
votes
1answer
635 views

Does $\mathsf{EXP}=\mathsf{NEXP}$ imply $\mathsf{E}=\mathsf{NE}$?

Does $\mathsf{EXP}=\mathsf{NEXP}$ imply $\mathsf{E}=\mathsf{NE}$?
11
votes
4answers
2k views

Consequences of $NP=coNP$ and $P\ne NP$?

We know that if $P=NP$ then the whole PH collapses. What if the polynomial hierarchy collapses partially ? (Or how to understand that PH could collapse above a certain point and not below ?) In ...
10
votes
0answers
353 views

VNP = VP versus complexity classes in Arithmetic Geometry

What is the implication of $VNP = VP$ to cryptography schemes such as Elliptic curve/Abelian Variety/Arithmetic Geometry based cryptography? Are there any papers or books that talk about sophisticated ...
29
votes
2answers
894 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 ...
42
votes
0answers
1k 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 ...
28
votes
3answers
2k 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 ...
19
votes
1answer
481 views

Problems in NP but not in Average-P/poly

The Karp–Lipton Theoem states that if $\mathsf{NP} \subset \mathsf{P/poly}$, then $\mathsf{PH}$ collapses to $\mathsf{\Sigma^P_2}$. Therefore, assuming separations between $\mathsf{\Sigma^P_2}$ and ...
18
votes
1answer
877 views

Consequences of UP equals NP

EDIT at 2011/02/08: After some references finding and reading, I decided to separate the original question into two separate ones. Here's the part concerning UP vs NP, for the syntactic and semantic ...
17
votes
4answers
711 views

If P = BQP, does this imply that PSPACE (= IP) = AM?

Recently, Watrous et al proved that QIP(3) = PSPACE a remarkable result. This was a surprising result to myself to say the least and it set me off thinking... I wondered what if Quantum Computers ...
30
votes
4answers
771 views

Hardness of approximation assuming NP != coNP

Two of the common assumptions for proving hardness of approximation results are $P \neq NP$ and Unique Games Conjecture. Are there any hardness of approximation results assuming $NP \neq coNP$ ? I am ...
26
votes
3answers
1k views

Consequences of existence of a strongly polynomial algorithm for linear programming?

One of the holy grails of algorithm design is finding a strongly polynomial algorithm for linear programming, i.e., an algorithm whose runtime is bounded by a polynomial in the number of variables and ...
26
votes
3answers
1k views

A decision problem which is not known to be in PH but will be in P if P=NP

Edit: As Ravi Boppana correctly pointed out in his answer and Scott Aaronson also added another example in his answer, the answer to this question turned out to be “yes” in a way which I had not ...
31
votes
3answers
953 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 ...
49
votes
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 ...
26
votes
2answers
2k 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 ($P \ne NP$) ...
29
votes
3answers
2k 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 ...
10
votes
0answers
311 views

Collapsing of exptime and alternation bounded turing machine

This question was already asked on math overflow, but I did not find any answer to my question (or let say the answer was that up to the knowledge of those people, no answer were known) Let C be a ...
28
votes
0answers
650 views

Does $EXP\neq ZPP$ imply 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 ...
23
votes
2answers
904 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 ...
24
votes
2answers
886 views

What are the consequences of Parity-L = P?

Parity-L is the set of languages recognized by a non-deterministic Turing machine which can only distinguish between an even number or odd number of "acceptance" paths (rather than a zero or non-zero ...
24
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 ...
20
votes
1answer
1k views

Consequences of Complete problems for NP intersects coNP

What are the consequences of having complete problems in $NP\cap coNP$?