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Questions tagged [conditional-results]

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

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37 votes
2 answers
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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 ($P \ne NP$) ...
Mohammad Al-Turkistany's user avatar
65 votes
5 answers
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 ...
Robin Kothari's user avatar
55 votes
2 answers
4k 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 "...
András Salamon's user avatar
40 votes
3 answers
4k 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 ...
Benno's user avatar
  • 403
20 votes
1 answer
1k 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 ...
Hsien-Chih Chang 張顯之's user avatar
30 votes
3 answers
2k 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 ...
Tsuyoshi Ito's user avatar
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38 votes
3 answers
5k 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 ...
Giorgio Camerani's user avatar
29 votes
2 answers
2k 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 ...
Niel de Beaudrap's user avatar
49 votes
4 answers
2k 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{...
argentpepper's user avatar
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31 votes
4 answers
5k 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 ...
Denis's user avatar
  • 8,903
28 votes
1 answer
2k views

Consequences of Complete problems for NP intersects coNP

What are the consequences of having complete problems in $NP\cap coNP$?
Marcos Villagra's user avatar
27 votes
4 answers
2k 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 ...
András Salamon's user avatar
22 votes
1 answer
1k 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 \mathsf{P}/\text{...
Martin Schwarz's user avatar
20 votes
1 answer
736 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 $\...
Sadeq Dousti's user avatar
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18 votes
1 answer
740 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 the ...
Sasho Nikolov's user avatar
36 votes
3 answers
2k 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 $$(\...
András Salamon's user avatar
36 votes
2 answers
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 ...
Giorgio Camerani's user avatar
33 votes
3 answers
2k 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 ...
Ian's user avatar
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32 votes
4 answers
889 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 ...
Shiva Kintali's user avatar
27 votes
1 answer
939 views

What are consequences of the collapse of CH?

I don't grasp the full complexity of the counting hierarchy $CH$. I understand $CH$ is in $PSPACE$, and contains $PH$ within its second level, due to the Toda's theorem. But, what would be important ...
neophyte's user avatar
  • 531
14 votes
1 answer
1k 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 $P/poly \neq NP/poly$...
Thomas Klimpel's user avatar
11 votes
1 answer
1k views

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

Does $\mathsf{EXP}=\mathsf{NEXP}$ imply $\mathsf{E}=\mathsf{NE}$?
argentpepper's user avatar
  • 2,281
10 votes
0 answers
363 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 ...
Arthur MILCHIOR's user avatar
4 votes
1 answer
224 views

Limits of variants of Independent Set?

Independent Set (IS) is the $\mathsf{NP}$-complete decision problem Input: graph $G$ with $v=|V(G)|$, integer $k$ Question: is there an independent set $S \subseteq V(G)$ with at least $k$ vertices? ...
András Salamon's user avatar
4 votes
1 answer
201 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 $\...
user13417's user avatar
3 votes
1 answer
217 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 couldn'...
dean's user avatar
  • 31