12
votes
Is there a theory that combines category theory/abstract algebra and computational complexity?
[Computational complexity and category theory] seem like such natural pairs.
Given the prominence of computational complexity as a research field, if they were such natural bedfellows, maybe somebody ...
- 10.6k
9
votes
Integer multiplication when one integer is fixed
I am not sure whether this is directly relevant to the question, but the following elementary result might be of interest. Given a fixed natural number $k$, the operation $n \to kn$ can be realized by ...
- 4,761
9
votes
Accepted
VC dimension of polynomials over tropical semirings?
I've realized that the answer to my question is - yes: the VC dimension of degree $\leq d$ polynomials on $n$ variables over any tropical semiring is at most a constant times $n^2\log(n+d)$. This can ...
- 6,655
8
votes
Accepted
Implications of a recent negative result to geometric complexity
It means that to separate permanent from determinant (a la GCT) one must either (a) use actual differences in multiplicities (and not merely their vanishing or non-vanishing) in order to get an ...
- 36.4k
7
votes
Accepted
Commutative matrix multiplication algorithms
In answer to the "Update": yes, for any $c$, the existence of an $O(n^c)$ non-commutative algorithm for matrix multiplication is equivalent to the existence of an $O(n^c)$ commutative algorithm for ...
- 36.4k
6
votes
Riemann Hypothesis and Complexity Theory
Valiant's classes are defined over some field. They can use arbitrary constants from that field. To draw some conclusion about Boolean complexity classes, one needs to replace these arbitrary ...
- 2,838
5
votes
Accepted
DET is $VQP-complete$ and also $DET\in VP$ Does that mean $VP=VQP$
One is used to this kind of argument because of what happens in $P$ and $NP$. However, one has to go back to why it is the case in general. If $P \in B$ is $A$-complete under $\prec$-reductions and $B ...
- 2,059
5
votes
Accepted
Complexity of counting integer roots of multivariate polynomials in a polyhedron?
The decision version of this problem is obviously in $\mathsf{NP}$, and Manders & Adleman showed that a specific case is NP-complete. Namely, even deciding whether there exists an integer $x \in [...
- 36.4k
5
votes
Accepted
Is $GCT$ necessarily a negative result program?
It depends a little what you mean exactly by "GCT". If you mean it more generally, the answer is certainly yes. If you mean it more specifically about multiplicity obstructions, this is a ...
- 36.4k
5
votes
Sorting using ring operations
This is more a comment than an answer, but the space in the comment box was too short. Or if it's an answer, it's one in the other direction: evidence that linear time is possible.
I think you're ...
- 50.7k
4
votes
Accepted
$NP \not\subseteq BPP \implies NP_{\mathbb{C}} \not\subseteq P_{\mathbb{C}}$
As proved in [1], Boolean languages computable in $\mathrm P_\mathbb C$ are in $\mathrm{BPP}$. (They state it for $\mathrm P_\mathbb R$ without inequality tests, which amounts to the same thing.) On ...
- 15.6k
4
votes
Accepted
Can reciprocal inputs speed up monotone computations?
I believe the answer to the Question 1 is negative.
We introduce an auxiliary circuit type: $X_k$-circuits have inputs $x_1, \dots, x_n$ and $1/x_k,\dots,1/x_n$ and have in addition to $+$ and $\...
- 565
3
votes
Accepted
Questions about P vs NP and geometric complexity theory
The short answer is no these are not known, though they are certainly not out of the question. There are no direct implications known to P vs NP, and we do not even have a conjecture (let alone ...
- 36.4k
3
votes
Accepted
In depth reduction of arithmetic formula why we get a $v$ st $\frac{s}3\leq |\Phi_v|\leq \frac{2s}{3}$
You do not cite the part of the survey that is actually relevant for getting the $s/3$ lower bound:
Starting from the root, walk down to the leaves by always taking the child with a larger sub-tree ...
- 2,059
3
votes
IPS upper bound for subset sum axiom
First, Kaveh is correct that the verification for IPS is randomized, so all it would show is $\mathsf{NP} \subseteq \mathsf{coAM}$ (not $\mathsf{NP} = \mathsf{coNP}$). However, this alone would still ...
- 36.4k
2
votes
Is there a theory that combines category theory/abstract algebra and computational complexity?
This answer about isomorphisms between formal languages combines algebraic results from the theory of codes with notions from category theory to investigate possible notions of equivalence and ...
2
votes
IPS upper bound for subset sum axiom
I think what you are missing is probably the complexity of the proof verification algorithm for IPS.
It is generally true that if we have a Cook-Reckhow proof system and have short proofs for a coNP-...
- 21.4k
2
votes
Accepted
What is the computational complexity of solutions over $\mathbb{Q}$ of polynomial equation with coeffiecents over $\mathbb{Z}$
As already pointed out by Emil Jeřábek in the comments, this is Hilbert's Tenth Problem over the rationals, whose computability is a notorious open question.
In this case, note that the Boolean ...
- 36.4k
2
votes
Decomposing outer product or general rank factorization over $\Bbb F_q$
There might be faster algorithms, but it is easy to compute such a factorization (for any $r$) from the reduced row-echelon form of $M$: set $M_2$ to be the RREF with zero rows removed, and $M_1$ to ...
- 1,419
1
vote
Accepted
Complexity of matrix diagonalization
Reducing to a tridiagonal matrix takes $O(n^3)$ independent of $\epsilon$. I believe the fastest algorithm after that is divide and conquer, which I believe is $O(n^2 \log(1/\epsilon))$, for a total ...
- 3,223
1
vote
VNP is closed under taking coefficients using Valiant's criterion
Here is a related statement which can be proven "algebraically" (as opposed to going to the boolean world of #P/poly).
Suppose $f(x_1,\ldots,x_n,y_1,\ldots,y_m) \in \text{VNP}$ for $m = \...
- 11
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