Questions tagged [algebraic-complexity]

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12
votes
1answer
669 views

automorphism in Cai-Furer-Immerman gadgets

In the famous counter example for graph isomorphism via Weisfeiler-Lehman (WL) method the following gadget was constructed in this paper by Cai, Furer and Immerman. They construct a graph $X_k = (V_k, ...
25
votes
2answers
2k views

Sum-of-squares proof system

Recently I have seen several articles on arxiv that refer to a proof system called sum-of-squares. Can someone explain what is a sum-of-squares proof and why such proofs are important/interesting? ...
5
votes
2answers
377 views

Complexity of algorithm to test if a graph is asymmetric

Counting the order of automorphism group of a graph is polynomial-time equivalent to graph isomorphism problem. But if we just want to know if the order is greater than 1, what is the complexity of ...
13
votes
1answer
832 views

Smallest known formula for the determinant

The smallest known formula for the determinant has size $n^{\mathcal O(\log n)}$ according to the folklore (or to Ran Raz in its paper Multi-Linear Formulas for Permanent and Determinant are of Super-...
13
votes
2answers
951 views

Gaussian Elimination in terms of Group Action

Gaussian elimination makes determinant of a matrix polynomial-time computable. The reduction of complexity in computing the determinant, which is otherwise sum of exponential terms, is due to ...
18
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2answers
1k views

Is there a theory that combines category theory/abstract algebra and computational complexity?

Category theory and abstract algebra deal with the way functions can be combined with other functions. Complexity theory deals with how hard a function is to compute. It's weird to me that I haven't ...
13
votes
1answer
451 views

Capacity of Uniquely Solvable Puzzle (USP)

In their seminal paper Group-theoretic algorithms for matrix multiplications, Cohn, Kleinberg, Szegedy and Umans introduce the concept of uniquely solvable puzzle (defined below) and USP capacity. ...
8
votes
1answer
239 views

Is tensor rank is in VNP?

Is it known if tensor rank of three dimensional tensors lies in VNP (non deterministic valiant class)? If yes, what is known about high dimensional tensor rank? In fact I am interested in much more ...
5
votes
1answer
432 views

Classical Matrix-Vector multiplication Complexity of standard matrices

Why are standard unitary transforms such as the Fourier and the Hadamard transforms believed to have a multiplicative complexity (number of multiplications) of $O(n^{1+\delta_{m}})$ and an additive ...
10
votes
2answers
1k views

Determinant of a generalized Vandermonde matrix

Moore matrix is similar to Vandermonde matrix but has a slightly modified definition. http://en.wikipedia.org/wiki/Moore_matrix What is the complexity of computing the determinant of a given $n \...
8
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1answer
517 views

Conditional results implying difficulty of improving upper/lower bounds for permanent

Let $A$ be a given square matrix. Is there any evidence that beating quadratic lower bounds for $B$ such that $\text{det}(B) = \text{per}(A)$ could be hard? Is there any plausible conjecture which ...
38
votes
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 ...
9
votes
1answer
609 views

Permanent of a $3 \times 3$ and $4 \times 4$ matrix from determinants

Let $A$ be a $3 \times 3$ or a $4 \times 4$ matrix with entries $a_{ij}$. Can someone provide me a matrix $B$ so that $\operatorname{per}(A) = \det(B)$? What is the smallest explicit $B$ that is known ...
5
votes
1answer
566 views

Permanent as projection of determinant and another permanent

I am looking for explicit examples where permanent of a given matrix $A$ is given by a determinant of a larger matrix $B$ (projection in the sense of Valiant). Is there any reference where I can find ...
6
votes
2answers
493 views

Iterative algorithms in algebraic complexity (Blum-Shub-Smale-Model)

I know the Blum-Shub-Smale model. It is claimed to provide a theoretical framework for algorithms in real and complex algebra and analysis. A very general question: Most algorithms compromise of ...
12
votes
1answer
280 views

Explicit polynomials in 1 variable with superlogarithmic circuit complexity lower bounds?

By counting arguments, one can show that there exist polynomials of degree n in 1 variable (i.e., something of the form $a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0)$ which have circuit complexity n. ...
15
votes
4answers
561 views

Are there known to exist functions with the following direct-sum property?

This question can be asked either in the framework of circuit complexity of Boolean circuits, or in the framework of algebraic complexity theory, or probably in lots of other settings. It is easy to ...
36
votes
5answers
2k views

Integer multiplication when one integer is fixed

$n$ is a parameter in the problem. For every $n$ we pick a random integer $a_n\in\{2^{n-1},2^{n-1}+1,\dots,2^n-1\}$ where $n\in\{1,2,\dots\}$. Problem: Given $n$ what is the complexity of ...
9
votes
1answer
1k views

True Bit Complexity of matrix multiplication is $O(n^{4})$

Matrix multiplication using regular (row - column inner product) technique takes $O(n^{3})$ multiplucations and $O(n^{3})$ additions. However assuming equal sized entries (number of bits in each entry ...
17
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3answers
638 views

Formal representation of rings in computations

While reading a paper about using algebraic methods to detect some induced subgraphs, it appears that edge ideal is an important tool connecting commutative algebra and graph theory. Since I'm not ...
15
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3answers
650 views

Hardness Guarantees for AES

Many public-key cryptosystems have some kind of provable security. For example, the Rabin cryptosystem is provably as hard as factoring. I wonder whether such kind of provable security exists for ...
10
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2answers
427 views

Lower bounds for linear satisfiability problem

In SODA 1995, Jeff Erickson showed lower bounds for linear satisfiability (checking if a some $r$-subset of $n$ real numbers satisfies a linear equation on $r$ variables). The proof method uses ...
37
votes
3answers
3k 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 ...
36
votes
5answers
2k views

Complexity of testing for a value versus computing a function

In general we know that the complexity of testing whether a function takes a particular value at a given input is easier than evaluating the function at that input. For example: Evaluating the ...

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