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30

There's an algorithm for multiplying an $N \times N^{0.172}$ matrix with an $N^{0.172} \times N$ matrix in $N^2 \operatorname{polylog}\left(N\right)$ arithmetic operations. The main identity used for it comes from Coppersmith's paper "Rapid multiplication of rectangular matrices", but the explanation for why it leads to $N^2 \operatorname{polylog}\left(N\... 16 The space usage is at most$O(n^2)$for all Strassen-like algorithms (i.e. those based on upper bounding the rank of matrix multiplication algebraically). See Space complexity of Coppersmith–Winograd algorithm However, I realized in my previous answer that I did not explain why the space usage is$O(n^2)$... so here goes something hand-wavy. Consider what a ... 15 I think the answer to your first question is also$\widetilde O(n^3 \log( \|A\| + \|b\|))$due to the following arguments: Edmonds' paper does not describe a variant of Gaussian elimination but it proves that any number computed in a step of the algorithm is a determinant of some submatrix of A. By Schrijver's book on Theory of Linear and Integer Programming ... 13 Take a look at https://www.sciencedirect.com/science/article/pii/S0022000008001141 "The complexity of satisfiability problems: Refining Schaefer's theorem" by Allender et al. which answer your questions: (1) Open (2)$\oplus L = L$(3) Yes 13 Well, one thing is I think that all the constructions we know of - and even the families of potential constructions that people have proposed (e.g., Cohn-Umans approaches, generalizations of Coppersmith-Winograd) - would "simply" produce a family of algorithms$A_\epsilon$running in time$O(n^{2+\epsilon})$. So to have a single algorithm which ran in$O(n^2 ...

12

I cannot say for sure if you will consider the following approach as better, but from a complexity-theoretic point of view there is a more efficient solution. The idea is to rephrase your question in the first-order theory of the rationals with addition and order. You have that $P_1$ is included in $P_2$ if and only if \begin{align*} \Phi := \forall \vec{x}....

12

For matrices of sizes $k = 2,3$ the Matrix Powering Positivity Problem is in $\mathsf{P}$ (cf. this paper to appear in STACS 2015)

12

Consider the problem $\text{MAX-LIN}(R)$ of maximizing the number of satisfied linear equations over some ring $R$, which is often NP-hard, for example in the case $R=\mathbb{Z}$ Take an instance of this problem, $Ax=b$ where $A$ is a $n\times m$ matrix. Let $k=m+1$. Construct a new linear system $\tilde{A}\tilde{x} = \tilde{b}$, where $\tilde{A}$ is a $kn ... 12 There is a very neat randomized algorithm by Cheung, Kwok, and Lau, which computes the rank of an$n\times m$matrix$A$,$n\le m$, in time$O(\mathrm{nnz}(A) + \min\{n^\omega, n\cdot\mathrm{nnz}(A)\})$. Here$\mathrm{nnz}(A)$is the number of nonzero elements of$A$, and$\omega$is the matrix multiplication exponent. Not quite what you are asking for, but ... 11 This is a paper that seems to improve upon the result. https://arxiv.org/pdf/1209.3995.pdf 10 The approach you're describing is the generating function approach. By solving systems of polynomial equations, one can calculate the number of words of given length (generated by any non-terminal), and through it the corresponding asymptotics. For regular languages one can directly use linear algebra to estimate the number of words of given length. The ... 9 It is known that the number of arithmetic operations necessary to compute the determinant of an$n\times n$matrix is$n^{\omega+o(1)}$, where$\omega$is the matrix multiplication constant. See for example this table on Wikipedia, as well as its footnotes and references. Note that the asymptotic complexity of matrix inversion is also the same as matrix ... 8 No. Suppose all$a_i$'s are$0$and all your$b_i$'s are equal; then the polytopes you can get by varying the$b_i$'s are essentially the hypersimplices. But the number of vertices of an$n$-dimensional hypersimplex can be any binomial coefficient$\binom{n}{k}$. In particular choosing$k=n/2$gives an exponential number of extreme points. 8 Yes, it is in$\mathsf{NC}^2$: Mulmuley, K. A fast parallel algorithm to compute the rank of a matrix over an arbitrary field. Combinatorica 7 (1987), no. 1, 101–104. The following (earlier) paper shows that solving a system of linear equations reduces to computing the rank, and thus, together with the above result, solving the system (in particular, ... 8 No. Consider block matrices$A = \left(\begin{matrix} 0 & 0 \\ 0 & X \end{matrix}\right)$(with symmetric$X$) and$B = \left(\begin{matrix} 0 & Y \\ 0 & 0 \end{matrix} \right)$. Computing$AB^T$from$A$,$B$and$AB$means computing$XY^T$from$X$and$Y$, since$AB = 0$8 The paper by Raghavendra is now also published and available here under the title: Correlation Decay and Tractability of CSPs, appeared in the 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). A related article has appeared in the Electronic Colloquium on Computational Complexity, Report No. 7 (2015), available here. 8 The revised conjecture is true, even under relaxed constraints on$S$and$t$—they may be arbitrary integer vectors (as long as the set$S$is finite). Notice that if we arrange the vectors from$S$into a matrix, the question simply asks about the solvability of the linear system $$Sx=t$$ in the integers, hence I will formulate the problem as such below. ... 8 When$k$is given as part of the input, the second problem is equivalent to the monochromatic Max-IP problem (given$S \subseteq \{0,1\}^d$, find$\max_{(a,b) \in S, a\ne b} a \cdot b$). Recently I and Ryan Williams have an (unpublic yet) work showing that when$d = O(\log n)$, OVP and a bichromatic version of Max-IP (given$A,B$, find$\max_{(a,b) \in A \...

8

The paper "Two algorithmic results for the traveling salesman problem" by Barvinok describes an $n^{\mathcal{O}(m)}$ algorithm for computing the permanent of a rank-m matrix. I don't know whether this can be improved to $\mathrm{poly}(n) 2^{\mathcal{O}(m)}$. (I originally posted this as a comment. I post it as an answer by lack of any other answers.)

7

I use the user17410 equivalent formulation: Input: $n$ vectors $X = \{ x_1, \dots, x_m \}$ over $\{0,1\}^n$, $n$ is part of the input Question: Are there two different subsets $A,B \subseteq X$ such that $$\sum_{x \in A} x = \sum_{x \in B} x$$ The hardness proof involve many intermediate reductions that follow the same "chain" used to prove the hardness of ...

7

EDIT: My original proof had a bug. I now believe that it is fixed. We reduce the problem of EQUAL SUM SUBSETS to this problem. EQUAL SUM SUBSETS is the problem of: given a set of $m$ integers, find two disjoint subsets which have the same sum. EQUAL SUM SUBSETS is known to be NP-complete. Suppose these bit strings were not vectors but representations of $n$...

7

The answer to (1) is yes (regardless of the properties D.W. asked for in the comments), depending on how $R$ is given: First, note that since $R$ is finite, the abelian group $(R,+)$ is of the form $\mathbb{Z}_{p_1^{k_1}} \oplus \dotsb \oplus \mathbb{Z}_{p_\ell^{k_\ell}}$, where $\ell \leq \log_2|R|$. Now, if $R$ is only given to you by generators and ...

6

This problem is NP-hard in general. This is equivalent to finding the sparsest vector $y\in \mathbb F^n$ such that $$Ay = -Ax$$ As finding a sparsest solution vector for an underdetermined system of linear equations is NP-hard, so is your problem.

6

Let me give some details for the Cauchy matrix construction, which is simple. Let $x = (x_1, \ldots, x_n)$, $y = (y_1, \ldots, y_m)$ be sequences of pairwise distinct numbers. The corresponding Cauchy matrix is $$C = \begin{pmatrix} \frac{1}{x_1 - y_1} & \frac{1}{x_1 - y_2} & \ldots & \frac{1}{x_1 - y_m}\\ \frac{1}{x_2 - y_1} & \frac{1}{x_2 -... 6 Yes, your problem is essentially equivalent to the one (General Position) in the Alexander Chistov, Hervé Fournier, Leonid Gurvits and Pascal Koiran paper. Consider an n \times m matrix A, n < m. Without loss of generality, assume that \text{rank}(A) = n and the first n columns of A are independent: A =[B\ |\ D], where B is a nonsingular ... 6 Isn't this graph just a collection of cycles? So we only need to compare that all the lengths in the two graphs match (which can be done by sorting the lengths). 6 Elaborating Paul's suggestion for a O(n \log n)-time algorithm: Input: Let u \in [m]^k and v \in [m]^n with k \leq n, where U=[m]=\{1,2,\cdots,m\}. Define polynomials$$p(x,y) = \sum_{i \in [n]} x^i y^{v_i} \qquad \text{and} \qquad q(x,y) = \sum_{j \in [k]} x^{k-j} y^{m-u_j}.$$Compute the polynomial$$r(x,y) = p(x,y) q(x,y).$$Then$$r(x,y) = \...

6

(what follows solves the question for columns, easily adapted to rows either by transposing or by doing a column echelon instead of a row one) You can do the following: put your matrix in reduced echelon form by the Gauss-Jordan elimination method. Note that this preserves $N(A)$. Let's say we work on a 5x5 matrix and that we get: \begin{bmatrix} 1 & ...

6

The problem is NP-complete, by reduction from the following problem: Given an $m\times n$ matrix $A$ with integer entries and an integer vector $b$ with $n$ entries, does there exist a 0-1 vector $x$ with $Ax=b$? For every coordinate $x_i$ of vector $x$, introduce $100(n+m)$ new equations $x_i+y_{i,k}=0$ with $k=1,\ldots,100(n+m)$, and introduce \$100(...

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