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Yaroslav Shitov

Publications and source records attributed to Yaroslav Shitov.

At least 37 records · Page 2Linked to original sources
arXiv2017

Which semifields are exact?

Every (left) linear function on a subspace of a finite-dimensional vector space over a (skew) field can be extended to a (left) linear function on the whole space. This paper explores the extent to what this basic fact of linear algebra is applicable to more general structures. Semifields with a similar property imposed on linear functions are called (left) exact, and we present a complete description of such semifields. Namely, we show that a semifield $S$ is left exact if and only if $S$ is either a skew field or an idempotent semiring. In particular, our result is new even for the tropical semiring and gives a solution to the problem posed by Wilding. Also, we point out several problems that require further investigation.

math.CO↗
arXiv2017

Low-Rank Matrix Approximation in the Infinity Norm

The low-rank matrix approximation problem with respect to the entry-wise $\ell_{\infty}$-norm is the following: given a matrix $M$ and a factorization rank $r$, find a matrix $X$ whose rank is at most $r$ and that minimizes $\max_{i,j} |M_{ij} - X_{ij}|$. In this paper, we prove that the decision variant of this problem for $r=1$ is NP-complete using a reduction from the problem `not all equal 3SAT'. We also analyze several cases when the problem can be solved in polynomial time, and propose a simple practical heuristic algorithm which we apply on the problem of the recovery of a quantized low-rank matrix.

cs.CC↗
arXiv2017

Three observations on spectra of zero-nonzero patterns

Using standard techniques from combinatorics, model theory, and algebraic geometry, we prove generalized versions of several basic results in the theory of spectrally arbitrary matrix patterns. Also, we point out a counterexample to a conjecture proposed recently by McDonald and Melvin.

math.CO↗
arXiv2017

Column subset selection is NP-complete

Let $M$ be a real $r\times c$ matrix and let $k$ be a positive integer. In the column subset selection problem (CSSP), we need to minimize the quantity $\|M-SA\|$, where $A$ can be an arbitrary $k\times c$ matrix, and $S$ runs over all $r\times k$ submatrices of $M$. This problem and its applications in numerical linear algebra are being discussed for several decades, but its algorithmic complexity remained an open issue. We show that CSSP is NP-complete.

math.CO↗
arXiv2017

A counterexample on spectra of zero patterns

An $n\times n$ zero pattern $S$, which is a matrix with entries $*$ and $0$, is called spectrally arbitrary with respect to a field $F$ if any monic polynomial $f$ of degree $n$ can be realized as the characteristic polynomial of a matrix obtained from $S$ by replacing the $*$'s with non-zero elements of $F$. We construct an $n\times n$ zero pattern that is spectrally arbitrary with respect to $\mathbb{C}$ and has $2n-1$ nonzero entries.

math.CO↗
arXiv2016

Counterexamples on spectra of sign patterns

An $n\times n$ sign pattern $S$, which is a matrix with entries $0,+,-$, is called spectrally arbitrary if any monic real polynomial of degree $n$ can be realized as a characteristic polynomial of a matrix obtained by replacing the non-zero elements of $S$ by numbers of the corresponding signs. A sign pattern $S$ is said to be a superpattern of those matrices that can be obtained from $S$ by replacing some of the non-zero entries by zeros. We develop a new technique that allows us to prove spectral arbitrariness of sign patterns for which the previously known "Nilpotent Jacobian" method does not work. Our approach leads us to solutions of numerous open problems known in the literature. In particular, we provide an example of a sign pattern $S$ and its superpattern $S'$ such that $S$ is spectrally arbitrary but $S'$ is not, disproving a conjecture proposed in 2000 by Drew, Johnson, Olesky, and van den Driessche.

math.CO↗
arXiv2016

How hard is the tensor rank?

We investigate the computational complexity of tensor rank, a concept that plays fundamental role in different topics of modern applied mathematics. For tensors over any integral domain, we prove that the rank problem is polynomial time equivalent to solving a system of polynomial equations over this integral domain. Our result gives a complete description of the algorithmic complexity of tensor rank and allows one to solve several known open problems. In particular, the tensor rank over $\mathbb{Z}$ turns out to be undecidable, which answers the question posed by Gonzalez and Ja'Ja' in 1980. We generalize our result and prove that the symmetric rank admits a similar description of computational complexity as the one we give for usual rank. In particular, computing the symmetric rank of a rational tensor is shown to be NP-hard, which proves a recent conjecture of Hillar and Lim. As a byproduct of our approach, we get a similar characterization of the algorithmic complexity of the minimal rank matrix completion problem, which gives a complete answer to the question discussed in 1999 by Buss, Frandsen, and Shallit.

math.CO↗
arXiv2016

On the complexity of failed zero forcing

Let $G$ be a simple graph whose vertices are partitioned into two subsets, called filled vertices and empty vertices. A vertex $v$ is said to be forced by a filled vertex $u$ if $v$ is a unique empty neighbor of $u$. If we can fill all the vertices of $G$ by repeatedly filling the forced ones, then we call an initial set of filled vertices a forcing set. We discuss the so-called failed forcing number of a graph, which is the largest cardinality of a set which is not forcing. Answering the recent question of Ansill, Jacob, Penzellna, Saavedra, we prove that this quantity is NP-hard to compute. Our proof also works for a related graph invariant which is called the skew failed forcing number.

math.CO↗
arXiv2016

Euclidean distance matrices and separations in communication complexity theory

A Euclidean distance matrix $D(α)$ is defined by $D_{ij}=(α_i-α_j)^2$, where $α=(α_1,\ldots,α_n)$ is a real vector. We prove that $D(α)$ cannot be written as a sum of $\left[2\sqrt{n}-2\right]$ nonnegative rank-one matrices, provided that the coordinates of $α$ are algebraically independent. This result allows one to solve several open problems in computation theory. In particular, we provide an asymptotically optimal separation between the complexities of quantum and classical communication protocols computing a matrix in expectation.

math.CO↗
arXiv2016

Coloring the power graph of a semigroup

Let $G$ be a semigroup. The vertices of the power graph $\mathcal{P}(G)$ are the elements of $G$, and two elements are adjacent if and only if one of them is a power of the other. We show that the chromatic number of $\mathcal{P}(G)$ is at most countable, answering a recent question of Aalipour et al.

math.CO↗
arXiv2016

The complexity of positive semidefinite matrix factorization

Let $A$ be a matrix with nonnegative real entries. The PSD rank of $A$ is the smallest integer $k$ for which there exist $k\times k$ real PSD matrices $B_1,\ldots,B_m$, $C_1,\ldots,C_n$ satisfying $A(i|j)=\operatorname{tr}(B_iC_j)$ for all $i,j$. This paper determines the computational complexity status of the PSD rank. Namely, we show that the problem of computing this function is polynomial-time equivalent to the existential theory of the reals.

math.CO↗
arXiv2016

Nonnegative rank depends on the field II

We provide an example of a $21\times 21$ matrix with nonnegative integer entries which can be written as a sum of $19$ nonnegative rank-one matrices but not as a sum of $19$ rational nonnegative rank-one matrices. This gives a solution for a problem posed by Cohen and Rothblum in 1993.

math.CO↗
arXiv2016

A matrix ring with commuting graph of maximal diameter

The commuting graph of a semigroup is the set of non-central elements; the edges are defined as pairs $(u,v)$ satisfying $uv=vu$. We provide an example of a field $F$ and an integer $n$ such that the commuting graph of $\operatorname{Mat}_n(F)$ has maximal possible diameter, equal to six.

math.CO↗
arXiv2015

On the characteristic polynomial of a supertropical adjoint matrix

Let $χ(A)$ denote the characteristic polynomial of a matrix $A$ over a field; a standard result of linear algebra states that $χ(A^{-1})$ is the reciprocal polynomial of $χ(A)$. More formally, the condition $χ^n(X) χ^k(X^{-1})=χ^{n-k}(X)$ holds for any invertible $n\times n$ matrix $X$ over a field, where $χ^i(X)$ denotes the coefficient of $λ^{n-i}$ in the characteristic polynomial $\det(λI-X)$. We confirm a recent conjecture of Niv by proving the tropical analogue of this result.

math.CO↗
arXiv2014

Diameter of the commuting graph of $\mathbb{R}^{n\times n}$

The vertices of commuting graph of $\mathbb{R}^{n\times n}$ are non-scalar matrices; the edges are defined as pairs $(u, v)$ satisfying $uv = vu$. For $n\geqslant 3$, the diameter of this graph is at least four; we give a short proof that it is exactly four.

math.CO↗
arXiv2014

Chess God's number grows exponentially

We give an example of two $n\times n$ chess positions, $A$ and $B$, such that (1) there is a sequence $σ$ of legal chess moves leading from $A$ to $B$; (2) the length of $σ$ cannot be less than $\exp Θ(n)$.

math.CO↗
arXiv2013

Sign patterns of rational matrices with large rank

Let $A$ be a real matrix. The term rank of $A$ is the smallest number $t$ of lines (that is, rows or columns) needed to cover all the nonzero entries of $A$. We prove a conjecture of Li et al. stating that, if the rank of $A$ exceeds $t-3$, there is a rational matrix with the same sign pattern and rank as those of $A$. We point out a connection of the problem discussed with the Kapranov rank function of tropical matrices, and we show that the statement fails to hold in general if the rank of $A$ does not exceed $t-3$.

math.CO↗