A recent conjecture of Caputo, Carlen, Lieb, and Loss, and, independently, of the author, states that the maximum of the permanent of a matrix whose rows are unit vectors in l_p is attained either for the identity matrix I or for a constant multiple of the all-1 matrix J.
The conjecture is known to be true for p = 1 (I) and for p \ge 2 (J).
We prove the conjecture for a subinterval of (1,2), and show the conjectured upper bound to be true within a subexponential factor (in the dimension) for all 1 < p < 2. In fact, for p bounded away from 1, the conjectured upper bound is true within a constant factor.
This leads to a mild (subexponential) improvement in deterministic approximation factor for the permanent. We present an efficient deterministic algorithm that approximates the permanent of a nonnegative n\times n matrix within exp{n - O(n/\log n)}.
We formulate conjectures regarding the maximum value and maximizing matrices of the permanent and of diagonal products on the set of stochastic matrices with bounded rank. We formulate equivalent conjectures on upper bounds for these functions for nonnegative matrices based on their rank, row sums and column sums. In particular we conjecture that the permanent of a singular nonnegative matrix is bounded by 1/2 times the minimum of the product of its row sums and the product of its column sums, and that the product of the elements of any diagonal of a singular nonnegative matrix is bounded by 1…
In this paper we consider four basic multidimensional matrix operations (outer product, Kronecker product, contraction, and projection) and two derivative operations (dot and circle products). We start with the interrelations between these operations and deduce some of their algebraic properties. Next, we study their action on $k$-stochastic matrices. At last, we prove several relations on the permanents of products of multidimensional matrices. In particular, we obtain that the permanent of the dot product of nonnegative multidimensional matrices is not less than the product of their permanen…
The principal permanent rank characteristic sequence is a binary sequence $r_0 r_1 \ldots r_n$ where $r_k = 1$ if there exists a principal square submatrix of size $k$ with nonzero permanent and $r_k = 0$ otherwise, and $r_0 = 1$ if there is a zero diagonal entry. A characterization is provided for all principal permanent rank sequences obtainable by the family of nonnegative matrices as well as the family of nonnegative symmetric matrices. Constructions for all realizable sequences are provided. Results for skew-symmetric matrices are also included.
Combinatorial formulas expressing cyclic rook polynomials and cyclic permanents of rectangular matrices in terms of expansions along rows are presented
We prove a conjecture due to Wanless about the permanent of Hadamard matrices in the particular case of Sylvester-Hadamard matrices. Namely we show that for all n greater or equal to 2, the dyadic valuation of the permanent of the Sylvester-Hadamard matrix of order n is equal to the dyadic valuation of n!. As a consequence, the permanent of the Sylvester-Hadamard matrix of order n doesn't vanish for n greater or equal to 2.
The permanent of a multidimensional matrix is the sum of products of entries over all diagonals. By Minc's conjecture, there exists a reachable upper bound on the permanent of 2-dimensional (0,1)-matrices. In this paper we obtain some generalizations of Minc's conjecture to the multidimensional case. For this purpose we prove and compare several bounds on the permanent of multidimensional (0,1)-matrices. Most estimates can be used for matrices with nonnegative bounded entries.
In this paper we give a generalization created by the author of the theory of rook polynomials and permanents of circulants, Toeplits matrices and their submatrices.
A nonnegative multidimensional matrix is called polystochastic if the sum of its entries over each line is equal to $1$. In this paper we overview known results on positiveness of the permanent of polystochastic matrices and prove that the permanent of every $4$-dimensional polystochastic matrix of order $4$ is greater than zero.
The trees, respectively unicyclic graphs, on $n$ vertices with the smallest Laplacian permanent are studied. In this paper, by edge-grafting transformations, the $n$-vertex trees of given bipartition having the second and third smallest Laplacian permanent are identified. Similarly, the $n$-vertex bipartite unicyclic graphs of given bipartition having the first, second and third smallest Laplacian permanent are characterized. Consequently, the $n$-vertex bipartite unicyclic graphs with the first, second and third smallest Laplacian permanent are determined.
It is well-known that the evaluation of the permanent of an arbitrary $(-1,1)$-matrix is a formidable problem. Ryser's formula is one of the fastest known general algorithms for computing permanents. In this paper, Ryser's formula has been rewritten for the special case of Sylvester Hadamard matrices by using their cocyclic construction. The rewritten formula presents an important reduction in the number of sets of $r$ distinct rows of the matrix to be considered. However, the algorithm needs a preprocessing part which remains time-consuming in general.
We show that the permanent of a matrix is a linear combination of determinants of block diagonal matrices which are simple functions of the original matrix. To prove this, we first show a more general identity involving α-permanents: for arbitrary complex numbers αand β, we show that the α-permanent of any matrix can be expressed as a linear combination of β-permanents of related matrices. Some other identities for the α-permanent of sums and products of matrices are shown, as well as a relationship between the α-permanent and general immanants. We conclude with a discussion of the computation…
In this paper we investigate the permanent of $(-1,1)$-matrices over fields of zero characteristics and our main goal is to provide a sharp upper bound for the value of the permanent of such matrices depending on matrix rank, solving Wang's problem posed in 1974 by confirming Kräuter conjecture formulated in 1985.
A $d$-dimensional matrix is called \emph{$1$-polystochastic} if it is non-negative and the sum over each line equals~$1$. Such a matrix that has a single $1$ in each line and zeros elsewhere is called a \emph{$1$-permutation} matrix. A \emph{diagonal} of a $d$-dimensional matrix of order $n$ is a choice of $n$ elements, no two in the same hyperplane. The \emph{permanent} of a $d$-dimensional matrix is the sum over the diagonals of the product of the elements within the diagonal. For a given order $n$ and dimension $d$, the set of $1$-polystochastic matrices forms a convex polytope that include…
We give upper and lower bounds for the spectral radius of a nonnegative matrix by using its average 2-row sums, and characterize the equality cases if the matrix is irreducible. We also apply these bounds to various nonnegative matrices associated with a graph, including the adjacency matrix, the signless Laplacian matrix, the distance matrix, the distance signless Laplacian matrix, and the reciprocal distance matrix.
We prove that the maximum determinant of an $n \times n $ matrix, with entries in $\{0,1\}$ and at most $n+k$ non-zero entries, is at most $2^{k/3}$, which is best possible when $k$ is a multiple of 3. This result solves a conjecture of Bruhn and Rautenbach. We also obtain an upper bound on the number of perfect matchings in $C_4$-free bipartite graphs based on the number of edges, which, in the sparse case, improves on the classical Bregman's inequality for permanents. This bound is tight, as equality is achieved by the graph formed by vertex disjoint union of 6-vertex cycles.
An $n\times n$ matrix $C$ is said to be {\it centrosymmetric} if it satisfies the relation $JCJ=C$, where $J$ is the $n\times n$ counteridentity matrix. Centrosymmetric matrices have a rich eigenstructure that has been studied extensively in the literature. Many results for centrosymmetric matrices have been generalized to wider classes of matrices that arise in a wide variety of disciplines. In this paper, we obtain interesting spectral properties for nonnegative centrosymmetric matrices. We show how to change one single eigenvalue, two or three eigenvalues of an $n\times n$ nonnegative centr…
Matrix factorization problems over various semirings naturally arise in different contexts of modern pure and applied mathematics. These problems are very hard in general and cause computational difficulties in applications. We give a survey of what is known on the algorithmic complexity of Boolean, fuzzy, tropical, nonnegative, and positive semidefinite factorizations, and we examine the behavior of the corresponding rank functions on matrices of bounded bandwidth. We show that the Boolean, fuzzy, and tropical versions of matrix factorization become polynomial time solvable when restricted to…
In this paper, we give upper and lower bounds for the spectral radius of a nonnegative irreducible matrix and characterize the equality cases. These bounds theoretically improve and generalize some known results of Duan et al.[X. Duan, B. Zhou, Sharp bounds on the spectral radius of a nonnegative matrix, Linear Algebra Appl. (2013), http://dx.doi.org/10.1016/j.laa.2013.08.026]. Finally, applying these bounds to various matrices associated with a graph, we obtain some new upper and lower bounds on various spectral radiuses of graphs, which generalize and improve some known results.
We would desire to have done the calculations of this paper in the measure on nxn matrices that weights uniformly all 0-1 matrices with row and column sum equal to r, other matrices given weight zero. Instead we work with all matrices that are the sum of r independent uniformly weighted permutation matrices, with the hope that the computations we perform give the same result in this measure. We derive the result for limiting expectations lim (1/n)ln(E(perm_m(A) perm_m'(A))) =lim (1/n)ln(E(perm_m(A)))+ +lim (1/n)ln(E(perm_m'(A))) Here the limit is n to infinity, r is fixed, and m and m' are tak…
A total weighting of a graph $G$ is a mapping $f$ which assigns to each element $z \in V(G) \cup E(G)$ a real number $f(z)$ as its weight. The vertex sum of $v$ with respect to $f$ is $ϕ_f(v)=\sum_{e \in E(v)}f(e)+f(v)$. A total weighting is proper if $ϕ_f(u) \ne ϕ_f(v)$ for any edge $uv$ of $G$. A $(k,k')$-list assignment is a mapping $L$ which assigns to each vertex $v$ a set $L(v)$ of $k$ permissible weights, and assigns to each edge $e$ a set $L(e)$ of $k'$ permissible weights. We say $G$ is $(k,k')$-choosable if for any $(k,k')$-list assignment $L$, there is a proper total weighting $f$ o…
We show that the permanent of an $n \times n$ matrix with iid Bernoulli entries $\pm 1$ is of magnitude $n^{({1/2}+o(1))n}$ with probability $1-o(1)$. In particular, it is almost surely non-zero.
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.
Let $A$ be an $n \times n$ positive definite Hermitian matrix with all eigenvalues between 1 and 2. We represent the permanent of $A$ as the integral of some explicit log-concave function on ${\Bbb R}^{2n}$. Consequently, there is a fully polynomial randomized approximation scheme (FPRAS) for the permanent of $A$.
Let $M$ and $M_n,n\ge1$ be nonnegative 2-by-2 matrices such that $\lim_{n\rightarrow\infty}M_n=M.$ It is usually hard to estimate the entries of $M_{k+1}\cdots M_{k+n}$ which are useful in many applications. In this paper, under a mild condition, we show that up to a multiplication of some positive constants, entries of $M_{k+1}\cdots M_{k+n}$ are asymptotically the same as $ξ_{k+1}\cdots ξ_{k+n},$ the product of the tails of a continued fraction which is related to the matrices $M_k,k\ge1,$ as $n\rightarrow\infty.$
Let $Λ_n^k$ denote the class of $(0,1)$ square matrices containing in each row and in each column exactly $k$ 1's. The minimal value of $k,$ for which the behavior of the permanent in $Λ_n^k$ is not quite studied, is $k=3.$ We give a simple algorithm for calculation upper magnitudes of permanent in $Λ_n^3$ and consider some extremal problems in a generalized class $Λ_n(α,β,γ),$ the matrices of which contain in each row and in each column nonzero elements $α,β,γ$ and $n-3$ zeros.
Let $A$ be a set in a prime field $\mathbb{F}_p$. In this paper, we prove that $d\times d$ matrices with entries in $A$ determine almost $|A|^{3+\frac{1}{45}}$ distinct determinants and almost $|A|^{2-\frac{1}{6}}$ distinct permanents when $|A|$ is small enough.
The independence number of a square matrix $A$, denoted by $α(A)$, is the maximum order of its principal zero submatrices. Let $S_n^{+}$ be the set of $n\times n$ nonnegative symmetric matrices with zero trace. Denote by $J_n$ the $n\times n$ matrix with all entries equal to one. Given any integer $n$, we prove that a linear map $ϕ: S_n^+\rightarrow S_n^+$ satisfies $$α(ϕ(X))= α(X) {\quad\rm for~ all\quad}X\in S_n^+$$ if and only if there is a permutation matrix $P$ such that $$ϕ(X)=H\circ(P^TXP)\quad { \rm for~ all\quad}X\in S_n^+,$$ where $H=ϕ(J_n-I_n)$ with all off-diagonal entries positive…
Using the relationship between totally nonnegative matrices and directed acyclic weighted planar networks, we show that $2\times 2$ minors of minor matrices of totally nonnegative matrices are also nonnegative. We give a combinatorial interpretation for the minors of minor matrices in terms of the weights of families of paths in a network.
We study the convex relaxation of a polynomial optimization problem, maximizing a product of linear forms over the complex sphere. We show that this convex program is also a relaxation of the permanent of Hermitian positive semidefinite (HPSD) matrices. By analyzing a constructive randomized rounding algorithm, we obtain an improved multiplicative approximation factor to the permanent of HPSD matrices, as well as computationally efficient certificates for this approximation. We also propose an analog of van der Waerden's conjecture for HPSD matrices, where the polynomial optimization problem i…
A matrix is $k$-nonnegative if all its minors of size $k$ or less are nonnegative. We give a parametrized set of generators and relations for the semigroup of $k$-nonnegative $n\times n$ invertible matrices in two special cases: when $k = n-1$ and when $k = n-2$, restricted to unitriangular matrices. For these two cases, we prove that the set of $k$-nonnegative matrices can be partitioned into cells based on their factorizations into generators, generalizing the notion of Bruhat cells from totally nonnegative matrices. Like Bruhat cells, these cells are homeomorphic to open balls and have a to…
The permanent-on-top conjecture states that the largest eigenvalue of the Schur power matrix of a positive semi-definite Hermitian matrix H is per(H). A counterexample has been found with the help of computers, but here, I present another counterexample that can be checked by hand. My method is to use linear representations of groups to connect the spectrum of the Schur power matrix with the spectra of the entrywise(Hadamard) product matrices of permanental compound matrices. By that, we are able to study the properties of the spectrum of the Schur power matrix through the entrywise(Hadamard) …
Let $A$ be a matrix with nonnegative real entries. A nonnegative factorization of size $k$ is a representation of $A$ as a sum of $k$ nonnegative rank-one matrices. The space of all such factorizations is a bounded semialgebraic set, and we prove that spaces arising in this way are universal. More presicely, we show that every bounded semialgebraic set $U$ is rationally equivalent to the set of nonnegative size-$k$ factorizations of some matrix $A$ up to a permutation of matrices in the factorization. We prove that, if $U\subset\mathbb{R}^n$ is given as the zero locus of a polynomial with coef…
The permanent of a non-negative matrix appears naturally in many information processing scenarios. Because of the intractability of the permanent beyond small matrices, various approximation techniques have been developed in the past. In this paper, we study the Bethe approximation of the permanent and add to the body of literature showing that this approximation is very well behaved in many respects. Our main technical tool are topological double covers of the normal factor graph whose partition function equals the permanent of interest, along with a transformation of these double covers.
We design a deterministic polynomial time $c^n$ approximation algorithm for the permanent of positive semidefinite matrices where $c=e^{γ+1}\simeq 4.84$. We write a natural convex relaxation and show that its optimum solution gives a $c^n$ approximation of the permanent. We further show that this factor is asymptotically tight by constructing a family of positive semidefinite matrices.
This paper defines the Iris function and provides two formulations of the matrix permanent. The first formulation, valid for arbitrary complex matrices, expresses the permanent of a complex matrix as a contour integral of a second order Iris function over the unit circle around zero. The second formulation is defined for the restricted set of matrices with complex or "Gaussian" integer elements. Using the second formulation, the paper shows that the computation of the permanent of an arbitrary $n\times n$ 0-1 matrix is bounded by $o\left( n^{27} \left( \log \left( {n^3} \right) \right)^6 \left…
Algebras generated by strictly positive matrices are described up to similarity, including the commutative, simple, and semisimple cases. We provide sufficient conditions for some block diagonal matrix algebras to be generated by a set of nonnegative matrices up to similarity. Also we find all realizable dimensions of algebras generated by two nonnegative semi-commuting matrices. The last result provides the solution to the problem posed by M. Kandić, K. Šivic (2017).
In this survey article, we describe recent work that connects three separate objects of interest: totally nonnegative matrices; quantum matrices; and matrix Poisson varieties.
Calculating the permanent of a (0,1) matrix is a #P-complete problem but there are some classes of structured matrices for which the permanent is calculable in polynomial time. The most well-known example is the fixed-jump (0,1) circulant matrix which, using algebraic techniques, was shown by Minc to satisfy a constant-coefficient fixed-order recurrence relation. In this note we show how, by interpreting the problem as calculating the number of cycle-covers in a directed circulant graph, it is straightforward to reprove Minc's result using combinatorial methods. This is a two step process: the…
The main result of the note describes certain optimal-score partitions, which can be interpreted as optimal resource allocations. This result is based on the fact that any nonnegative square matrix whose column sums are the same as the corresponding row sums can be represented as the sum of circuit matrices.
We prove that for any $λ> 1$, fixed in advance, the permanent of an $n \times n$ complex matrix, where the absolute value of each diagonal entry is at least $λ$ times bigger than the sum of the absolute values of all other entries in the same row, can be approximated within any relative error $0 < ε< 1$ in quasi-polynomial $n^{O(\ln n - \ln ε)}$ time. We extend this result to multidimensional permanents of tensors and discuss its application to weighted counting of perfect matchings in hypergraphs.
A square matrix of order n with $n\geq 2$ is called permutative matrix when all its rows (up to the frst one) are permutations of precisely its frst row. In this paper recalling spectral results for partitioned into $2$-by-$2$ symmetric blocks matrices sufcient conditions on a given complex list to be the list of the eigenvalues of a nonnegative permutative matrix are given. In particular, we study NIEP and PNIEP when some complex elements into the considered lists have no zero imaginary part. Realizability regions for nonnegative permutative matrices are obtained.
Computing the distribution of permanents of random matrices has been an outstanding open problem for several decades. In quantum computing, "anti-concentration" of this distribution is an unproven input for the proof of hardness of the task of boson-sampling. We study permanents of random i.i.d. complex Gaussian matrices, and more broadly, submatrices of random unitary matrices. Using a hybrid representation-theoretic and combinatorial approach, we prove strong lower bounds for all moments of the permanent distribution. We provide substantial evidence that our bounds are close to being tight a…
We present a deterministic algorithm, which, for any given 0< epsilon < 1 and an nxn real or complex matrix A=(a_{ij}) such that | a_{ij}-1| < 0.19 for all i, j computes the permanent of A within relative error epsilon in n^{O(ln n -ln epsilon)} time. The method can be extended to computing hafnians and multidimensional permanents.
In this note we settle two open problems in the theory of permanents by using recent results from other areas of mathematics. Bapat conjectured that certain quotients of permanents, which generalize symmetric function means, are concave. We prove this conjecture by using concavity properties of hyperbolic polynomials. Motivated by problems on random point processes, Shirai and Takahashi raised the problem: Determine all real numbers $α$ for which the $α$-permanent (or $α$-determinant) is nonnegative for all positive semidefinite matrices. We give a complete solution to this problem by using re…
We investigate the tropical analogues of totally positive and totally nonnegative matrices. These arise when considering the images by the nonarchimedean valuation of the corresponding classes of matrices over a real nonarchimedean valued field, like the field of real Puiseux series. We show that the nonarchimedean valuation sends the totally positive matrices precisely to the Monge matrices. This leads to explicit polyhedral representations of the tropical analogues of totally positive and totally nonnegative matrices. We also show that tropical totally nonnegative matrices with a finite perm…
