Nonnegative Integer Matrices Research Articles

Improved estimates for the number of non-negative integer matrices with given row and column sums

The number of non-negative integer matrices with given row and column sums features in a variety of problems in mathematics and statistics but no closed-form expression for it is known, so we rely on approximations. In this paper, we describe a new such approximation, motivated by consideration of the statistics of matrices with non-integer numbers of columns. This estimate can be evaluated in time linear in the size of the matrix and returns results of accuracy as good as or better than existing linear-time approximations across a wide range of settings. We show that the estimate is asymptotically exact in the regime of sparse tables, while empirically performing at least as well as other linear-time estimates in the regime of dense tables. We also use the new estimate as the starting point for an improved numerical method for either counting or sampling matrices with given margins using sequential importance sampling. Code implementing our methods is available.

Asymptotic Properties of Random Contingency Tables with Uniform Margin

Let $C\geq 2$ be a positive integer. Consider the set of $n\times n$ non-negative integer matrices whose row sums and column sums are all equal to $Cn$ and let $X=(X_{ij})_{1\leq i,j\leq n}$ be uniformly distributed on this set. This $X$ is called the random contingency table with uniform margin. In this paper, we study various asymptotic properties of $X=(X_{ij})_{1\leq i,j\leq n}$ as $n\to\infty$.

A Structure-Tensor Approach to Integer Matrix Completion in Indivisible Resource Allocation

Indivisible resource allocation motivates us to study the matrix completion concerning the class of (0,1)-matrices with prescribed row/column sums and preassigned zeros. We illustrate and generalize the (0,1)-matrix completion in the following two scenarios: a demand-response application involving nonnegative integer matrices with different bounds across rows and an edge caching matching problem allowing row and column sums to vary within separately designated bounds. The applications require analytic characterizations of the supply adequacy and cause large-scale matrix completion instances. Remarkably, we derive a structure tensor and use its nonnegativity to establish a necessary and sufficient condition under which the considered matrix class is nonempty. The tensor condition can characterize the adequacy of a supply for a prescribed demand and facilitate identifying the minimum supplement to the supply so that the augmented supply becomes adequate when the adequacy gap is nonzero. Notably, we design a tensor-based combinatorial algorithm to construct a required matrix, representing a feasible resource allocation. Numerical simulations justify the efficiency of our approach.

Algorithmic aspects of arithmetical structures

Arithmetical structures on graphs were first introduced in [11]. Later in [3] they were further studied in the setting of square non-negative integer matrices. In both cases, necessary and sufficient conditions for the finiteness of the set of arithmetical structures were given.More precisely, an arithmetical structure on a non-negative integer matrix L with zero diagonal is a pair (d,r)∈N+n×N+n such that(Diag(d)−L)rt=0t and gcd⁡(r1,…,rn)=1. Thus, arithmetical structures on L are solutions of the polynomial Diophantine equationfL(X):=det⁡(Diag(X)−L)=0. Therefore, it is of interest to ask for an algorithm that compute them.We present an algorithm that computes arithmetical structures on a square integer non-negative matrix L with zero diagonal. In order to do this we introduce a new class of Z-matrices, which we call quasi M-matrices.

Balanced strong shift equivalence, balanced in-splits, and eventual conjugacy

AbstractWe introduce the notion of balanced strong shift equivalence between square non-negative integer matrices, and show that two finite graphs with no sinks are one-sided eventually conjugate if and only if their adjacency matrices are conjugate to balanced strong shift equivalent matrices. Moreover, we show that such graphs are eventually conjugate if and only if one can be reached by the other via a sequence of out-splits and balanced in-splits, the latter move being a variation of the classical in-split move introduced by Williams in his study of shifts of finite type. We also relate one-sided eventual conjugacies to certain block maps on the finite paths of the graphs. These characterizations emphasize that eventual conjugacy is the one-sided analog of two-sided conjugacy.

English

We investigate the formal matrix ring over $R$ defined by a certain system of factors. We give a method for constructing formal matrix rings from non-negative integer matrices. We also show that the principal factor matrix of a binary system of factors determine the structure of the system.

Sets of Integer Matrix Pairs Derived from Row Rank Inequalities and Their Preservers

In this paper, we consider the row rank inequalities derived from comparisons of the row ranks of the additions and multiplications of nonnegative integer matrices and construct the sets of nonnegative integer matrix pairs which is occurred at the extreme cases for the row rank inequalities. We characterize the linear operators that preserve these extreme sets of nonnegative integer matrix pairs

A note on the nonzero spectra of irreducible matrices

In this note we extend the necessary and sufficient conditions of Boyle–Handleman [M. Boyle and D. Handelman, The spectra of nonnegative matrices via symbolic dynamics, Ann. Math. 133 (1991), pp. 249–316] and Kim–Ormes–Roush [K.H. Kim, N.S. Ormes, and F.W. Roush, The spectra of nonnegative integer matrices via formal power series, J. Am. Math. Soc. 13 (2000), pp. 773–806] for a nonzero eigenvalue multiset of primitive matrices over ℝ+ and Z +, respectively, to irreducible matrices.

Properties of the nonsymmetric Robinson–Schensted–Knuth algorithm

We introduce a generalization of the Robinson–Schensted–Knuth insertion algorithm for semi-standard augmented fillings whose basement is an arbitrary permutation σ∈S n . If σ is the identity, then our insertion algorithm reduces to the insertion algorithm introduced by the second author (Sémin. Lothar. Comb. 57:B57e, 2006) for semi-standard augmented fillings and if σ is the reverse of the identity, then our insertion algorithm reduces to the original Robinson–Schensted–Knuth row insertion algorithm. We use our generalized insertion algorithm to obtain new decompositions of the Schur functions into nonsymmetric elements called generalized Demazure atoms (which become Demazure atoms when σ is the identity). Other applications include Pieri rules for multiplying a generalized Demazure atom by a complete homogeneous symmetric function or an elementary symmetric function, a generalization of Knuth’s correspondence between matrices of non-negative integers and pairs of tableaux, and a version of evacuation for composition tableaux whose basement is an arbitrary permutation σ.

Self-Dual Interval Orders and Row-Fishburn Matrices

Recently, Jelínek derived that the number of self-dual interval orders of reduced size $n$ is twice the number of row-Fishburn matrices of size $n$ by using generating functions. In this paper, we present a bijective proof of this relation by establishing a bijection between two variations of upper-triangular matrices of nonnegative integers. Using the bijection, we provide a combinatorial proof of the refined relations between self-dual Fishburn matrices and row-Fishburn matrices in answer to a problem proposed by Jelínek.

An asymptotic formula for the number of non-negative integer matrices with prescribed row and column sums

We count mxn non-negative integer matrices (contingency tables) with prescribed row and column sums (margins). For a wide class of smooth margins we establish a computationally efficient asymptotic formula approximating the number of matrices within a relative error which approaches 0 as m and n grow.

ASYMPTOTIC ENUMERATION OF SYMMETRIC INTEGER MATRICES WITH UNIFORM ROW SUMS

AbstractWe investigate the number of symmetric matrices of nonnegative integers with zero diagonal such that each row sum is the same. Equivalently, these are zero-diagonal symmetric contingency tables with uniform margins, or loop-free regular multigraphs. We determine the asymptotic value of this number as the size of the matrix tends to infinity, provided the row sum is large enough. We conjecture that one form of our answer is valid for all row sums. An example appears in Figure 1.

A finiteness theorem for W-graphs

A W-graph for a Coxeter group W is a combinatorial structure that encodes a module for the group algebra of W, or more generally, a module for the associated Iwahori–Hecke algebra. Of special interest are the W-graphs that encode the action of the Hecke algebra on its Kazhdan–Lusztig basis, as well as the action on individual cells. In previous work, we isolated a few basic features common to the W-graphs in Kazhdan–Lusztig theory and used these to define the class of “admissible” W-graphs. The main result of this paper resolves one of the basic question about admissible W-graphs: there are only finitely many admissible W-cells (i.e., strongly connected admissible W-graphs) for each finite Coxeter group W. Ultimately, the finiteness depends only on the fact that admissible W-graphs have nonnegative integer edge weights. Indeed, we formulate a much more general finiteness theorem for “cells” in finite-dimensional algebras which in turn is fundamentally a finiteness theorem for nonnegative integer matrices satisfying a polynomial identity.

Binary ranks and binary factorizations of nonnegative integer matrices

A matrix is binary if each of its entries is either 0 or 1. The binary rank of a nonnegative integer matrix A is the smallest integer b such that A = BC, where B and C are binary matrices, and B has b columns. In this paper, bounds for the binary rank are given, and nonnegative integer matrices that attain the lower bound are characterized. Moreover, binary ranks of nonnegative integer matrices with low ranks are determined, and binary ranks of nonnegative integer Jacobi matrices are estimated.

Matrices with prescribed row and column sums

This is a survey of the recent progress and open questions on the structure of the sets of 0–1 and non-negative integer matrices with prescribed row and column sums. We discuss cardinality estimates, the structure of a random matrix from the set, discrete versions of the Brunn–Minkowski inequality and the statistical dependence between row and column sums.

A function approximation approach to the segmentation step in IMRT planning

In the segmentation step of intensity modulated radiation therapy planning, given intensity profiles have to be decomposed into a number of leaf positions of a multileaf collimator (MLC) such that the superposition of the corresponding field shapes is close to the desired profile. Until now, these decomposition problems have been formulated as discrete optimization problems where the profiles are nonnegative integer matrices. The segments are modeled as 0-1-matrices, 1 indicating that radiation is transmitted through this part of the field and 0 for the areas that are covered by the leaves of the MLC. But in physical reality, radiation has a penumbra at the boundary of the segment causing a decline of the intensity, that is not modeled in these formulations. This paper embeds the segmentation task into the wider context of function approximation and models both profiles and segments as real-valued functions of two variables. This leads to convex optimization problems whose objective is to minimize the approximation error between the profile and the superposition of the real weighted segments. Thus, a more realistic model of radiation is used and may enable an improvement in treatment quality.

Asymptotic Estimates for the Number of Contingency Tables, Integer Flows, and Volumes of Transportation Polytopes

We prove an asymptotic estimate for the number of m × n non-negative integer matrices (contingency tables) with prescribed row and column sums and, more generally, for the number of integer-feasible flows in a network. Similarly, we estimate the volume of the polytope of m × n non-negative real matrices with prescribed row and column sums. Our estimates are solutions of convex optimization problems, and hence can be computed efficiently. As a corollary, we show that if row sums R = (r 1 , …, r m ) and column sums C = (c 1 , …, c n ) with r 1 + ⋯ + r m = c 1 + ⋯ + c n = Nare sufficiently far from constant vectors, then, asymptotically, in the uniform probability space of the m × nnon-negative integer matrices with the total sum N of entries, the event consisting of the matrices with row sums R and the event consisting of the matrices with column sums C are positively correlated.

Brunn–Minkowski inequalities for contingency tables and integer flows

We establish approximate log-concavity for a wide family of combinatorially defined integer-valued functions. Examples include the number of non-negative integer matrices (contingency tables) with prescribed row and column sums (margins), as a function of the margins and the number of integer feasible flows in a network, as a function of the excesses at the vertices. As a corollary, we obtain approximate log-concavity for the Kostant partition function of type A. We also present an indirect evidence that at least some of the considered functions might be genuinely log-concave.

Crystal graphs for Lie superalgebras and Cauchy decomposition

We discuss Cauchy type decompositions of crystal graphs for general linear Lie superalgebras. More precisely, we consider bicrystal graph structures on various sets of matrices of non-negative integers, and obtain their decompositions with explicit combinatorial isomorphisms.

Improved bounds for sampling contingency tables

AbstractWe study the problem of sampling contingency tables (nonnegative integer matrices with specified row and column sums) uniformly at random. We give an algorithm which runs in polynomial time provided that the row sums ri and the column sums cj satisfy ri = Ω(n3/2m log m), and cj = Ω(m3/2n log n). This algorithm is based on a reduction to continuous sampling from a convex set. The same approach was taken by Dyer, Kannan, and Mount in previous work. However, the algorithm we present is simpler and has weaker requirements on the row and column sums. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 21: 135–146, 2002