On the rank of a random binary matrix

Abstract

We study the rank of a random n × m matrix A n,m;k with entries from GF (2), and exactly k unit entries in each column, the other entries being zero. The columns are chosen independently and uniformly at random from the set of all n k such columns. We obtain an asymptotically correct estimate for the rank as a function of the number of columns m in terms of c, n, k, and where m = cn/k. The matrix A n,m;k forms the vertex-edge incidence matrix of a k-uniform random hypergraph H. The rank of A n,m;k can be expressed as follows. Let |C 2 | be the number of vertices of the 2-core of H, and |E(C 2 )| the number of edges. Let m * be the value of m for which |C 2 | = |E(C 2 )|. Then w.h.p. for m < m * the rank of A n,m;k is asymptotic to m, and for m ≥ m * the rank is asymptotic to m − |E(C 2 )| + |C 2 |.In addition, assign i.i.d. U [0, 1] weights X i , i ∈ 1, 2, ...m to the columns, and define the weight of a set of columns S as X(S) = j∈S X j . Define a basis as a set of n − 1(k even) linearly independent columns. We obtain an asymptotically correct estimate for the minimum weight basis. This generalises the well-known result of Frieze [On the value of a random minimum spanning tree problem, Discrete Applied Mathematics, (1985)] that, for k = 2, the expected length of a minimum weight spanning tree tends to ζ(3) ∼ 1.202.

The rank of sparse random matrices

“…The same rank formula was obtained independently in [4]. Extending the results from [4,14], Theorem 1.1 shows that the rank of the random matrix with these degrees over any field F is given by…”

“…The principal new proof ingredient is the asymptotically precise analysis of the second moment by means of the study of the sub-grids of the integer lattice induced by the constraints as sketched in Section 2.4. This issue that was absent in the prior literature on variations on random k-XORSAT [6,10,15] and on other random constraint satisfaction problems [12,13]. However, in the study of the random regular matrix from Example 1.5 Huang [24] faced a similar issue in the special case d = k constant and χ = 1 deterministically.…”

“…Example 1.3 (fixed row sums). Motivated by the minimum spanning tree problem in weighted random graphs, Cooper, Frieze and Pegden [14] studied the rank of the random matrix with degree distributions k = k ≥ 3 fixed and d ∼ Po(d) over the field F 2 . The same rank formula was obtained independently in [4].…”

The Satisfiability Threshold For Random Linear Equations

“…In the case q = 2 this result was recently obtained independently by Cooper, Frieze and Pegden [17], using a different approach.…”

О Ранге Случайной Двоичной Матрицы С Заданными Весами Независимых Строк

“…Поэтому интерес представляет изучение частных постановок, например матриц из независимых случайных строк (или столбцов). Наибольший интерес вызывал случай разреженных матриц, когда число единиц в каждой из строк невелико по сравнению с длиной строк (см., например, [14,15] и ссылки в этих работах). Так, в [14] исследовано асимптотическое поведение ранга двоичной матрицы A n,m,k размера n × m, столбцы которой выбираются независимо и равновероятно из множества всех C k n столбцов с k единицами, а n, m → ∞ таким образом, что n существенно меньше m. Показано наличие своего рода порогового эффекта в поведении ранга матрицы при n, m → ∞ и фиксированном k в зависимости от соотношения между параметрами m, n, k.…”