Inclusion of forbidden minors in random representable matroids

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

Frieze²,

Pegden³

2019

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.

Section: Introductionmentioning

confidence: 99%

On the rank of a random binary matrix

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

Frieze²,

Pegden³

2019

“…This then implies that f k −1 ( ) approaches f k ( ) as k grows. Suppose now that we condition on the row sums 1…”

Section: Proof Of Lemma 21mentioning

confidence: 99%

“…(Remember that r i is row i of B −1 and that u i is row i of D.) For a column x of A m , let R (x) be the column x restricted to the p rows of R. Let c 1 be the first column of the target matrix M and let c be a random candidate column. Let v satisfy Dv = m 1 and v = 0 if |A | < 1 n. Assume also that v has at most p ones and that k ≥ p. There are always such solutions. Then we have…”

Section: 9mentioning

confidence: 99%

“…(For X ∉ , the contraction can be defined by ∕X ∶= (∕Y) ⧵ X, where Y is any basis of X.) Theorem 1.1 is related to the result of Altschuler and Yang [1]. They prove that if matrix M is an n(m) × m matrix with random entries in GF q and m − n(m) → ∞ then w.h.p.…”

Section: Introductionmentioning

confidence: 97%

“…(We have reversed the roles of m, n from their statement.) However, the results of [1] rely heavily on the fact that premultiplying a uniform random matrix in this model by a nonsingular matrix yields another uniform random matrix. Our model lacks this property.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Minors of a random binary matroid

Random Struct Algorithms

Frieze

Pegden

2019

Let A be an n × m matrix over GF 2 where each column consists of k ones, and let M be an arbitrary fixed binary matroid. The matroid growth rate theorem implies that there is a constant C M such that m ≥ C M n 2 implies that the binary matroid induced by A contains M as a minor. We prove that if the columns of A = A n,m,k are chosen randomly, then there are constants k M , L M such that k ≥ k M and m ≥ L M n implies that A contains M as a minor with high probability.KEYWORDS binary, matroids, minors, random 1 Random Struct Alg. 2019;55:865-880.wileyonlinelibrary.com/journal/rsa

On the Rank of a Random Binary Matrix

Frieze

Pegden

2019

Electron. J. Combin.

We study the rank of a random $n \times m$ matrix $\mathbf{A}_{n,m;k}$ 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 ${n \choose k}$ 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 $\mathbf{A}_{n,m;k}$ forms the vertex-edge incidence matrix of a $k$-uniform random hypergraph $H$. The rank of $\mathbf{A}_{n,m;k}$ 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 $\mathbf{A}_{n,m;k}$ is asymptotic to $m$, and for $m \ge m^*$ the rank is asymptotic to $m-|E(C_2)|+|C_2|$. In addition, assign i.i.d. $U[0,1]$ weights $X_i, i \in {1,2,...m}$ to the columns, and define the weight of a set of columns $S$ as $X(S)=\sum_{j \in S} X_j$. Define a basis as a set of $n-𝟙 (k\text{ even})$ 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 $\zeta(3)\sim 1.202$.