Minors of a random binary matroid

Cooper, Colin; Frieze, Alan; Pegden, Wesley

doi:10.1002/rsa.20881

Cited by 2 publications

(6 citation statements)

References 25 publications

(41 reference statements)

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“…The first task here is to prove (6). Let B n,k denote a minimum weight basis and let W n,k denote its weight.…”

Section: Minimum Weight Basismentioning

confidence: 99%

“…In a recent paper [6], we studied the binary matroid M n,m;k induced by the columns of A n,m;k . It was shown that for any fixed binary matroid M, there were constants k M , L M such that if k ≥ k M and m ≥ L m n then w.h.p.…”

Section: Introductionmentioning

confidence: 99%

“…M n,m;k contains M as a minor. The paper [6] contributes to the theory of random matroids as developed by [1], [2], [10], [12], [13]. In this paper we study a related aspect of A n,m;k , namely its rank, and improve on results from Cooper [4].…”

Section: Introductionmentioning

confidence: 99%

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On the rank of a random binary matrix

Cooper

Frieze²,

Pegden³

2019

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

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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.

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“…The first task here is to prove (6). Let B n,k denote a minimum weight basis and let W n,k denote its weight.…”

Section: Minimum Weight Basismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the rank of a random binary matrix

Cooper

Frieze²,

Pegden³

2019

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

Self Cite

View full text Add to dashboard Cite

show abstract

“…In a recent paper [7], we studied the binary matroid M n,m;k induced by the columns of A n,m;k . It was shown that for any fixed binary matroid M , there were constants k M , L M such that if k k M and m L m n then w.h.p.…”

Section: Introductionmentioning

confidence: 99%

“…M n,m;k contains M as a minor. The paper [7] contributes to the theory of random matroids as developed by [1], [3], [11], [13], [14]. In this paper we study a related aspect of A n,m;k , namely its rank, and improve on results from Cooper [5].…”

Section: Introductionmentioning

confidence: 99%

On the Rank of a Random Binary Matrix

Cooper

Frieze

Pegden

2019

Electron. J. Combin.

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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 $\zeta(3)\sim 1.202$.

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Minors of a random binary matroid

Cited by 2 publications

References 25 publications

On the rank of a random binary matrix

On the rank of a random binary matrix

On the Rank of a Random Binary Matrix

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