On the Number of Bases of Almost All Matroids

Pendavingh, RA Rudi; Pol, JG Jorn van der

doi:10.1007/s00493-016-3594-4

Cited by 9 publications

(16 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We finish our paper by showing that asymptotically almost all matroids have a constant-competitive matroid secretary algorithm. We require two recent results of Pendavingh and van der Pol [20,21].…”

Section: Asymptotic Resultsmentioning

confidence: 99%

See 1 more Smart Citation

The Matroid Secretary Problem for Minor-Closed Classes and Random Matroids

Huynh

Nelson

2020

SIAM J. Discrete Math.

View full text Add to dashboard Cite

We prove that for every proper minor-closed class M of F p -representable matroids, there exists a O(1)-competitive algorithm for the matroid secretary problem on M. This result relies on the extremely powerful matroid minor structure theory being developed by Geelen, Gerards and Whittle.We also note that for asymptotically almost all matroids, the matroid secretary algorithm that selects a random basis, ignoring weights, is (2 + o(1))-competitive. In fact, assuming the conjecture that almost all matroids are paving, there is a (1 + o (1))competitive algorithm for almost all matroids. Conjecture 1 (Babaioff, Immorlica, and Kleinberg [1]). For all matroids M, there is a O(1)-competitive matroid secretary algorithm for M. Babaioff, Immorlica, and Kleinberg [1] gave a O(log r)-competitive algorithm for all matroids M, where r is the rank of M. This was improved to a O( √ log r)-competitive algorithm by Chakraborty and Lachish [2]. The state-of-the-art is a O(log log r)competitive algorithm, first obtained by Lachish [17]. Using different tools, Feldman, Svensson, and Zenklusen [6] give a simpler O(log log r)-competitive algorithm for all matroids.

show abstract

Section: Asymptotic Resultsmentioning

confidence: 99%

“…The veracity of Theorem 3 is not terribly surprising, although given the limited tools in asymptotic matroid theory, it is a bit surprising that we can prove it. Indeed, the proof of Theorem 3 relies on recent breakthrough results of Pendavingh and van der Pol [20,21].…”

Section: Introductionmentioning

confidence: 99%

The Matroid Secretary Problem for Minor-Closed Classes and Random Matroids

Huynh

Nelson

2020

SIAM J. Discrete Math.

View full text Add to dashboard Cite

show abstract

“…And Nelson [19] showed that almost all matroids are nonrepresentable. Pendavingh and van der Pol [20] considered random matroids of rank r and showed that almost all r-sets will be bases in this model.…”

Section: Introductionmentioning

confidence: 99%

Minors of a random binary matroid

Cooper

Frieze

Pegden

2019

Random Struct Algorithms

View full text Add to dashboard Cite

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

show abstract

“…These are shown in Figure 2.2. Later work by the same authors [23] proved that the same result holds for all uniform matroids.…”

Section: Counting Matroids In Minor-closed Classes (Pendavingh and Van Der Pol's Work)mentioning

confidence: 72%

“…Some progress on asymptotic questions has been made recently, especially by Pendavingh and van der Pol who showed (amongst other things) that asymptotically almost all matroids are k-connected [23], are not a truncation [23], contain a large uniform minor [23], and contain minors isomorphic to each of a small collection of matroids [24].…”

Section: Chapter 1 Introductionmentioning

confidence: 99%

Randomness in classes of matroids

Critchlow¹

View full text Add to dashboard Cite

<p>This thesis is inspired by the observation that we have no good random model for matroids. That stands in contrast to graphs, which admit a number of elegant random models. As a result we have relatively little understanding of the properties of a "typical" matroid. Acknowledging the difficulty of the general case, we attempt to gain a grasp on randomness in some chosen classes of matroids. Firstly, we consider sparse paving matroids, which are conjectured to dominate the class of matroids (that is to say, asymptotically almost all matroids would be sparse paving). If this conjecture were true, then many results on properties of the random sparse paving matroid would also hold for the random matroid. Sparse paving matroids have limited richness of structure, making counting arguments in particular more feasible than for general matroids. This enables us to prove a number of asymptotic results, particularly with regards to minors. Secondly, we look at Graham-Sloane matroids, a special subset of sparse paving matroids with even more limited structure - in fact Graham-Sloane matroids on a labelled groundset can be randomly generated by a process as simple as independently including certain bases with probability 0.5. Notably, counting Graham-Sloane matroids has provided the best known lower bound on the total number of matroids, to log-log level. Despite the vast size of the class we are able to prove severe restrictions on what minors of Graham-Sloane matroids can look like. Finally we consider transversal matroids, which arise naturally in the context of other mathematical objects - in particular partial transversals of set systems and partial matchings in bipartite graphs. Although transversal matroids are not in one-to-one correspondence with bipartite graphs, we shall link the two closely enough to gain some useful results through exploiting the properties of random bipartite graphs. Returning to the theme of matroid minors, we prove that the class of transversal matroids of given rank is defined by finitely many excluded series-minors. Lastly we consider some other topics, including the axiomatisability of transversal matroids and related classes.</p>

show abstract

On the Number of Bases of Almost All Matroids

Cited by 9 publications

References 18 publications

The Matroid Secretary Problem for Minor-Closed Classes and Random Matroids

The Matroid Secretary Problem for Minor-Closed Classes and Random Matroids

Minors of a random binary matroid

Randomness in classes of matroids

Contact Info

Product

Resources

About