On the number of matroids
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On the number of matroids. Nikhil Bansal (TU Eindhoven) Rudi Pendavingh (TU Eindhoven) Jorn van der Pol (TU Eindhoven). Matroids. Matroid (U,C): U = Universe [n], C: collection of independent sets Subset closed: I independent, then also .

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On the number of matroids

On the number of matroids

Nikhil Bansal (TU Eindhoven)

Rudi Pendavingh (TU Eindhoven)

Jorn van der Pol (TU Eindhoven)


Matroids

Matroids

Matroid(U,C): U = Universe [n], C: collection of independent sets

  • Subset closed: I independent, then also .

  • Exchange: |I|>|I’| then some s.t also independent.

    How does a typical matroid look ?

    Can we generate themrandomly?

    How many matroids m(n) on n elements?

    Clearly, m(n)

    Knuth’74: m(n)

    (explicit construction of a class called sparse paving matroids)

    i.e. log log m(n) [n – 3/2 log n – O(1), n]


Narrowing the gap

Narrowing the gap

Why bother about this tiny 3/2 log n gap?

log log scale a bit deceptive.

x vs.

Conjecture: Most matroids are sparse paving (various versions)

Knuth’s bound perhaps close to optimal

Often counting ->

Sampling and generating matroids.

m(n)

Better bound


Known results

Known results

Knuth 74

Easy Upper bound:

Pf: Any rank r matroid is specified by its bases (max. indep. sets)

Number of rank r matroidsm(n,r) .

So,

m(n) (n+1) (can just focus on rank r=n/2)

Piff 73: ]


Our result

Our Result

Thm: log log m(n) Knuth’s lower bound + 1+o(1) .

(Piff had ½ log n gap)

Knuth 74: m

We show: m

Need only the most basic matroid facts.

Main Tool: Bounding number of stable sets in a graph.


Outline

Outline

  • Knuth’s lower bound construction

  • Counting stable sets

  • The final upper bound


Knuth s lower bound

Knuth’s Lower bound

Matroid of rank r can be specified by r-sets that are non-bases.

Johnson Graph: J(n,r) V: r-subsets of [n] |V|=.

Edge (u,v): if

Fact: If non-bases form a stable set in J(n,r), then get a matroid.

These are called sparse paving matroids.

(various nice properties)


Knuth s bound

Knuth’s bound

Sparse Paving Matroids : precisely the stable sets of J(n,r).

For graph G: = size of max stable set.

i(G) = # stable sets. Note: .

J(n,r) is a regular graph of degree d = for .

So,

Knuth:

Proof: Color vertex by j if

Gives proper n-coloring.


Rest of the talk

Rest of the talk

Goal: Show log log m(n) Knuth’s lower bound + 1+o(1)

(Necessary) first step: Show this for sparse paving matroids

log log s(n) Knuth’s lower bound + 1+o(1)

( same as bounding i(G) for J(n,r))

The ideas developed there will be useful for bounding m(n).


Bounding s n

Bounding s(n)

Claim: Max stable set in J(n,n/2) (2/n) N N = # of vertices

Fact: If is smallest eigenvalue of adj. matrix of a d-regular graph.

Then, (proof later)

Johnson graphs: for J(n/n/2) So,

Naïve bound: i(G) + + … +

Recall, knuth Lower’s bound:

Niavely: i(J(n,n/2)) (note: base of exponent)


Better bound

Better Bound

Refined bound: i(J(n,n/2))

Morally: All independent sets are subsets of few large independent sets.

Examples: n-Hypercube vertices

Naïve bound on i(G) =

Right answer: [Saphozhenko’83] (1+o(1)

Entropy Method [Kahn’01]: Any d-regular bipartite graph

Tight: Disjoint copies of (n/2d copies)

Holds even for general graphs [Zhao’10]


Our result1

Our Result

Thm: In any d-regular graph G with min eigenvalue

i(G)

Idea: Encode an independent set using few bits of information.

Eg: Bipartite graphs: Our bound:

Our approach closest to Alon, Balogh, Morris, Samotij[arxiv’12]

(their bound not useful for our purposes)

This encoding idea is later used to encode matroids.


Rest of the talk1

Rest of the talk

Encoding for independent sets.

Encoding for Matroids.


A useful lemma

A useful lemma

G: d-regular with min eigenvalue . For any vertex subset A.

2

Proof:

Split

Corollary:

Corollary: If |A| + N, then G[A] has a vertex of degree

(For random set A of size , expected degree

A


Encoding a stable set

Encoding a stable set

Associate to an independent set I of G the pair of vertices (S,A), s.t.

  • |A|

  • A is completely determined by S.

    I can be uniquely specified by (S, ).

    Key Point: A is completely determined by S.

    Number of possibilities for I = (gives the result)

S

I

A

G


Encoding a stable set1

Encoding a stable set

Input: Independent set I.

Initialize: A = V and S =

While |A| >

Let v = largest degree vertex in G[A] (ties in lex order)

If v in I, add to S and set

Else discard v and set A = A\{v}

.


Encoding a stable set2

Encoding a stable set

Input: Independent set I.

Initialize: A = V and S =

While |A| >

Let v = largest degree vertex in G[A] (ties in lex order)

If v in I, add to S and set

Else discard v and set A = A\{v}

.


Encoding a stable set3

Encoding a stable set

Input: Independent set I.

Initialize: A = V and S =

While |A| >

Let v = largest degree vertex in G[A] (ties in lex order)

If v in I, add to S and set

Else discard v and set A = A\{v}

.


Encoding a stable set4

Encoding a stable set

Input: Independent set I.

Initialize: A = V and S =

While |A| >

Let v = largest degree vertex in G[A] (ties in lex order)

If v in I, add to S and set

Else discard v and set A = A\{v}

Observe: S completely determines A.


Encoding a stable set5

Encoding a stable set

Claim:

Pf: Alg in phase j if ]

A vertex in phase jhas at least j neighbors in G[A].

Must pick vertices in S. Sum over j=d,…1.

Phases:

d

d-1

1

2


Encoding matroids

Encoding Matroids

Matroid can be specified by listing r-sets that are non-bases.

Want a more compact representation (fewer bits).

Idea: r-set Y is dependent iff some X s.t. || > rk(X)

(i.e. X acts as a witness that Y contains a dependency).

E.g. X=Y trivially works. But not very efficient.

Want small witness set { (X,rk(X)) } that works for all Y.


On the number of matroids

Key Lemma: For a dependent r-set X, we can associate sets that are witnesses for all non-bases Y in .

If rank(X) < r-1. Witness = (X,rk(X))


On the number of matroids

Key Lemma: For a dependent r-set X, we can associate sets that are witnesses for all non-bases Y in .

If rank(X) = r-1. Then X has a unique circuit C.

Witness = (Cl(X), C)

Cl(X) : closure all z s.t. rank(X U z) = rank(X)

Proof:

(as Y=X-x+y)

Case 1: If rk(X+y) = r-1, then

= |Y|> rk(Cl(X))

Case 2: rk(X-x) r-2 < |X-x|

So X-x contains a circuit C’. But C’=C by uniqueness.

So (as Y=X-x+y). Hence || = |C| > rk(C)


Finish up

Finish up

Given a matroid, let K = set of non-bases.

Apply stable set procedure to K obtain (S, A) with

  • S and A small as before, S determined by A.

    Encoding: {witness of } for each

    List

    1) Witness for all non-bases in

    2) List of remaining non-bases.


Questions

Questions

Would be nice to reduce the gap to o(1).

The reason for +1+o(1) gap

Do not understand the size of max. stable set in J(n,n/2)

N/n (explicit construction) vs. 2N/n (eigenvalue methods)

Studied a lot in coding. Simulations suggest closer to N/n

Perhaps a new method for certifying that

would also bound m(n).


Thank you

Thank You


Narrowing the gap1

Narrowing the gap

Why bother about this tiny 3/2 log n gap?

log log scale a bit deceptive.

x vs.

Conjectures (various quantitative versions):

Most matroids are sparse paving. s(n): sparse paving matroids

  • m_n/s_n \rightarrow 1

  • log logm_n = log logs_n + o(1)

  • log logm_n = log logs_n + O(log log n)

    Perhaps counting -> Sampling and generating matroids.


Encoding matroids1

Encoding Matroids

If rank(X) = r-1. Then X has a unique circuit C.

Witness = (Cl(X), C)

Cl(X) : closure all z s.t. rank(X U z) = rank(X)

Proof this witness works:

(as Y=X-x+y)

Case 1: rk(X+y) = r-1, but then then

Case 2: rk(X-x) r-2 < |X-x|

So X-x contains a circuit C’. But then C=C’ by uniqueness.

So Y=X-x+y contains C.


Finish up1

Finish up

Given a matroid, let K = set of non-bases.

Apply stable set procedure to K obtain (S, A) with

  • S and A small as before, S determined by A.

    For a non-basis X, we have a witness for non-bases in the neighborhood of X.

    Encoding: (X, witness of X) for each

    List


Knuth s lower bound1

Knuth’s Lower bound

Matroid of rank r can be specified by r-sets that are non-bases.

Johnson Graph: J(n,r) V: r-subsets of [n] |V|=.

Edge (u,v): if

Claim: If non-bases form a stable set in J(n,r), then get a matroid.

These are called sparse paving matroids.

Proof: Base Exchange: M is matroidiff for every two bases B,B’ and e B\B’ there exists f in B’ such that B-e+f is base.

B

not a base

and not a base

B

B’


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