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
(i) Subset closed: I independent, then 𝐼’ ⊂ 𝐼 also .
(ii) Exchange: |I|>|I’| then some 𝑥 ∈ 𝐼\I ′ s.t. 𝐼’ ∪ 𝑥 also independent.
How does a typical matroid look ?
Can we generate them randomly?
How many matroids m(n) on n elements?
𝑛
Clearly, m(n) ≤ 22
1 𝑛
𝑛 𝑛/2
2𝑛
𝑛1.5
Knuth’74: m(n) ≥ 2
≈ 2
(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
Why bother about this tiny 3/2 log n gap?
log log scale a bit deceptive.
x vs. 𝑥 2
log log 𝑥 2 = log 2 log 𝑥 = 1 + log log 𝑥
Conjecture: Most matroids are sparse paving (various versions)
Knuth’s bound perhaps close to optimal
Often counting ->
Sampling and generating matroids.
Better bound
m(n)
22
𝑛
Known results
3
2
Knuth 74: log log 𝑚 𝑛 ≥ 𝑛 – log 𝑛 − 𝑂 1
Easy Upper bound: log log 𝑚 𝑛 ≤ 𝑛 – ½ log 𝑛 + 𝑂 1
Pf: Any rank r matroid is specified by its bases (max. indep. sets)
Number of rank r matroids m(n,r) ≤ 2
So, log log 𝑚 𝑛, 𝑟 ≤ log
m(n) ≤(n+1) max 𝑚 𝑛, 𝑟
𝑟
𝑛
𝑛/2
𝑛
𝑟
.
≤ 𝑛 – ½ log 𝑛 + 𝑂 1
(can just focus on rank r=n/2)
Piff 73: log log 𝑚(𝑛) ≤ 𝑛 – log 𝑛 + 𝑂 1
[Best so far]
Our Result
Thm: log log m(n) ≤ Knuth’s lower bound + 1+o(1) .
(Piff had ≈ ½ log n gap)
Knuth 74: m 𝑛 ≥ 2
We show: m 𝑛 ≤ 2
1 𝑛
𝑛 𝑛/2
2+𝑜(1) 𝑛
𝑛/2
𝑛
Need only the most basic matroid facts.
Main Tool: Bounding number of stable sets in a graph.
Outline
• Knuth’s lower bound construction
• Counting stable sets
• The final upper 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|=
𝑛
𝑟
.
Edge (u,v): if 𝑢 ∩ 𝑣 = 𝑟 − 1
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
Sparse Paving Matroids : precisely the stable sets of J(n,r).
For graph G: let 𝛼 𝐺 = size of max stable set.
i(G) = # stable sets.
Note: 𝑖 𝐺 ≥ 2𝛼 𝐺 .
J(n,r) is a regular graph of degree d = 𝑟 𝑛 − 𝑟 ≈
So, 𝛼 ≥
1
𝑛
𝑑+1 𝑛/2
≈
2𝑛
𝑛2.5
𝑛2
4
1
𝑛
Knuth: 𝛼 ≥ 𝑛/2
𝑛
Proof: Color vertex 𝑥1 , … , 𝑥𝑛 by j if
Gives proper n-coloring.
𝑖𝑖
𝑛
2
for 𝑟 = .
𝑥𝑖 = 𝑗 mod 𝑛.
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)
Claim: Max stable set in J(n,n/2) ≤ (2/n) N
N = # of vertices
Fact: If −𝜆 is smallest eigenvalue of adj. matrix of a d-regular graph.
Then, 𝛼 𝐺 ≤
𝜆𝑁
𝑑+𝜆
(proof later)
Johnson graphs: 𝜆 ≈
Naïve bound: i(G) ≤
≈
𝑛
2
So, 𝛼 𝐺 ≤ ≈
for J(n/n/2)
𝑁
0
+
𝑐𝑛
𝑁
1
2𝑁/𝑛
+…+
𝑎𝑠
Recall, knuth Lower’s bound: 2𝑁/𝑛
Niavely: i(J(n,n/2)) ≈ 𝑛2𝑁/𝑛
2
𝑛
𝑁
𝑁
2𝑁/𝑛
𝑁
𝑘
≈
𝑒𝑁 𝑘
𝑘
(note: base of exponent)
Better Bound
Refined bound:
i(J(n,n/2)) ≤ 2(2+𝑜(1))𝑁/𝑛
Morally: All independent sets are subsets of few large independent sets.
Examples: n-Hypercube
𝛼 𝐺 =
𝑁
2
𝑁 = 2𝑛 vertices
Naïve bound on i(G) =
1
2
2𝑁
Right answer: [Saphozhenko’83] (1+o(1) 2𝑒 2𝑁/2
𝑁 𝑁
+
2 2𝑑
Entropy Method [Kahn’01]: Any d-regular bipartite graph 2
Tight: Disjoint copies of 𝐾𝑑,𝑑 (n/2d copies)
Holds even for general graphs [Zhao’10]
Our Result
Thm: In any d-regular graph G with min eigenvalue −𝜆
𝑁 (ln2 𝑑)/𝑑
i(G) ≤ 2
2𝑁 𝜆/(𝑑+𝜆)
Idea: Encode an independent set using few bits of information.
Eg: Bipartite graphs: 𝜆 = −𝑑
Our bound:
2 𝑑)/𝑑
𝑁/2+
𝑁
(ln
2
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 talk
Encoding for independent sets.
Encoding for Matroids.
A useful lemma
G: d-regular with min eigenvalue −𝜆. For any vertex subset A.
2𝑒 𝐺 𝐴
𝑑𝐴
𝑁
≥ 𝐴
𝐴
𝑁
A
= 𝜒 𝐴 𝑇𝐺 𝜒 𝐴
Proof: 𝑒 𝐺 𝐴
Split 𝜒 𝐴 =
−𝜆 1−
𝐴
𝑁
Corollary: 𝛼 𝐺 ≤
1, … , 1 + other stuff ⊥ (1, … , 1)
𝜆
𝑁
𝑑+𝜆
Corollary: If |A| ≥
≥𝜖 𝑑+𝜆
𝜆𝑁
𝑑+𝜆
+ 𝜖N, then G[A] has a vertex of degree
(For random set A of size 𝜖𝑁, expected degree ≈ 𝜖𝑑)
Encoding a stable set
Associate to an independent set I of G the pair of vertices (S,A), s.t.
1) 𝑆 ⊂ 𝐼 ⊂ 𝑆 ∪ 𝐴
2)
3)
𝜆𝑁
ln 𝑑+1
S
|A| ≤
.
𝑆 ≤𝑁
𝑑+𝜆
𝑑
A is completely determined by S.
I can be uniquely specified by (S, 𝐴 ∩ 𝐼).
I
A
Key Point: A is completely determined by S.
Number of possibilities for I =
𝑁
|𝑆|
2𝐴
(gives the result)
G
Encoding a stable set
Input: Independent set I.
Initialize: A = V and S = ∅.
𝜆
While |A| >
𝑁
𝑑+𝜆
Let v = largest degree vertex in G[A] (ties in lex order)
If v in I, add to S and set 𝐴 = 𝐴 \ (𝑣 ∪ 𝑁(𝑣))
Else discard v and set A = A\{v}
.
Encoding a stable set
Input: Independent set I.
Initialize: A = V and S = ∅.
𝜆
While |A| >
𝑁
𝑑+𝜆
Let v = largest degree vertex in G[A] (ties in lex order)
If v in I, add to S and set 𝐴 = 𝐴 \ (𝑣 ∪ 𝑁(𝑣))
Else discard v and set A = A\{v}
.
Encoding a stable set
Input: Independent set I.
Initialize: A = V and S = ∅.
𝜆
While |A| >
𝑁
𝑑+𝜆
Let v = largest degree vertex in G[A] (ties in lex order)
If v in I, add to S and set 𝐴 = 𝐴 \ (𝑣 ∪ 𝑁(𝑣))
Else discard v and set A = A\{v}
.
Encoding a stable set
Input: Independent set I.
Initialize: A = V and S = ∅.
𝜆
While |A| >
𝑁
𝑑+𝜆
Let v = largest degree vertex in G[A] (ties in lex order)
If v in I, add to S and set 𝐴 = 𝐴 \ (𝑣 ∪ 𝑁(𝑣))
Else discard v and set A = A\{v}
Observe: S completely determines A.
Encoding a stable set
Phases:
d
Claim: 𝑆 ≤
…
d-1
ln 𝑑+1
𝑑
2
1
𝑁
Pf: Alg in phase j if 𝐴 ∈ [
𝑗+𝜆 𝑁 𝑗−1+𝜆 𝑁
,
]
𝑑+𝜆
𝑑+𝜆
A vertex in phase j has at least j neighbors in G[A].
Must pick ≤
𝑁
𝑗+1 𝑑+𝜆
vertices in S. Sum over j=d,…1.
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. |𝑌 ∩ 𝑋| > 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.
Key Lemma: For a dependent r-set X, we can associate ≤
two sets that are witnesses for all non-bases Y in 𝑋 ∪ 𝑁 𝑋 .
If rank(X) < r-1.
Witness = (X,rk(X))
Key Lemma: For a dependent r-set X, we can associate ≤
two sets that are witnesses for all non-bases Y in 𝑋 ∪ 𝑁 𝑋 .
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: 2𝑟 − 2 ≥ rk 𝑋 + rk 𝑌 ≥ rk 𝑋 ∩ 𝑌 + rk 𝑋 ∪ 𝑌
= rk 𝑋 − 𝑥 + rk 𝑋 + 𝑦
(as Y=X-x+y)
Case 1: If rk(X+y) = r-1, then 𝑦 ∈ 𝐶𝑙 𝑋 .
So, 𝑌 ∩ 𝐶𝑙 𝑋 = |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 𝐶 ∈ 𝑌 (as Y=X-x+y).
Hence |𝑌 ∩ 𝐶| = |C| > rk(C)
Finish up
Given a matroid, let K = set of non-bases.
Apply stable set procedure to K obtain (S, A) with
1) S and A small as before, S determined by A.
2) 𝑆 ⊂ 𝐾
Encoding:
{witness of 𝑋 ∪ 𝑁 𝑋 } for each 𝑋 ∈ 𝑆.
List 𝐾 ∩ 𝐴
1) Witness for all non-bases in 𝑆 ∪ 𝑁 𝑆 .
2) List of remaining non-bases.
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 𝛼 𝐺 ≈
would also bound m(n).
𝑁
𝑛
Thank You
Narrowing the gap
Why bother about this tiny 3/2 log n gap?
log log scale a bit deceptive.
x vs. 𝑥 2
log log 𝑥 2 = log 2 log 𝑥 = 1 + log log 𝑥
Conjectures (various quantitative versions):
Most matroids are sparse paving.
s(n): sparse paving matroids
1) m_n/s_n \rightarrow 1
2) log log m_n = log log s_n + o(1)
3) log log m_n = log log s_n + O(log log n)
Perhaps counting -> Sampling and generating matroids.
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:
2𝑟 − 2 ≥ 𝑟𝑘 𝑋 + 𝑟𝑘 𝑌 ≥ 𝑟𝑘 (𝑋 ∩
Finish up
Given a matroid, let K = set of non-bases.
Apply stable set procedure to K obtain (S, A) with
1) S and A small as before, S determined by A.
2) 𝑆 ⊂ 𝐾 ⊂ (𝑆 ∪ 𝑁 𝑆 ∪ 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 𝑋 ∈ 𝑆.
List 𝐾 ∩ 𝐴
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|=
𝑛
𝑟
.
Edge (u,v): if 𝑢 ∩ 𝑣 = 𝑟 − 1
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 matroid iff for every two bases B,B’
and e ∈ B\B’ there exists f in B’ such that B-e+f is base.
𝑒1 𝑒2 …
B
B 𝑓1
𝑓2 …
B’
𝐵 − e1 + 𝑓1 not a base
and 𝐵 − 𝑒1 + 𝑓2 not a base