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On the number of matroids Nikhil Bansal (TU Eindhoven) Rudi Pendavingh (TU Eindhoven) Jorn van der...
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Transcript of On the number of matroids Nikhil Bansal (TU Eindhoven) Rudi Pendavingh (TU Eindhoven) Jorn van der...
On the number of matroids
Nikhil Bansal (TU Eindhoven)Rudi Pendavingh (TU Eindhoven)Jorn van der Pol (TU Eindhoven)
MatroidsMatroid (U,C): U = Universe [n], C: collection of independent sets(i) Subset closed: I independent, then also .(ii) Exchange: |I|>|I’| then some s.t also independent. How does a typical matroid look ? Can we generate them randomly?
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
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)
22𝑛Better bound
Known resultsKnuth 74
Easy Upper bound:
Pf: Any rank r matroid is specified by its bases (max. indep. sets)Number of rank r matroids m(n,r) .So,
m(n) (n+1) (can just focus on rank r=n/2)Piff 73: ]
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
• Knuth’s lower bound construction• Counting stable sets• The final upper bound
Knuth’s Lower boundMatroid 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 boundSparse 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
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 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 BoundRefined bound: i(J(n,n/2)) Morally: All independent sets are subsets of few large independent sets.
Examples: n-Hypercube verticesNaïve 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 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 talk
Encoding for independent sets.Encoding for Matroids.
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
Associate to an independent set I of G the pair of vertices (S,A), s.t.
1) 2) |A| 3) 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
G
I
A
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 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 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 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 set
Claim: Pf: Alg in phase j if ]
A vertex in phase j has at least j neighbors in G[A].Must pick vertices in S. Sum over j=d,…1.
Phases: d 12d-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. || > 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 sets that are witnesses for all non-bases Y in .
If rank(X) < r-1. Witness = (X,rk(X))
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
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.
Encoding: {witness of } for each List 1) Witness for all non-bases in 2) List of remaining non-bases.
Questions
Would be nice to reduce the gap to o(1).
The reason for +1+o(1) gapDo 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
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
1) m_n/s_n \rightarrow 12) 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: (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 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.
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 boundMatroid 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 matroid iff 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 baseand not a base
B B’