Matroids from Lossless Expander Graphs Nick Harvey U. Waterloo TexPoint fonts used in EMF. Read the...
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Transcript of Matroids from Lossless Expander Graphs Nick Harvey U. Waterloo TexPoint fonts used in EMF. Read the...
Matroids fromLossless Expander Graphs
Nick HarveyU. Waterloo
Maria-Florina BalcanGeorgia Tech
Matroids
• Ground Set V• Family of Independent Sets I• Axioms:• ; 2 I “nonempty”
• J ½ I 2 I ) J 2 I “downwards closed”
• J, I 2 I and |J|<|I| ) 9x2InJ s.t. J+x 2 I “maximum-size sets can be found
greedily”
• Rank function: r(S) = max { |I| : I2I and IµS }
Partition Matroid· 2 · 2
A1 A2
• This is a matroid• In general, if V = A1 [ [ Ak, then
is a partition matroid
V
. .
Intersecting Ai’s
a b c d e f g h i j k l
· 2 · 2
A1 A2
• Topic of This Talk:What if Ai’s intersect? Then I is not a matroid.
• For example, {a,b,k,l} and {f,g,h} are both maximal sets in I.
V
A fix
a b c d e f g h i j k l
· 2 · 2
A1 A2
• After truncating the rank to 3, then {a,b,k,l}I.• Checking a few cases shows that I is a matroid.
V
A general fix (for two Ai’s)
a b c d e f g h i j k l
· b1 · b2
A1 A2
• This works for any A1,A2 and bounds b1,b2
(unless b1+b2-|A1ÅA2|<0)
• Summary: There is a matroid that’s like a partition matroid, if bi’s large relative to |A1ÅA2|
V
The Main Question• Let V = A1[[Ak and b1,,bk2N
• Is there a matroid s.t.• r(Ai) · bi 8i
• r(S) is “as large as possible” for SAi (this is not formal)
• If Ai’s are disjoint, solution is partition matroid
• If Ai’s are “almost disjoint”, can we find a matroid that’s “almost” a partition matroid?
Next: formalize this
Lossless Expander Graphs
• Definition:G =(U[V, E) is a (D,K,²)-lossless expander if– Every u2U has degree D– |¡ (S)| ¸ (1-²)¢D¢|S| 8SµU with |S|·K,
where ¡ (S) = { v2V : 9u2S s.t. {u,v}2E }
“Every small left-set has nearly-maximalnumber of right-neighbors”
U V
Lossless Expander Graphs
• Definition:G =(U[V, E) is a (D,K,²)-lossless expander if– Every u2U has degree D– |¡ (S)| ¸ (1-²)¢D¢|S| 8SµU with |S|·K,
where ¡ (S) = { v2V : 9u2S s.t. {u,v}2E }
“Neighborhoods of left-vertices areK-wise-almost-disjoint”
Why “lossless”?Spectral techniques cannot obtain ² < 1/2.
U V
Trivial Example: Disjoint Neighborhoods
• Definition:G =(U[V, E) is a (D,K,²)-lossless expander if– Every u2U has degree D– |¡ (S)| ¸ (1-²)¢D¢|S| 8SµU with |S|·K,
where ¡ (S) = { v2V : 9u2S s.t. {u,v}2E }
• If left-vertices have disjoint neighborhoods, this gives an expander with ²=0, K=1
U V
Main Theorem: Trivial Case
• Suppose G =(U[V, E) has disjoint left-neighborhoods.• Let A={A1,…,Ak} be defined by A = { ¡(u) : u2U }.
• Let b1, …, bk be non-negative integers.
• Theorem:
is family of independent sets of a matroid.
I = f I : jI \ [ j 2 J A j j ·X
j 2 J
bj 8J gI = f I : jI \ A j j · bj 8j g
A1
A2
· b1
· b2U V
Main Theorem• Let G =(U[V, E) be a (D,K,²)-lossless expander• Let A={A1,…,Ak} be defined by A = { ¡(u) : u2U }
• Let b1, …, bk satisfy bi ¸ 4²D 8i
A1
· b1
A2
· b2
Main Theorem• Let G =(U[V, E) be a (D,K,²)-lossless expander• Let A={A1,…,Ak} be defined by A = { ¡(u) : u2U }
• Let b1, …, bk satisfy bi ¸ 4²D 8i
• “Wishful Thinking”: I is a matroid, whereI =f I : jI \ [ j 2 J A j j ·
X
j 2 J
bj 8J g
Main Theorem• Let G =(U[V, E) be a (D,K,²)-lossless expander• Let A={A1,…,Ak} be defined by A = { ¡(u) : u2U }
• Let b1, …, bk satisfy bi ¸ 4²D 8i
• Theorem: I is a matroid, whereI =f I : jI \ [ j 2 J A j j ·
X
j 2 J
bj ¡³ X
j 2 J
jA j j ¡ j [ j 2 J A j j´
8J s.t. jJ j · K
^ jI j · ²DK g
Main Theorem• Let G =(U[V, E) be a (D,K,²)-lossless expander• Let A={A1,…,Ak} be defined by A = { ¡(u) : u2U }
• Let b1, …, bk satisfy bi ¸ 4²D 8i
• Theorem: I is a matroid, where
• Trivial case: G has disjoint neighborhoods,i.e., K=1 and ²=0.
I =f I : jI \ [ j 2 J A j j ·X
j 2 J
bj ¡³ X
j 2 J
jA j j ¡ j [ j 2 J A j j´
8J s.t. jJ j · K
^ jI j · ²DK g
= 0
= 1
= 0
= 1
• Paving matroids can also be constructed by the main theorem
• A paving matroid is a matroid of rank D where every circuit has cardinality either D or D+1
Application: Paving Matroids
;
V
A1
A2A3
Ak
• Paving matroids can also be constructed by the main theorem
• A paving matroid is a matroid of rank D where every circuit has cardinality either D or D+1
• Sketch:– Let A={A1,...,Ak} be the circuits of cardinality D– A is a code of constant weight D and distance ¸ 4– This gives a (D,K,²)-expander with K=2 and ²=1-2/D
• Plugging this into the main theorem gives it(Actually, you need a more precise version from our paper)
Application: Paving Matroids
LB for Learning Submodular Functions
;
VA2
A1
• Similar idea to paving matroid construction,except we need “deeper valleys”
• If there are many valleys, the algorithm can’t learn all of them
n1/3
log2 n
LB for Learning Submodular Functions• Let G =(U[V, E) be a (D,K,²)-lossless expander, where Ai =
¡(ui) and– |V|=n − |U|=nlog n
– D = K = n1/3 − ² = log2(n)/n1/3
• Such graphs exist by the probabilistic method• Sketch:– Delete each node in U with prob. ½, then use main theorem
to get a matroid– If ui2U was not deleted then r(Ai) · bi = 4²D = O(log2 n)
– Claim: If ui deleted then Ai 2 I (Needs a proof) ) r(Ai) = |Ai| = D = n1/3
– Since # Ai’s = |U| = nlog n, no algorithm can learna significant fraction of r(Ai) values in polynomial time
I =f I : jI \ Cj · f (C) 8C 2 C gLemma: Let I be defined by
where f : C ! Z is some function. For any I 2 I, letT(I ) = f C : jI \ Cj = f (C) g
be the “tight sets” for I. Suppose thatC1 2 T(I ) ^ C2 2 T(I ) =) C1 [ C2 2 T(I )
Then I is independent sets of a matroid.
Proof: Let J,I 2 I and |J|<|I|. Must show 9x2InJ s.t. J+x 2 I.Let C be the maximal set in T(J). Then |IÅC| · f(C) = |JÅC|.Since |I|>|J|, 9x in In(C [ J).We must have J+x 2 I,because every C’3x has C’T(J).So |(J+x) Å C’|·f(C’). So J+x 2 I.
C JI
x
Concluding Remarks• A new family of matroids that give a common
generalization of partition & paving matroids
• Useful if you want...– a partition matroid, but the sets are not a partition– a paving matroid with deeper “valleys”
• Matroids came from analyzing learnability of submodular functions. – Imply a (n1/3) lower bound– Nearly matches O(n1/2) upper bound
Open Questions
• Other applications of these matroids?• n1/2 lower bound for learning submodular functions?• Are these matroids “maximal” s.t. |IÅAi|·bi?• Are these matroids linear?