1. Matroids from Lossless Expander Graphs
Maria-Florina Balcan Georgia Tech
Nick Harvey U. Waterloo
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2. 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 }

3. Partition Matroid . . This is a matroid
· 2
· 2 V A1 A2
This is a matroid
In general, if V = A1 [  [ Ak, then is a partition matroid . .

4. Intersecting Ai’s
· 2
· 2 V a b c d e f g h i j k l 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.

5. A fix After truncating the rank to 3, then {a,b,k,l}I.
· 2
· 2 V a b c d e f g h i j k l A1 A2
After truncating the rank to 3, then {a,b,k,l}I.
Checking a few cases shows that I is a matroid.

6. A general fix (for two Ai’s)
· b1
· b2 V a b c d e f g h i j k l 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|

7. 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 SAi (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

8. Lossless Expander GraphsU V
Definition: G =(U[V, E) is a (D,K,²)-lossless expander if
Every u2U has degree D
|¡ (S)| ¸ (1-²)¢D¢|S| SµU with |S|·K, where ¡ (S) = { v2V : 9u2S s.t. {u,v}2E }
“Every small left-set has nearly-maximal number of right-neighbors”
This is basically a result of Kahale.

9. Lossless Expander GraphsU V
Definition: G =(U[V, E) is a (D,K,²)-lossless expander if
Every u2U has degree D
|¡ (S)| ¸ (1-²)¢D¢|S| SµU with |S|·K, where ¡ (S) = { v2V : 9u2S s.t. {u,v}2E }
“Neighborhoods of left-vertices are K-wise-almost-disjoint”
Why “lossless”? Spectral techniques cannot obtain ² < 1/2.

10. Trivial Example: Disjoint NeighborhoodsU V
Definition: G =(U[V, E) is a (D,K,²)-lossless expander if
Every u2U has degree D
|¡ (S)| ¸ (1-²)¢D¢|S| SµU 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

11. Main Theorem: Trivial Case
· b1
· b2 V U A2
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.

12. Main Theorem A1 A2 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
· b2 A2

13. 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, where

14. 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
Bound on |I| should be min { sum b_j - (error for K) : |J| = K/2 }
This is >= (4 eps D) K / 2 – eps D K = epsilon D K

15. 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.
= 0
= 0
= 1
Wait: epsilon * K is undefined!
But it’s OK, we can take epsilon = 1/4D. G is certainly still a 1/4D expander.
The lower bound on b_i is now 1.
And now that term goes to infinity.
= 1

16. Application: Paving Matroids
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 A1 A2 ; V A3
Distance = 2D – 2|Ai n Aj|
=> |Ai n Aj| = D – Distance/2
Expansion is (1-eps) 2 D = 2D – 2|Ai n Aj|
Epsilon = |Ai n Aj| / D = 1 – Distance / (2D)
Epsilon*D = D- Distance/2 Ak

17. Application: Paving Matroids
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)
Distance = 2D – 2|Ai n Aj|
=> |Ai n Aj| = D – Distance/2
Expansion is (1-eps) 2 D = 2D – 2|Ai n Aj|
Epsilon = |Ai n Aj| / D = 1 – Distance / (2D)
Epsilon*D = D- Distance/2

18. LB for Learning Submodular Functions; V A1
log2 n A2
Similar idea to paving matroid construction, except we need “deeper valleys”
If there are many valleys, the algorithm can’t learn all of them

19. 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 learn a significant fraction of r(Ai) values in polynomial time

20. Lemma: Let I be defined by
where f : C ! Z is some function. For any I 2 I, let
be the “tight sets” for I. Suppose that
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 J I x

21. 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

22. 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?