# Matroids from Lossless Expander Graphs

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Matroids from Lossless Expander Graphs
Maria-Florina Balcan Georgia Tech Nick Harvey U. Waterloo TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAA

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

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.

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.

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|

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

Lossless Expander Graphs
U 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.

Lossless Expander Graphs
U 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.

Trivial Example: Disjoint Neighborhoods
U 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

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.

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

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

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

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

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

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

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

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

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

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?