Matroids from Lossless Expander Graphs

Matroids fromLossless Expander Graphs Maria-FlorinaBalcanGeorgia Tech Nick HarveyU. 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” • J½I2I)J2I“downwards closed” • J, I2I and |J|<|I| )9x2InJs.t. J+x2I“maximum-size sets can be found greedily” • Rank function: r(S) = max { |I| : I2I and IµS }

Partition Matroid · 2 · 2 V A1 A2 • This is a matroid • In general, if V = A1[[Ak, thenis 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 · 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 matroids.t. • r(Ai) · bi8i • 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 • Definition:G =(U[V, E) is a (D,K,²)-lossless expanderif • 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 expanderif • 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” U V 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 expanderif • 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

Main Theorem: Trivial Case A1 ·b1 ·b2 V U • 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. 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 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¸ 4²D8i • “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¸ 4²D8i • Theorem: 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¸ 4²D8i • Theorem: I is a matroid, where • Trivial case: G has disjoint neighborhoods,i.e., K=1 and ²=0. = 0 = 0 = 1 = 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 V A1 ; A2 A3 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)

LB for Learning Submodular Functions n1/3 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 V log2 n A2 ;

LB for Learning Submodular Functions • Let G =(U[V, E) be a (D,K,²)-lossless expander, where Ai = ¡(ui) and • |V|=n −|U|=nlogn • 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(log2n) • Claim: If ui deleted then Ai2I(Needs a proof) )r(Ai) = |Ai| = D = n1/3 • Since # Ai’s = |U| = nlogn, no algorithm can learna 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 2I, let be the “tight sets” for I. Suppose that Then I is independent sets of a matroid. Proof: Let J,I2I and |J|<|I|. Must show 9x2InJs.t. J+x2I. 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+x2I,because every C’3x has C’T(J). So |(J+x) ÅC’|·f(C’). So J+x2I. J C 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?

Matroids from Lossless Expander Graphs