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

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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| ) 9 x 2 I n J s.t. J+x 2 I “maximum-size sets can be found greedily” Rank function: r(S) = max { |I| : I 2I and I µ S }

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Partition Matroid · 2 A1A1 A2A2 This is a matroid In general, if V = A 1 [ [ A k, then is a partition matroid V..

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Intersecting A i ’s abcdefghijkl · 2 A1A1 A2A2 Topic of This Talk: What if A i ’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

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A fix abcdefghijkl · 2 A1A1 A2A2 After truncating the rank to 3, then {a,b,k,l} I. Checking a few cases shows that I is a matroid. V

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A general fix (for two A i ’s) abcdefghijkl · b 1 · b 2 A1A1 A2A2 This works for any A 1,A 2 and bounds b 1,b 2 (unless b 1 +b 2 -|A 1 Å A 2 |<0) Summary: There is a matroid that’s like a partition matroid, if b i ’s large relative to |A 1 Å A 2 | V

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The Main Question Let V = A 1 [ [ A k and b 1, ,b k 2 N Is there a matroid s.t. r(A i ) · b i 8 i r(S) is “as large as possible” for S A i (this is not formal) If A i ’s are disjoint, solution is partition matroid If A i ’s are “almost disjoint”, can we find a matroid that’s “almost” a partition matroid? Next: formalize this

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

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Lossless Expander Graphs Definition: G =(U [ V, E) is a (D,K, ² )-lossless expander if – Every u 2 U has degree D – | ¡ (S)| ¸ (1- ² ) ¢ D ¢ |S| 8 S µ U with |S| · K, where ¡ (S) = { v 2 V : 9 u 2 S s.t. {u,v} 2 E } “Neighborhoods of left-vertices are K-wise-almost-disjoint” Why “lossless”? Spectral techniques cannot obtain ² < 1/2. UV

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Trivial Example: Disjoint Neighborhoods Definition: G =(U [ V, E) is a (D,K, ² )-lossless expander if – Every u 2 U has degree D – | ¡ (S)| ¸ (1- ² ) ¢ D ¢ |S| 8 S µ U with |S| · K, where ¡ (S) = { v 2 V : 9 u 2 S s.t. {u,v} 2 E } If left-vertices have disjoint neighborhoods, this gives an expander with ² =0, K= 1 UV

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Main Theorem: Trivial Case Suppose G =(U [ V, E) has disjoint left-neighborhoods. Let A ={A 1,…,A k } be defined by A = { ¡ (u) : u 2 U }. Let b 1, …, b k be non-negative integers. Theorem: is family of independent sets of a matroid. A1A1 A2A2 · b1· b1 · b2· b2 U V

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Main Theorem Let G =(U [ V, E) be a (D,K, ² )-lossless expander Let A ={A 1,…,A k } be defined by A = { ¡ (u) : u 2 U } Let b 1, …, b k satisfy b i ¸ 4 ² D 8 i A1A1 · b1· b1 A2A2 · b2· b2

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Main Theorem Let G =(U [ V, E) be a (D,K, ² )-lossless expander Let A ={A 1,…,A k } be defined by A = { ¡ (u) : u 2 U } Let b 1, …, b k satisfy b i ¸ 4 ² D 8 i “Wishful Thinking”: I is a matroid, where

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Main Theorem Let G =(U [ V, E) be a (D,K, ² )-lossless expander Let A ={A 1,…,A k } be defined by A = { ¡ (u) : u 2 U } Let b 1, …, b k satisfy b i ¸ 4 ² D 8 i Theorem: I is a matroid, where

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Main Theorem Let G =(U [ V, E) be a (D,K, ² )-lossless expander Let A ={A 1,…,A k } be defined by A = { ¡ (u) : u 2 U } Let b 1, …, b k satisfy b i ¸ 4 ² D 8 i Theorem: I is a matroid, where Trivial case: G has disjoint neighborhoods, i.e., K= 1 and ² =0. = 0 = 1 = 0 = 1

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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 A1A1 A2A2 A3A3 AkAk

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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 ={A 1,...,A k } 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

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LB for Learning Submodular Functions ; V A2A2 A1A1 Similar idea to paving matroid construction, except we need “deeper valleys” If there are many valleys, the algorithm can’t learn all of them n 1/3 log 2 n

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LB for Learning Submodular Functions Let G =(U [ V, E) be a (D,K, ² )-lossless expander, where A i = ¡ (u i ) and – |V|=n − |U|=n log n – D = K = n 1/3 − ² = log 2 (n)/n 1/3 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 u i 2 U was not deleted then r(A i ) · b i = 4 ² D = O(log 2 n) – Claim: If u i deleted then A i 2 I (Needs a proof) ) r(A i ) = |A i | = D = n 1/3 – Since # A i ’s = |U| = n log n, no algorithm can learn a significant fraction of r(A i ) values in polynomial time

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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 9 x 2 I n J s.t. J+x 2 I. Let C be the maximal set in T(J). Then |I Å C| · f(C) = |J Å C|. Since |I|>|J|, 9 x in I n (C [ J). We must have J+x 2 I, because every C’ 3 x has C’ T(J). So |(J+x) Å C’| · f(C’). So J+x 2 I. C J I x

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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 (n 1/3 ) lower bound – Nearly matches O(n 1/2 ) upper bound

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Open Questions Other applications of these matroids? n 1/2 lower bound for learning submodular functions? Are these matroids “maximal” s.t. |I Å A i | · b i ? Are these matroids linear?

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