Query Lower Bounds for Matroids via Group Representations Nick Harvey
Matroids Definition A matroid is a pair (S, B ) where B ⊂ 2 S s.t. Example: B = { spanning trees of graph G } Sets in B are called bases Rank of matroid is |A| for any A ∈ B Exchange Property Let A and B ∈ B a ∈ A\B, b ∈ B\A s.t. B+a-b ∈ B
Matroids Definition A matroid is a pair (S, B ) where B ⊂ 2 S s.t. Example: B = { spanning trees of graph G } Applications –Generalize Graph Problems –Approximation Algorithms –Network Coding Exchange Property Let A and B ∈ B a ∈ A\B, b ∈ B\A s.t. B+a-b ∈ B b1⊕b2b1⊕b2 b1⊕b2b1⊕b2 b1⊕b2b1⊕b2 s t b1b1 b2b2
Multicast Network Coding Goal: Multicast source sinks at maximum rate Algorithm: Can construct optimal solution in P Flow: [Jaggi et al. ’05], Matroids: [H., Karger, Murota ’05] Source Sinks a b a b ab a+b
Reversibility of Network Codes Sources Sinks a b ab a+b sasa sbsb tbtb tata G
Reversibility of Network Codes Flow: G feasible G rev feasible Coding: feasible G s.t. G rev not feasible [Dougherty, Zeger ’06] Sources Sinks a b ab a+b sasa sbsb tbtb tata G rev
Constructed from Fano and non-Fano matroids Reversibility of Network Codes [Dougherty, Zeger ’06]
Discrete Optimization Problems Matroid Intersection Bipartite Matching Non-Bip. Matching Network Flow Submodular Function Minimization Submodular Flow Matroid Matching Minimum Spanning Tree Matroid Greedy Algorithm Spanning Tree Packing Min-cost Arboresence Matroid Intersection Matroid Intersection Given matroids M 1 =(S, B 1 ) and M 2 =(S, B 2 ), is B 1 ⋂ B 2 = ∅ ?
n = # elements r = rank Unweighted Edmonds/Lawler ’68- ’75 O(nr 2 ) oracle Cunningham ’86O(nr 1.5 ) oracle Gabow-Xu ’89-’96O(nr 1.62 ) Harvey ’06O(nr 1.38 ) Matroid Intersection Algorithms Linear Matroids W = max weight Weighted Lawler/Edmonds ~’75O(nr 2 ) oracle Shigeno-Iwata ’95 O ( nr 1.5 log(rW) ) oracle Gabow-Xu ’89-’96 O ( nr 1.77 log(rW) ) Harvey ’07O(nr 1.38 W) Linear Matroids
n = # elements r = rank Unweighted Edmonds/Lawler ’68- ’75 O(nr 2 ) oracle Cunningham ’86O(nr 1.5 ) oracle Gabow-Xu ’89-’96O(nr 1.62 ) Harvey ’06O(nr 1.38 ) Linear Matroids Weighted Are these algorithms optimal? W = max weight Lawler/Edmonds ~’75O(nr 2 ) oracle Shigeno-Iwata ’95 O ( nr 1.5 log(rW) ) oracle Gabow-Xu ’89-’96 O ( nr 1.77 log(rW) ) Harvey ’07O(nr 1.38 W) Linear Matroids
Algorithm Computational Lower Bounds Strong lower bounds in unrestricted computational models are beyond our reach –5n - o(n) is best-known lower bound on circuit size for an explicit boolean function. [Iwama et al. ’05] –We believe 3SAT requires 2 (n) time, but best-known result is (n). A super-linear lower bound for any natural problem in P is hopeless. Data
Query Lower Bounds Strong lower bounds can be proven in concrete computational models –Sorting in comparison model –Monotone graph properties [Rivest-Vuillemin ’76] –Volume of convex body Deterministic [Elekes ’86] Randomized [Rademacher-Vempala ’06] Our work –Matroid intersection, Submodular Function Minimization B Out In Data Black Box Algorithm Queries
Query Model for Matroids (Independence Oracle) Out In Matroid (S, B ) Algorithm “Yes” if B ∈ B s.t. T ⊆ B “No” otherwise T⊆ST⊆S Example: if B = { spanning trees of graph G }, then query asks if T is an acyclic subgraph of G
Algorithms O(nr 2 ) queries [Lawler ’75] Matroid Intersection Complexity Are (nr 2 ) queries necessary and sufficient to solve matroid intersection? D. J. A. Welsh, “Matroid Theory”, 1976.
Algorithms O(nr 2 ) queries [Lawler ’75] Matroid Intersection Complexity O(nr 1.5 ) queries [Cunningham ’86] Can one prove any non-trivial lower bound on # queries to solve matroid intersection? Are (nr 2 ) queries necessary and sufficient to solve matroid intersection? D. J. A. Welsh, “Matroid Theory”, 1976.
Algorithms O(nr 2 ) queries [Lawler ’75] # queries Rank r 0nn/2 O(nr 1.5 ) queries [Cunningham ’86] Trivial LB Cunningham UB Matroid Intersection Complexity n 2n 0
Algorithms O(nr 1.5 ) queries [Cunningham ’86] # queries Rank r 0nn/2 Trivial LB Cunningham UB Via Dual Matroids Matroid Intersection Complexity n 2n 0
Algorithms O(nr 1.5 ) queries [Cunningham ’86] # queries Rank r 0nn/2 Trivial LB Cunningham UB Via Dual Matroids Optimal UB? Matroid Intersection Complexity n 2n 0
Matroid Intersection Complexity Algorithms O(nr 1.5 ) queries [Cunningham ’86] # queries Rank r 0n n 2n 0 n/2 Trivial LB Cunningham UB Via Dual Matroids Optimal UB? 1.58n New LB [Harvey ’08]
A family M of matroids, each of rank n/2 # oracle queries for any deterministic algorithm on inputs from M is: (log 2 3) n - o(n) > 1.58n Lower Bound [Harvey ’08] Hard Instances Communication Complexity Rank Computation AliceBob M =M =
Hard Instances Bipartite Matching in Almost-2-Regular Graphs Is there a perfect matching? Four vertices have degree 1
Hard Instances Bipartite Matching in Almost-2-Regular Graphs Is there a perfect matching? No: if path from 1 to 2 Yes: otherwise Four vertices have degree 1
Alice given ∈ S n and Bob given ∈ S n Elements 1 and 2 are not in the same cycle Permutation Permutation ’ 2’ 3’ 4’ 5’ 6’ In-Same-Cycle Problem: Are elements 1 and 2 in the same cycle of composition -1 º ? Permutation Formulation
LB from Rank Computation Let C be a matrix with rows and columns indexed by permutations in S n C , = where G = { : 1 & 2 are in the same cycle of } C is adjacency matrix of Cayley graph for S n with generators G 1 0 if -1 º ∈ G otherwise Main Result: rank C = (Moreover, it’s diagonalizable, all eigenvalues are integers, and they can be explicitly computed.) Corollary: # queries log rank C = (log 2 3) n - o(n).
Main Proof A Tour of Algebraic Combinatorics Step 0: Young Tableaux Step 1: Decomposing G Step 2: Decomposing C Step 3: Block diagonalizing R Step 4: Diagonalizing X i Wrap-up G = { (1,2) } × X 3 × X 4 … × X n
Young Diagrams Young diagram of shape =( 1, 2,..., k ) Standard Young Tableau of shape Main Result: rank C = # of SYT with n boxes such that 3 1 Row i has i boxes 1 2 … k >0 # boxes = n = ∑ i i Place numbers {1,..,n} in boxes Rows increase → Columns increase ↓ (and some other minor conditions)
Main Proof Step 0: Young Tableaux Step 1: Decomposing G Step 2: Decomposing C Step 3: Block diagonalizing R Step 4: Diagonalizing X i Wrap-up G = { (1,2) } × X 3 × X 4 … × X n
Decomposing S n Claim: ∈ S n = ◦ ( n, -1 ( n ) ), where ∈ S n-1 Example: Let ∈ S 6 be Then ◦ ( 7, 3 ) ∈ S 7 is ˜ ˜ ˜ ˜
Decomposing S n Restatement: Let X i = { (j,i) : 1 j i }. Then S n = X 2 × … × X n (2,2) = = ◦ (1,3)◦ (2,4)◦ (3,5)◦ (5,6)◦ (3,7) Claim: ∈ S n = ◦ ( n, -1 ( n ) ), where ∈ S n-1 ˜ ˜
Decomposing G Let G = { : 1 & 2 are in the same cycle } Claim: Let X i = { (j,i) : 1 j i }. Then G = { (1,2) } × X 3 × X 4 … × X n. 1 & 2 remain in the same cycle (1,2) = = ◦ (1,3)◦ (4,4)◦ (2,5)◦ (5,6)◦ (3,7)
Main Proof Step 0: Young Tableaux Step 1: Decomposing G Step 2: Decomposing C Step 3: Block diagonalizing R Step 4: Diagonalizing X i Wrap-up G = { (1,2) } × X 3 × X 4 … × X n
Decomposing C Regular Representation Recall: C is defined C , = Definition: R( ) is defined R( ) , = Thus: C = ∑ ∈ G R( ) 1 0 if -1 º ∈ G otherwise 1 0 if -1 º = otherwise
Main Proof Step 0: Young Tableaux Step 1: Decomposing G Step 2: Decomposing C Step 3: Block diagonalizing R Step 4: Diagonalizing X i Wrap-up G = { (1,2) } × X 3 × X 4 … × X n C = ∑ ∈ G R( )
Decomposing R “Fourier Transform” R()R()BR( )B -1 Irreducible Representations Young Tableaux change-of-basis matrix B block-diagonalizing R( ) ′ ′′
Main Proof Step 0: Young Tableaux Step 1: Decomposing G Step 2: Decomposing C Step 3: Block diagonalizing R Step 4: Diagonalizing X i Wrap-up G = { (1,2) } × X 3 × X 4 … × X n C = ∑ ∈ G R( )
Let X i = { (j,i) : 1 j i } Let Y (X i ) = ∑ ∈ X i BR( )B -1, restricted to irreducible block R()R() BR( )B -1 Young Tableaux ′ ′′ Diagonalizing X i Jucys-Murphy Elements
Let X i = { (j,i) : 1 j i } Let Y (X i ) = ∑ ∈ X i BR( )B -1, restricted to irreducible block ∑∈Xi R()∑∈Xi R() ∑ ∈ X i BR( )B -1 Young Tableaux ′ ′′ Y (X i ) Diagonalizing X i Jucys-Murphy Elements
Let X i = { (j,i) : 1 j i } Let Y (X i ) = ∑ ∈ X i BR( )B -1, restricted to irreducible block Fact: Y (X i ) is diagonal (and entries known) ∑∈Xi R()∑∈Xi R() ∑ ∈ X i BR( )B -1 Young Tableaux ′ ′′ Y (X i )
Diagonalizing X i Jucys-Murphy Elements Let X i = { (j,i) : 1 j i } Let Y (X i ) = ∑ ∈ X i BR( )B -1, restricted to irreducible block Fact: Y (X i ) is diagonal (and entries known) Y (X 4 ) = Young Tableau SYT t1t1 t2t2 t3t3 t2t2 t3t3 t1t1 t2t2 t3t3 t1t1 Let X i = { (j,i) : 1 j i } Let Y (X i ) = ∑ ∈ X i BR( )B -1, restricted to irreducible block Fact: Y (X i ) is diagonal, and entry Y (X i ) t j,t j is c-r+1, where i is in row r and col c of t j. Content Value
Diagonalizing X i Jucys-Murphy Elements Let X i = { (j,i) : 1 j i } Let Y (X i ) = ∑ ∈ X i BR( )B, restricted to irreducible block Fact: Y (X i ) is diagonal (and entries known) Y (X 4 ) = Young Tableau SYT t1t1 t2t2 t3t3 t2t2 t3t3 t1t1 t2t2 t3t3 t1t1 Let X i = { (j,i) : 1 j i } Let Y (X i ) = ∑ ∈ X i BR( )B -1, restricted to irreducible block Fact: Y (X i ) is diagonal, and entry Y (X i ) t j,t j is c-r+1, where i is in row r and col c of t j. Content Value
Main Proof Step 0: Young Tableaux Step 1: Decomposing G Step 2: Decomposing C Step 3: Block diagonalizing R Step 4: Diagonalizing X i Wrap-up G = { (1,2) } × X 3 × X 4 … × X n C = ∑ ∈ G R( )
Wrap-up Diagonalizing C Y ({ (1,2) }) ∙ Y (X 3 ) … Y (X n ) is diagonal Y ({ (1,2) } × X 3 × … × X n ) is diagonal Y ( G ) is diagonal ∑ ∈ G BR( )B -1 is diagonal BCB -1 is diagonal (homomorphism) (Step 1) (Step 3) (Step 2) (Step 4)
Y ( G ) t j,t j 0 Y (X i ) t j,t j 0 i 3 If i 3 is in row c and col r of t j, then c-r+1 0 rank C = # SYT with 3 1 (and 2 below 1,...) Wrap-up What are eigenvalues of C? c-r Content Value Content Values SYT t j No i 3 can go here 3 1 Y ( G )=Y ({ (1,2) } × X 3 × … × X n )
Main Proof Step 0: Young Tableaux Step 1: Decomposing G Step 2: Decomposing C Step 3: Block diagonalizing R Step 4: Diagonalizing X i Wrap-up G = { (1,2) } × X 3 × X 4 … × X n C = ∑ ∈ G R( ) QED