Understanding Matroid Theory: Bounds, Applications, and Complexity

Query Lower Boundsfor Matroidsvia Group Representations Nick Harvey

Matroids • Definition A matroid is a pair (S,B ) where B ⊂2S 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

b1 b2 t s b1⊕b2 b1⊕b2 b1⊕b2 Matroids • Definition A matroid is a pair (S,B ) where B ⊂2S 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

Multicast Network Coding Source b a • Goal: Multicast sourcesinks at maximum rate • Algorithm: Can construct optimal solution in PFlow: [Jaggi et al. ’05],Matroids: [H., Karger, Murota ’05] a b a+b a b a+b a+b Sinks

Reversibility of Network Codes sb sa Sources a b G a b a+b a+b a+b ta Sinks tb

Reversibility of Network Codes tb ta Sinks a+b a+b Grev a b a+b b a sa Sources sb • Flow: G feasible  Grev feasible • Coding:  feasible G s.t. Grevnot feasible[Dougherty, Zeger ’06]

Reversibility of Network Codes [Dougherty, Zeger ’06] Constructed from Fano andnon-Fano matroids

Discrete Optimization Problems • Matroid IntersectionGiven matroids M1=(S,B 1) and M2=(S,B2),is B 1⋂B 2=∅? NetworkFlow SubmodularFlow MinimumSpanningTree SubmodularFunctionMinimization MatroidIntersection MatroidIntersection BipartiteMatching Spanning TreePacking MatroidGreedyAlgorithm Non-Bip.Matching MatroidMatching Min-costArboresence

Matroid Intersection Algorithms Unweighted n = # elements r = rank Linear Matroids Weighted W = max weight Linear Matroids

Are these algorithms optimal? Unweighted n = # elements r = rank Linear Matroids Weighted W = max weight Linear Matroids

Computational Lower Bounds • Strong lower bounds in unrestricted computational models are beyond our reach • 5n - o(n) is best-known lower bound on circuit sizefor an explicit boolean function. [Iwama et al. ’05] • We believe 3SAT requires 2(n) time, butbest-known result is (n). • A super-linear lower bound for anynatural problem in P is hopeless. Data Algorithm

B Data Algorithm In Queries Out Black Box Query Lower Bounds • Strong lower bounds can be proven inconcrete 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

Query Model for Matroids(Independence Oracle) • Example: if B = { spanning trees of graph G },then query asks if T is an acyclic subgraph of G T⊆S Algorithm In Matroid(S,B) “Yes” if B∈Bs.t. T⊆B “No” otherwise Out

Matroid Intersection Complexity O(nr2) queries[Lawler ’75] Algorithms Are (nr2) queries necessary and sufficientto solve matroid intersection? D. J. A. Welsh, “Matroid Theory”, 1976.

Matroid Intersection Complexity O(nr2) queries[Lawler ’75] O(nr1.5) queries[Cunningham ’86] Algorithms No Are (nr2) queries necessary and sufficientto solve matroid intersection? D. J. A. Welsh, “Matroid Theory”, 1976. Can one prove any non-trivial lower boundon # queries to solve matroid intersection?

Matroid Intersection Complexity O(nr2) queries[Lawler ’75] O(nr1.5) queries[Cunningham ’86] Algorithms 2n Cunningham UB Trivial LB n # queries 0 0 n/2 n Rank r

Matroid Intersection Complexity O(nr1.5) queries[Cunningham ’86] Algorithms 2n Cunningham UB Trivial LB n Via Dual Matroids # queries 0 0 n/2 n Rank r

Matroid Intersection Complexity O(nr1.5) queries[Cunningham ’86] Algorithms Optimal UB? 2n Cunningham UB Trivial LB n Via Dual Matroids # queries 0 0 n/2 n Rank r

Matroid Intersection Complexity O(nr1.5) queries[Cunningham ’86] Algorithms Optimal UB? 2n Cunningham UB 1.58n Trivial LB n Via Dual Matroids # queries New LB [Harvey ’08] 0 0 n/2 n Rank r

Lower Bound [Harvey ’08] • A family Mof matroids, each of rank n/2 • # oracle queries for any deterministic algorithm on inputs from M is: (log2 3) n - o(n) > 1.58n CommunicationComplexity RankComputation Hard Instances M= Alice Bob

Hard InstancesBipartite Matching in Almost-2-Regular Graphs 1 Four verticeshave degree 1 3 2 4 • Is there a perfect matching?

Hard InstancesBipartite Matching in Almost-2-Regular Graphs 1 Four verticeshave degree 1 3 2 4 • Is there a perfect matching? • No: if path from 1 to 2 • Yes: otherwise

Permutation Formulation • Alice given  ∈Sn and Bob given  ∈Sn • In-Same-Cycle Problem:Are elements 1 and 2 in the same cycleof composition -1º? 1 1’ Permutation 2 2’ Permutation-1 3 3’ Elements 1 and 2are not in the same cycle 4 4’ 5 5’ 6 6’

if -1 º  ∈G 1 otherwise 0 Main Result: rank C= (Moreover, it’s diagonalizable, all eigenvalues are integers, and they can be explicitly computed.) LB from Rank Computation • Let C be a matrix with rows and columnsindexed by permutations in Sn C, = where G= {  : 1 & 2 are in the same cycle of  } • C is adjacency matrix of Cayley graphfor Sn with generators G Corollary: # queries  log rank C =(log2 3)n - o(n).

Main ProofA Tour of Algebraic Combinatorics • Step 0: Young Tableaux • Step 1: Decomposing G • Step 2: Decomposing C • Step 3: Block diagonalizing R • Step 4: Diagonalizing Xi • Wrap-up 3 7 2 1 4 5 6 G= {(1,2)} ×X3×X4 … ×Xn

Young Diagrams Row i has i boxes 12…k>0 # boxes = n = ∑ii • 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 Place numbers {1,..,n} in boxes Rows increase → Columns increase ↓ 1 2 6 8 3 5 9 4 7 10 11 (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 Xi • Wrap-up 3 7 2 1 4 5 6 G= {(1,2)} ×X3×X4 … ×Xn

3 2 1 5 4     ˜ ˜ ˜ ˜ 6 Decomposing Sn • Claim: ∈Sn  = ◦ (n,-1(n)), where ∈Sn-1 • Example: Let∈S6 be • Then ◦(7, 3)∈S7 is 3 7 2 1 4 5 6

  ˜ ˜ Decomposing Sn • Claim: ∈Sn  = ◦ (n,-1(n)), where ∈Sn-1 • Restatement: Let Xi = { (j,i) : 1ji }.Then Sn = X2× … ×Xn.  = (2,2) ◦ (1,3) ◦ (2,4) ◦(3,5) ◦ (5,6) ◦ (3,7) 3 7 2 1 5 4 6

DecomposingG • Let G = {  : 1 & 2 are in the same cycle } • Claim: Let Xi = { (j,i) : 1ji }.Then G = {(1,2)} × X3 × X4… ×Xn.  = (1,2) ◦ (1,3) ◦ (4,4) ◦(2,5) ◦ (5,6) ◦ (3,7) 3 7 2 1 4 5 6 1 & 2 remain in the same cycle

Main Proof • Step 0: Young Tableaux • Step 1: Decomposing G • Step 2: Decomposing C • Step 3: Block diagonalizing R • Step 4: Diagonalizing Xi • Wrap-up G = {(1,2)}×X3×X4 …×Xn 3 7 2 1 4 5 6

if -1 º  ∈G if -1 º  =  1 1 otherwise otherwise 0 0 Decomposing CRegular Representation • Recall:C is defined • C, = • Definition: R() is defined • R(),= • Thus:C = ∑∈GR()

Main Proof • Step 0: Young Tableaux • Step 1: Decomposing G • Step 2: Decomposing C • Step 3: Block diagonalizing R • Step 4: Diagonalizing Xi • Wrap-up G = {(1,2)}×X3×X4 …×Xn 3 7 2 1 4 5 6 C = ∑∈G R()

Decomposing R“Fourier Transform”  change-of-basis matrix B block-diagonalizing R() YoungTableaux R() BR()B-1 1  1 1 1 1 ′ 1 1 1 1 ′′ IrreducibleRepresentations

Diagonalizing XiJucys-Murphy Elements • Let Xi = {(j,i): 1ji } • Let Y(Xi) = ∑∈XiBR()B-1,restricted to irreducible block  R() YoungTableaux BR()B-1  ′ ′′

Diagonalizing XiJucys-Murphy Elements • Let Xi = {(j,i): 1ji } • Let Y(Xi) = ∑∈XiBR()B-1,restricted to irreducible block  ∑∈XiR() YoungTableaux ∑∈XiBR()B-1  ′ Y(Xi) ′′

Diagonalizing XiJucys-Murphy Elements • Let Xi = {(j,i): 1ji } • Let Y(Xi) = ∑∈XiBR()B-1,restricted to irreducible block  • Fact: Y(Xi) is diagonal (and entries known) ∑∈XiR() YoungTableaux ∑∈XiBR()B-1  ′ Y(Xi) ′′

Diagonalizing XiJucys-Murphy Elements • Let Xi = {(j,i): 1ji } • Let Y(Xi) = ∑∈XiBR()B-1,restricted to irreducible block  • Fact: Y(Xi) is diagonal, and entry Y(Xi)tj,tjis c-r+1, where i is in row r and col c of tj. • Let Xi = {(j,i): 1ji } • Let Y(Xi) = ∑∈XiBR()B-1,restricted to irreducible block  • Fact: Y(Xi) is diagonal (and entries known) Content Value YoungTableau  t1 t2 t3 t1 Y(X4) = t2 1 2 3 1 2 4 1 3 4 SYT 4 3 2 t3 t1 t2 t3

Diagonalizing XiJucys-Murphy Elements • Let Xi = {(j,i): 1ji } • Let Y(Xi) = ∑∈XiBR()B-1,restricted to irreducible block  • Fact: Y(Xi) is diagonal, and entry Y(Xi)tj,tjis c-r+1, where i is in row r and col c of tj. • Let Xi = {(j,i): 1ji } • Let Y(Xi) = ∑∈XiBR()B,restricted to irreducible block  • Fact: Y(Xi) is diagonal (and entries known) Content Value YoungTableau  t1 t2 t3 t1 Y(X4) = t2 1 2 3 1 2 4 1 3 4 SYT 4 3 2 t3 t1 t2 t3

Wrap-upDiagonalizing C Y({(1,2)})∙Y(X3) … Y(Xn) is diagonal  Y({(1,2)}×X3×…×Xn) is diagonal  Y(G ) is diagonal  ∑∈GBR()B-1 is diagonal  BCB-1 is diagonal (Step 4) (homomorphism) (Step 1) (Step 3) (Step 2)

Wrap-upWhat are eigenvalues of C? Y(G )tj,tj 0  Y(Xi)tj,tj 0 i3 If i3 is in row c and col r of tj, then c-r+10  rank C = # SYT with 3  1 (and 2 below 1,...) Y(G )=Y({(1,2)}×X3×…×Xn) Content Value Content Values SYT tj c-r+1 1 2 3 4 1 0 No i3 cango here 0 1 2 0 2 0 -1 0 1 0 0  3  1 -2 1 0 0

Main Proof • Step 0: Young Tableaux • Step 1: Decomposing G • Step 2: Decomposing C • Step 3: Block diagonalizing R • Step 4: Diagonalizing Xi • Wrap-up G = {(1,2)}×X3×X4 …×Xn 3 7 2 1 4 5 6 C = ∑∈G R() QED

Understanding Matroid Theory: Bounds, Applications, and Complexity