Query Lower Bounds for Matroids via Group Representations Nick Harvey

Query Lower Boundsfor 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 PropertyLet 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 PropertyLet A and B∈B

a∈A\B, b∈B\A s.t. B+a-b∈B

b1⊕b2

b1⊕b2b1⊕b2

s tb1 b2

Multicast Network Coding

• Goal: Multicast sourcesinks 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

a b

a+b a+b

a+b

Reversibility of Network Codes

Sources

Sinks

a b

a b

a+b a+b

a+b

sa sb

tbta

G

Reversibility of Network Codes

• Flow: G feasible Grev feasible

• Coding: feasible G s.t. Grev not feasible[Dougherty, Zeger ’06]

Sources

Sinks

a b

a b

a+b a+b

a+b

sasb

tbta

Grev

Constructed fromFano and

non-Fano matroids

Reversibility of Network Codes [Dougherty, Zeger ’06]

Discrete Optimization Problems

MatroidIntersectionMatroidIntersection

BipartiteMatchingBipartiteMatching

Non-Bip.MatchingNon-Bip.Matching

NetworkFlowNetworkFlow

SubmodularFunctionMinimization

SubmodularFunctionMinimization

SubmodularFlowSubmodularFlow

MatroidMatchingMatroidMatching

MinimumSpanningTree

MinimumSpanningTree

MatroidGreedyAlgorithm

MatroidGreedyAlgorithm

Spanning TreePackingSpanning TreePacking

Min-costArboresenceMin-costArboresence

MatroidIntersectionMatroidIntersection

• Matroid IntersectionGiven matroids M1=(S,B 1) and M2=(S,B 2),

is B 1 ⋂ B 2 = ∅?

n = # elementsr = rank

Unweighted

Edmonds/Lawler ’68-’75

O(nr2) oracle

Cunningham ’86 O(nr1.5) oracle

Gabow-Xu ’89-’96 O(nr1.62)

Harvey ’06 O(nr1.38)

Matroid Intersection Algorithms

Linear Matroids

W = max weight

Weighted

Lawler/Edmonds ~’75 O(nr2) oracle

Shigeno-Iwata ’95 O(nr1.5 log(rW)) oracle

Gabow-Xu ’89-’96 O(nr1.77 log(rW))Harvey ’07 O(nr1.38 W)

Linear Matroids

n = # elementsr = rank

Unweighted

Edmonds/Lawler ’68-’75

O(nr2) oracle

Cunningham ’86 O(nr1.5) oracle

Gabow-Xu ’89-’96 O(nr1.62)

Harvey ’06 O(nr1.38)

Linear Matroids

Weighted

Are these algorithms optimal? Are these algorithms optimal?

W = max weightLawler/Edmonds ~’75 O(nr2) oracle

Shigeno-Iwata ’95 O(nr1.5 log(rW)) oracle

Gabow-Xu ’89-’96 O(nr1.77 log(rW))Harvey ’07 O(nr1.38 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, butbest-known result is (n).

• A super-linear lower bound for anynatural 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

BOut

InData

Black Box

AlgorithmQueries

Query Model for Matroids(Independence Oracle)

Out

InMatroid

(S,B ) Algorithm

“Yes” if B∈B s.t. T⊆B “No” otherwise

T⊆S

• Example: if B = { spanning trees of graph G },then query asks if T is an acyclic subgraph of G

Algorithms O(nr2) queries[Lawler ’75]

Matroid Intersection Complexity

Are (nr2) queries necessary and sufficientto solve matroid intersection?D. J. A. Welsh, “Matroid Theory”, 1976.


Matroid Intersection Complexity O(nr1.5) queries

[Cunningham ’86]

Can one prove any non-trivial lower boundon # queries to solve matroid intersection?

Are (nr2) queries necessary and sufficientto solve matroid intersection?D. J. A. Welsh, “Matroid Theory”, 1976.


# qu

erie

s

Rank r0 nn/2

O(nr1.5) queries[Cunningham ’86]

Trivial LB

Cunningham UB


n

2n

0

Algorithms O(nr1.5) queries[Cunningham ’86]

# qu

erie

s

Rank r0 nn/2

Trivial LB

Cunningham UB

Via Dual Matroids


n

2n

0


# qu

erie

s

Rank r0 nn/2

Trivial LB

Cunningham UB

Via Dual Matroids

Optimal UB?


n

2n

0



# qu

erie

s

Rank r0 n

n

2n

0n/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:

(log2 3) n - o(n) > 1.58n

Lower Bound[Harvey ’08]

Hard Instances CommunicationComplexity

RankComputation

Alice Bob

M =0 1 0 1

1 0 1 0

0 1 0 1

1 0 1 1

Hard InstancesBipartite Matching in Almost-2-Regular Graphs

1

3

2

4

• Is there a perfect matching?

Four verticeshave degree 1

Hard InstancesBipartite Matching in Almost-2-Regular Graphs

1

32

4

• Is there a perfect matching?• No: if path from 1 to 2• Yes: otherwise

Four verticeshave degree 1

• Alice given ∈ Sn and Bob given ∈ Sn

Elements 1 and 2are not in the same cycle

Permutation

Permutation -1

1

2

3

4

5

6

1’

2’

3’

4’

5’

6’

• In-Same-Cycle Problem:Are elements 1 and 2 in the same cycleof composition -1

º ?

Permutation Formulation

LB from Rank Computation• Let C be a matrix with rows and columns

indexed by permutations in Sn

C, =

where G = { : 1 & 2 are in the same cycle of }• C is adjacency matrix of Cayley graph

for Sn with generators G

10

if -1 º ∈ Gotherwise

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

1

3

56

24

7

11

11

11

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

12…k>0# boxes = n = ∑i i

Place numbers {1,..,n} in boxesRows increase →

Columns increase ↓

11 22 66 8833 55 9944

111177 1010

(and some other minor conditions)

Main Proof

• Step 0: Young Tableaux





• Wrap-up

1

3

56

24

7

11

11

11

Decomposing Sn

• Claim: ∈Sn = ◦ ( n, -1(n) ), where ∈Sn-1

• Example: Let ∈S6 be

• Then ◦ ( 7, 3 ) ∈ S7 is

1

3

56

2

4

24

7

1

3

56

Decomposing Sn

• Restatement: Let Xi = { (j,i) : 1ji }.

Then Sn = X2 × … × Xn.

13

56

2

4

7

(2,2) = ◦ (1,3) ◦ (2,4) ◦ (3,5) ◦ (5,6)◦ (3,7)

• Claim: ∈Sn = ◦ ( n, -1(n) ), where ∈Sn-1

DecomposingG• Let G = { : 1 & 2 are in the same cycle }

• Claim: Let Xi = { (j,i) : 1ji }.

Then G = {(1,2)} × X3 × X4 … × Xn.

1 & 2 remain in the same cycle

13

56

24

7

(1,2) = ◦ (1,3) ◦ (4,4)◦ (2,5) ◦ (5,6)◦ (3,7)

Main Proof• Step 0: Young Tableaux





• Wrap-up

1

3

56

24

7

11

11

11

G = { (1,2) } × X3 × X4 …× Xn

Decomposing CRegular Representation

• Recall: C is defined

C, =

Definition: R() is defined

R(), =

• Thus: C = ∑∈G R()

1

0if -1 º ∈ Gotherwise

1

0

if -1 º = otherwise

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1

1

1

1

1

1






• Wrap-up

1

3

56

24

7

11

11

11

G = { (1,2) } × X3 × X4 …× Xn

C = ∑∈G R()

Decomposing R“Fourier Transform”

1

1

1

1

1

1

1

1

1

R() BR()B-1

IrreducibleRepresentations

YoungTableaux

change-of-basis matrix B block-diagonalizing R()

′

′′






• Wrap-up

1

3

56

24

7

11

11

11

G = { (1,2) } × X3 × X4 …× Xn

C = ∑∈G R()

• Let Xi = { (j,i) : 1ji }

• Let Y(Xi) = ∑∈Xi BR()B-1,

restricted to irreducible block

1

1

1

1

1

1

R()

BR()B-1 YoungTableaux

′

′′

Diagonalizing XiJucys-Murphy Elements

• Let Xi = { (j,i) : 1ji }


restricted to irreducible block

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

∑∈Xi R() ∑∈Xi


′

′′

Y(Xi)



• Let Xi = { (j,i) : 1ji }


restricted to irreducible block • Fact: Y(Xi) is diagonal (and entries known)

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

∑∈Xi R() ∑∈Xi


′

′′

Y(Xi)


• Let Xi = { (j,i) : 1ji }



Y(X4) =

YoungTableau

SYT

t1

t2 t3

t2 t3t1t2t3

t1

• Let Xi = { (j,i) : 1ji }


restricted to irreducible block • Fact: Y(Xi) is diagonal, and entry Y(Xi)tj,tj

is c-r+1, where i is in row r and col c of tj.Content Value


• Let Xi = { (j,i) : 1ji }

• Let Y(Xi) = ∑∈Xi BR()B,


03

3Y(X4) =

YoungTableau

SYT

t1

t2 t3

t2 t3t1t2t3

t1

• Let Xi = { (j,i) : 1ji }


restricted to irreducible block • Fact: Y(Xi) is diagonal, and entry Y(Xi)tj,tj

is c-r+1, where i is in row r and col c of tj.Content Value






• Wrap-up

1

3

56

24

7

11

11

11

G = { (1,2) } × X3 × X4 …× Xn

C = ∑∈G R()

Wrap-upDiagonalizing C

Y({(1,2)}) ∙ Y(X3) … Y(Xn) is diagonal

Y({(1,2)}×X3×…×Xn) 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 )tj,tj 0

Y(Xi)tj,tj 0 i3

If i3 is in row c and col r of tj, then c-r+10

rank C = # SYT with 3 1 (and 2 below 1,...)

Wrap-upWhat are eigenvalues of C?

1 2 3 4

0 1 2

-1

-2

0 1

c-r+1

1 0

0

0

0

Content Value

Content Values

1

0

0

0

2

SYT tjNo i3 can

go here

3 1

Y(G )=Y({(1,2)}×X3×…×Xn)






• Wrap-up

1

3

56

24

7

11

11

11

G = { (1,2) } × X3 × X4 …× Xn

C = ∑∈G R()

QED

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Query Lower Bounds for Matroids via Group Representations Nick Harvey