1. An Algebraic Algorithm for Weighted Linear Matroid Intersection
Nick Harvey

2. What is Matroid Intersection?
Network Flow
Submodular Flow
Minimum Spanning Tree
Submodular Function Minimization
Bipartite Matching
Matroid Intersection
Spanning Tree Packing
Matroid Greedy Algorithm
Non-Bip. Matching
Matroid Matching
Minimum Arboresence
A central problem in discrete optimization

3. Matroid intersection has many uses
Edge connectivity [Gabow ’91]
Uniprocessor scheduling [Stallman ’91]
Survivable network design [Balakrishnan-Magnanti-Mirchandani ’98]
k-Delivery TSP [Chalasani-Motwani ’99]
Constrained MST [Hassin-Levin ’04]
Multicast Network Codes [Harvey-Karger-Murota ’05]
Bounded-Degree MST [Goemans ’06]

4. An Example Problem Does this matrix have full rank?2 x1 5 x2 x3 x4 9 4 3 8 x5 x6 1
Does this matrix have full rank?
Can one replace xi’s with numbers s.t. rank is maximized?
Solvable via Matroid Intersection [Murota ’93]

5. What is a Matroid? a b c 1
rank = 2
Linearly independent sets of columns I = { ∅, {a}, {b}, {c}, {a,b}, {b,c}, {a,c} }

6. What is a Matroid? a b c
rank =
Linearly independent sets of columns I = { ∅, {a}, {b}, {c}, {a,b}, {b,c}, {a,c} }
What is rank? Is a ∈ span({b,c})?
What properties of linear independence are needed to answer these questions?

7. Independence Properties
Linearly independent sets of columns I = { ∅, {a}, {b}, {c}, {a,b}, {b,c}, {a,c} }
Properties:
∅ ∈ I
If A ⊆ B ∈ I then A ∈ I
If A, B ∈ I and |A| < |B| then ∃ b ∈ B such that A+b ∈ I

8. Definition A matroid is a pair (S,I) with I ⊆ 2S satisfying the axioms
If A ⊆ B ∈ I then A ∈ I
If A, B ∈ I and |A| < |B| then ∃ b ∈ B such that A+b ∈ I

9. Matroid Problems Given M=(S,I) Find A ∈ I maximizing |A|
Find A ∈ I maximizing wt(A)
Given M1=(S,I1) and M2=(S,I2)
Find A ∈ I1 ⋂ I2 maximizing |A|
Find A ∈ I1 ⋂ I2 maximizing wt(A)
Given M1=(S,I1), M2=(S,I2), M3=(S,I3)
Find A ∈ I1 ⋂ I2 ⋂ I3 maximizing |A|
Matroid Greedy Algorithm
Matroid Intersection
Weighted Matroid Intersection
NP-hard!

10. Two types of algorithms
Oracle Algorithms Access matroid by oracle queries “Is A ∈ I?”
Linear Matroid Algorithms Input is a matrix M s.t. I is the set of linearly indep. columns of M

11. Weighted Linear Matroid Intersection
Find set of columns S such that AS and BS are both linearly independent and wt(S) is maximized AS 1
A =
weights: 1 9 8 5 BS 7 6 3 1 9 2 4 8 5
B =

12. Oracle Algorithm History
Access matroid by oracle queries “Is A ∈ I?”
Edmonds ’65-’70
Augmenting Paths
O(nr2)
Lawler ’75, Edmonds ’79
Cunningham ’86
“Blocking Flows”
O(nr1.5)
Shigeno-Iwata ’95
Dual Approximation
≈O(nr1.5 log(rW))
n = |S| r = rank(S) W = max weight
Grey row = unweighted algorithm

13. Linear Matroid Algorithm History
Given a matrix M s.t. I is set of indep. columns
Cunningham ’86
“Blocking Flows”
O(nr2 log r)
Gabow-Xu ’89-’96
“Blocking Flows” & Fast Matrix Multiplication
O(nr1.77 log W)
Harvey ’06
Fast Matrix Multiplication
O(nr-1)
This Paper
Fast Linear System Solving for Polynomial Matrices
O(nr-1 W1+ε) † †
matrix M has size n x r W = max weight
Grey row = unweighted algorithm
† Randomized, and assumes matroids can be represented over same field

14. Anatomy of a Weighted Optimization Algorithm
e.g: Primal-Dual Method
repeat
adjust dual
find best primal using items allowed by dual
until primal is optimal
Can any fast primal alg be used here?
Want primal alg to work incrementally
Sadly, algebraic method is not incremental

15. Polynomial Matrices
Used in PRAM algs for matching [KUW ’86], [MVV ’87] a b c d e f g h 2 a e 2
x1y2
x2y2
x3y5
x4y0
x5y1
x6y0
x7y1
x8y1 5 b f 1 c g 1 1 d h

16. Max Weight of a Matching
Polynomial Matrices
Used in PRAM algs for matching [KUW ’86], [MVV ’87] a b c d e f g h 2 a e 2
3y2
1y2
6y5
2y0
4y1
3y0
1y1 5 b f 1 c g 1 1 d h
Determinant = 12y4 + 72y3
Max Weight of a Matching

17. Polynomial Matrices
Matrix M, size n x n, each entry a degree W polynomial
Computing determinant:
On PRAM: O(log2(nW)) time
Sequentially: O(n5 W2) time [naive alg]
 Can compute max weight of a perfect matching in O(n W1+ε) time
O(n+1 W) time [interpolation]
O(n W1+ε) time [Storjohann ’03]
 Can compute max weight of a matroid intersection in O(nr-1 W1+ε) time

18. Weighted Linear Matroid Intersection1
A =
weights: 1 9 8 5 7 6 3 1 9 2 4 8 5
B =

19. Weighted Linear Matroid Intersection1 9 8 5

20. Weighted Linear Matroid IntersectionBT A 1 9 8 5 1 5 9

21. Weighted Linear Matroid IntersectionY BT y1 y0 A y9 y8 y5 y1 y5 y9

22. Weighted Linear Matroid Intersection
Claim 1: max weight of intersection is max exponent of y in det( A Y BT ) Y BT y1 y0 A y9 y8 y5 y1 y5 y9

23. Weighted Linear Matroid Intersection
Claim 2: computing det( A Y BT ) takes time O(nr-1 W1+ε) Y BT y1 y0 A y9 y8 y5 y1 y5 y9

24. Using Storjohann for Optimization Problems
Can compute weight(OPT). How to find OPT?
Sankowski found a method for bipartite matching [Sankowski ’06]
Define a family of perturbed instances. Compute weight(OPT) for all perturbed instances.
Using these weights, compute optimum dual for original instance.
Given optimum dual, compute OPT.
 Find optimal bip matching in O(n W1+ε) time

25. Extending Sankowski’s Method
Fast algorithm (using Storjohann)
[Sankowski ’06]
Network Flow
Submodular Flow
Minimum Spanning Tree
Submodular Function Minimization
Bipartite Matching
Matroid Intersection
Spanning Tree Packing
Matroid Greedy Algorithm
Non-Bip. Matching
Matroid Matching
Minimum Arboresence

26. Extending Sankowski’s Method
Fast algorithm (using Storjohann)
log W
O(nr-1 W1+ε)
[This Paper]
[Sankowski ’06]
Matroid Intersection
Bipartite Matching
Non-Bip. Matching
Network Flow
Submodular Flow
O(n W1+ε)
log W
Minimum Spanning Tree
Submodular Function Minimization
Spanning Tree Packing
Matroid Greedy Algorithm
Matroid Matching
Minimum Arboresence