Algebraic%20Structures%20and%20Algorithms%20for%20Matching%20and%20Matroid%20Problems

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Algebraic Structures and Algorithmsfor Matching
and Matroid Problems

• Nick Harvey

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Perfect Matching

• Given graph G(V,E)
• Perfect matching is M?E such that each vertex
  incident with exactlyone edge in M

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Matching History
Dense Graphs mn2
Edmonds 65 Blossom Alg. O(n2 m)
Micali-Vazirani 80-90 Blossoms Blocking Flow O(vn m)
Mucha-Sankowski 04 Fast Matrix Multiplication O(n?)
O(n4)
O(n2.5)
O(n2.38)
(n vertices, m edges)

• All are non-trivial!
• Mucha 05 Our algorithm is quite complicated
  and heavily relies on graph theoretic results and
  techniques. It would be nice to have a strictly
  algebraic, and possibly simpler, matching
  algorithm.

• All are non-trivial
• Mucha-Sankowski uses sophisticateddynamic
  connectivity data structures (Holm et al. 01)to
  maintain canonical partition (Cheriyan 97)

(?lt2.38 is exponent for matrix multiplication)
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Matching Algorithms
Dense Graphs mn2
Edmonds 65 Blossom Alg. O(n4)
Micali-Vazirani 80-90 Blossoms Blocking Flow O(n2.5)
Mucha-Sankowski 04 Fast Matrix Multiplication O(n2.38)
This Work Purely Algebraic,Clean Divide-and-Conquer,Fast Matrix Multiplication O(n2.38)

• Conceptually simple
• Implemented in 200 lines of MATLAB

Randomized, but Las Vegas via Cheriyan 97
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(Linear) Matroid Intersection

• Find largest set of columns S such thatAS and BS
  are both linearly independent

AS
1 1 1 0 1 0 1 0
1 0 0 1 1 1 0 1
0 0 1 1 0 1 1 0
0 1 0 1 1 0 1 1
A
BS
7 6 6 3 7 1 9 0
9 2 9 2 4 9 6 1
3 3 4 7 0 4 3 3
1 4 8 4 2 2 5 3
B
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Matroid Intersection

• Find largest set of columns S such thatAS and BS
  are both linearly independent
• Why?
• Many problems are a special case
• Bipartite matching
• Arboresence (directed spanning tree),
• Powerful tool
• Bounded-Degree Sp. Tree Goemans 06

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Matroid Intersection History
Edmonds 65-70, Lawler 75 Augmenting Paths O(mn2) oracle
Cunningham 86 Blocking Flows O(mn2 log n)
Gabow-Xu 89-96 Blocking Flows Fast Matrix Mult. O(mn1.62)
(for matrices of size n x m)
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Matroid Intersection History
Edmonds 65-70, Lawler 75 Augmenting Paths O(mn2) oracle
Cunningham 86 Blocking Flows O(mn2 log n)
Gabow-Xu 89-96 Blocking Flows Fast Matrix Mult. O(mn1.62)
This paper Purely Algebraic,Fast Matrix Mult. O(mn?-1)




O(mn1.38)

• Essentially optimal
• Best known alg to compute rank takes O(mn?-1)
• Computing rank reduces to matroid intersection

Assumes matroids are linear
Randomized, and assumes matroids can be
represented over same field
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Generic Matching Algorithm

• For each e ? E
• If e is contained in a perfect matching
• Add e to solution
• Delete endpoints of e
• Else
• Delete edge e

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Generic Matching Algorithm
Key step

• For each e ? E
• If e is contained in a perfect matching
• Add e to solution
• Delete endpoints of e
• Else
• Delete edge e

• How can we test this?
• Randomization and linear algebra play key role

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Matching Outline

• Implementing Generic Algorithm
• Tutte Matrix Properties
• Rabin-Vazirani Algorithm
• Rank-1 Updates
• Rabin-Vazirani with Rank-1 Updates
• Our Recursive Algorithm (overview)
• Our Recursive Algorithm (fast updates)

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Matching Tutte Matrix

• Let G(V,E) be a graph
• Define variable xu,v ?u,v?E
• Define a skew-symmetric matrix T s.t.

xu,v
if u,v?E
Tu,v
0
otherwise
0 -xa,b -xa,c
xa,b 0 -xb,c
xa,c xb,c 0
a
c
b
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Properties of Tutte Matrix

• Lemma Tutte47 G has a perfect matchingiff T
  is non-singular.
• Lemma RV89 G V\u,v has a perfect
  matching iff (T-1)u,v ? 0.

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Properties of Tutte Matrix

• Lemma Tutte47 G has a perfect matchingiff T
  is non-singular.
• Lemma RV89 G V\u,v has a perfect
  matching iff (T-1)u,v ? 0.

u
(T-1)u,v ? 0
v
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Properties of Tutte Matrix

• Lemma Tutte47 G has a perfect matchingiff T
  is non-singular.
• Lemma RV89 G V\u,v has a perfect
  matching iff (T-1)u,v ? 0.

G V\u,v has perfectmatching
(T-1)u,v ? 0
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Properties of Tutte Matrix

• Lemma Tutte47 G has a perfect matchingiff T
  is non-singular.
• Lemma RV89 G V\u,v has a perfect
  matching iff (T-1)u,v ? 0.

G V\u,v has perfectmatching
(T-1)u,v ? 0
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Properties of Tutte Matrix

• Lemma Tutte47 G has a perfect matchingiff T
  is non-singular.
• Lemma RV89 G V\u,v has a perfect
  matching iff (T-1)u,v ? 0.

u
G V\u,v has perfectmatching
(T-1)u,v ? 0
v
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Properties of Tutte Matrix

• Lemma Tutte47 G has a perfect matchingiff T
  is non-singular.
• Lemma RV89 G V\u,v has a perfect
  matching iff (T-1)u,v ? 0.
• Computing T-1 very slow Contains variables!
• Lemma Lovász79 These results hold w.h.p. if
  we randomly choose values for xu,vs.

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Matching Algorithm
Rabin-Vazirani 89

• Compute T-1
• For each u,v ? E
• If T-1u,v ? 0
• Add u,v to matching
• Recurse on G V\u,v

(Takes O(n?) time)

• Total time O(n?1) time
• Natural question Can we recomputeT-1 quickly in
  each iteration?

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Matching Outline

• Tutte Matrix Properties
• Rabin-Vazirani Algorithm
• Rank-1 Updates
• Rabin-Vazirani with Rank-1 Updates
• Our Recursive Algorithm (overview)
• Our Recursive Algorithm (fast updates)

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Rank-1 Updates

• Let T be a n x n matrix
• T uvT is called a rank-1 update of T

1 1 1 1
1 1 1
1 1

1
1
2
3

n


1 2 3 n
T
u
vT
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Rank-1 Updates

• Let T be a n x n matrix
• T uvT is called a rank-1 update of T
• Computing rank-1 update takes O(n2) time

1 1 1 1
1 1 1
1 1

1
1 2 3 n
2 4 6 2n
3 6 9 3n

n 2n 3n n2

T
uvT
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Rank-1 Updates

• Let T be a n x n matrix
• T uvT is called a rank-1 update of T
• Claim If you modify one entry of T,then T-1 is
  modified by a rank-1 update.

T
T-1
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Rank-1 Updates

• Let T be a n x n matrix
• T uvT is called a rank-1 update of T
• Claim If you modify one entry of T,then T-1 is
  modified by a rank-1 update.
• Claim If you delete O(1) rows and columns of T
  then T-1 is affected by O(1) rank-1 updates.
• ? T-1 can be updated in O(n2) time

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Matching Outline

• Tutte Matrix Properties
• Rabin-Vazirani Algorithm
• Rank-1 Updates
• Rabin-Vazirani with Rank-1 Updates
• Our Recursive Algorithm (overview)
• Our Recursive Algorithm (fast updates)

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RV Algorithm Rank-1 Updates
Mucha-Sankowski 04

• Compute T-1
• For each u,v ? E
• If T-1u,v ? 0
• Add u,v to matching
• Update T-1 (via rank-1 updates)
• Recurse on G V\u,v

• Total runtime O(n3) time
• Can updates use Fast Matrix Multiplication?
• Bipartite graphs Yes
• Non-bip. graphs Yes, but non-trivial

MS04
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RV Algorithm Rank-1 Updates
Mucha-Sankowski 04

• Compute T-1
• For each u,v ? E
• If T-1u,v ? 0
• Add u,v to matching
• Update T-1 (via rank-1 updates)
• Recurse on G V\u,v

• Total runtime O(n3) time
• Can updates use Fast Matrix Multiplication?

New approach An unusual recursion!
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Matching Outline

• Tutte Matrix Properties
• Rabin-Vazirani Algorithm
• Rank-1 Updates
• Rabin-Vazirani with Rank-1 Updates
• Our Recursive Algorithm (overview)
• Our Recursive Algorithm (fast updates)

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New Recursive Approach
(Here c4 parts)

• Partition into c parts V1,,Vc
  (arbitrarily)
• For each pair of parts Va,Vb (arbitrary
  order)
• Recurse on GVa ? Vb

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New Recursive Approach

• Partition into c parts V1,,Vc
  (arbitrarily)
• For each pair of parts Va,Vb (arbitrary
  order)
• Recurse on GVa ? Vb

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New Recursive Approach

• Partition into c parts V1,,Vc
  (arbitrarily)
• For each pair of parts Va,Vb (arbitrary
  order)
• Recurse on GVa ? Vb

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New Recursive Approach

• Partition into c parts V1,,Vc
  (arbitrarily)
• For each pair of parts Va,Vb (arbitrary
  order)
• Recurse on GVa ? Vb
• Base case 2 vertices

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New Recursive Approach

• Partition into c parts V1,,Vc
  (arbitrarily)
• For each pair of parts Va,Vb (arbitrary
  order)
• Recurse on GVa ? Vb
• Base case 2 vertices u,v
• If T-1u,v ? 0, add u,v to matching, update T-1

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Recursion F.A.Q.

• Why not just recurse on G Va ?
• Edges between parts would be missed
• Claim Our recursion examines every pair of
  vertices ? examines every edge
• Why does algorithm work?
• It implements Rabin-Vazirani Algorithm!
• Isnt this horribly slow?
• No well see recurrence next

• Partition into c parts V1,,Vc
  (arbitrarily)
• For each pair of parts Va,Vb (arbitrary
  order)
• Recurse on GVa ? Vb
• Base case 2 vertices u,v
• If T-1u,v ? 0, add u,v to matching, update T-1

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Final Matching Algorithm

• If base case with 2 vertices u,v
• If T-1u,v ? 0, add u,v to matching
• Else
• Partition into c parts V1,,Vc
• For each pair Va,Vb
• Recurse on GVa ? Vb
• Apply updates to current subproblem

s size of subproblem
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Final Matching Algorithm

• If base case with 2 vertices u,v
• If T-1u,v ? 0, add u,v to matching
• Else
• Partition into c parts V1,,Vc
• For each pair Va,Vb
• Recurse on GVa ? Vb
• Apply updates to current subproblem

s size of subproblem
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Final Matching Algorithm

• If base case with 2 vertices u,v
• If T-1u,v ? 0, add u,v to matching
• Else
• Partition into c parts V1,,Vc
• For each pair Va,Vb
• Recurse on GVa ? Vb
• Apply updates to current subproblem

Assume O(s?) time
s size of subproblem
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Time Analysis

• Basic Divide-and-Conquer
• If then
• Since ,just
  choose c large enough!

c 13 is large enough if ? 2.38
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Handling Updates
T
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Handling Updates
u
v
u
v
u
u
v
v
T
T-1

• Delete vertices u and v

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Handling Updates (Naively)
u
v
u
v
u
u
v
v
T
T-1

• Delete vertices u and v ? clear rows / columns
• Causes rank-1 updates to T-1
• Algorithm still takes ?(n3) time

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Matching Outline

• Tutte Matrix Properties
• Rabin-Vazirani Algorithm
• Rank-1 Updates
• Rabin-Vazirani with Rank-1 Updates
• Our Recursive Algorithm (overview)
• Our Recursive Algorithm (fast updates)

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Just-in-time Updates
u
v
u
v
T-1

• Dont update entire matrix!
• Just update parent in recursion tree
• Updates outside of parent are postponed

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Postponed Updates
u
v
u
v
T-1

• Accumulate batches of updates
• Claim New updates can be applied with matrix
  multiplication and inversion

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Final Matching Algorithm

• If base case with 2 vertices u,v
• If T-1u,v ? 0, add u,v to matching
• Else
• Partition into c parts V1,,Vc
• For each pair Va,Vb
• Recurse on GVa ? Vb
• Apply updates to current subproblem

Invariant Before / after child
subproblem,parents submatrix is completely
updated
? in every base case, T-1u,v is up-to-date!
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Graph G
children
Just apply postponed updates
All edges chosen in base cases
Only take edges that can be extended to a perfect
matching in the whole graph. This decision is
possible because invariant ensures that T-1u,v is
up-to-date.
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Matching Summary

• Can compute a perfect matching inO(n?)
  O(n2.38) time
• Algorithm uses only simple randomization, linear
  algebra and divide-and-conquer
• Easy to implement
• Extensions for (by existing techniques)
• maximum matchings
• Las Vegas

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More Extensions

• Same philosophy applies to manymatching-like
  problems
• Design matrix capturing problem
• Design algorithm to test if element in OPT
• Design method to efficiently do updates

• Design algebraic structure capturing problem

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More Extensions

• Same philosophy applies to manymatching-like
  problems
• Design matrix capturing problem

• Design algebraic structure capturing problem

Bipartite Matroid Matching Basic
Path-Matching O(n?) algorithm
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Open Problems

• Fast, deterministic methods to choosevalues for
  indeterminates?
• Implications for Matching in NC
• Scaling algorithms for weighted problems?
• O(n2) algorithms for matching bycompletely
  different techniques?
• e.g., via Karger-Levine randomized max flow