Weighted Linear Matroid Parity

Satoru Iwata

(RIMS, Kyoto University)

Extensions of Matching and Matroids

• Matroid Parity [Matroid Matching]

Lawler (1971), Lovász (1978)

• Jump Systems Delta-Matroids

Bouchet & Cunningham (1995)

• Path-Matching Even Factors

Cunningham & Geelen (1997)

W. H. Cunningham: Matching, Matroids, and Extensions, MPB 91 (2002), 515-542.

Y. Kobayashi & K. Takazawa: Even Factors, Jump Systems, and Discrete Convexity, JCTB 99 (2009), 139-161.

Linear Matroid Parity

Luu },{A

Partitioned into Pairs V

U Line

Parity Set: Union of Lines

Find an Independent Parity Set of Maximum Size

:)(A Maximum Size of an Independent Parity Set

)(2

1:)( AA Maximum Number of Lines in a Base

Linear Matroid Parity

k

i

i KLKA

1 2

/dimdimmin)(

],[ span: ii VRAL

Min-Max Theorem Lovász (1978)

:,...,1 kVV Partition of into Parity Sets V

A span:K Linear Subspace of

Linear Matroid Parity

Algorithms

Lovász (1978) Polynomial

Gabow & Stallmann (1986)

Orlin & Vande Vate (1990)

Orlin (2008)

Cheung, Lau, Leung (2011)

Randomized

)( 4nr

)( 3nr

)( 3nr

)( 2nr

|| |,| VnUr

Applications of Linear Matroid Parity

• Unique Solvability of RCG Circuits

Milic (1974)

• Pinning Down Planar Skeleton Structures

Lovász (1980)

• Maximum Genus Cellular Embedding

Furst, Gross, McGeoch (1988)

• Maximum Number of Disjoint S -paths

Lovász (1980), Schrijver (2003)

Weighted Linear Matroid Parity

A

V

U

B

B

wBw

)(:)(

RLw :

Find a Parity Base of Minimum Weight

Minimum Weight Common Base of Two Linear Matroids

Minimum Weight Perfect Matching in General Graphs

Main Result and Tools

• Polynomial Matrix Formulation.

• Combinatorial Relaxation by Murota (1990).

• Augmenting Path Algorithm of

Gabow and Stallmann (1986).

A Combinatorial, Deterministic, Strongly Polynomial Algorithm

Running Time Bound: )( 3nr

Alternating Matrix

000

000

000

000

000

000

976

985

834

732

621

541

ttt

ttt

ttt

ttt

ttt

ttt

T

M Mvu

uvTMT),(

)sgn(:Pf

2)(Pfdet TT

)(TGGraph

))((2 rank TGT

:)( Maximum Matching Size

1e

2e

3e

4e5e

6e

7e8e9e

Matrix Formulation

A

1D

O

A O

O A

2D

2nD

0

0

D

Linear Matroid Parity

nAA ˆrank )(

:)(A Maximum Size of an Independent Parity Set

: Indeterminate

Geelen & I. (2005)

Polynomial Matrix Formulation

A

1D

O

)(ˆ A O

O A

2D

2nD

0

0)(

)(

w

w

D

Weighted Linear Matroid Parity

)(ˆPfdeg)()( AwAL

:)(A Minimum Weight of a Parity Base

: Indeterminate

Augmenting Path Algorithm

BA

:B

),( LFVGB

B

Base

}base :,\,|),{( vuBBVvBuvuF

**

*

**

Gabow & Stallmann (1986)

u

u v

Source Line

Augmenting Path Algorithm

BA

B

*

Augmenting Path Algorithm

bud

blossom

tip

Augmenting Path Algorithm

blossom

b

b

s

s

t

t

),,( tsbz

* * 0

* * ?

transform

z

Augmenting Path Algorithm

k

i

i KLKA

1 2

/dimdimmin)(

Primal-Dual Algorithm

RVp :

R:q

},...,,{ 21 kHHH

:B ),( LFVGB Base

Laminar Family of Blossoms

Dual Variables

(DF1)

(DF2)

(DF3)

Luuwupup },{ ),()()(

.),( ,)()( FvuQupvp uv

).,,( ),()( tsbzspzp

H

uv HqvuHQ )(),,(:

1

1

0

H

Initial Steps

(DF1)

(DF2)

(DF3)

Luuwupup },{ ),()()(

.),( ,)()( FvuQupvp uv

).,,( ),()( tsbzspzp

.: Set so that (DF1) is satisfied. p

Find a base that minimizes B

Bu

upBp ).()(

Extract the set of tight edges in *F .F

Search for an augmenting path in ).,( **LFVGB

Dual Update

(DF1)

(DF2)

(DF3)

Luuwupup },{ ),()()(

.),( ,)()( FvuQupvp uv

).,,( ),()( tsbzspzp

Blossoms

Blossoms

bud (proper) tip

(DF1)

(DF2)

(DF3)

Luuwupup },{ ),()()(

.),( ,)()( FvuQupvp uv

).,,( ),()( tsbzspzp

t

v

u

),,( tuv

tip H

1),,( vuH

Augmentation

(DF1)

(DF2)

(DF3)

Luuwupup },{ ),()()(

.),( ,)()( FvuQupvp uv

).,,( ),()( tsbzspzp

Augmentation

(DF1)

(DF2)

(DF3)

Luuwupup },{ ),()()(

.),( ,)()( FvuQupvp uv

).,,( ),()( tsbzspzp

Optimality

A

1D

O

O

O A

2D

2nD

)(ˆPfdeg A Maximum Weight of a Perfect Matching

)(ˆ A

0

)(w

B UBV \

Optimality

)(ˆPfdeg A Maximum Weight of a Perfect Matching

)( 0)(

)( )()()( subject to

)()( Minimize

,

SSz

EeewSzuy

Szvy

SeSeu

Wv S

Dual Objective Value

)( )(:)(

)( )(:)(

Uuupuy

Vuupuy

VUW

Dual Linear Program

Optimality

v

v

t

ss

t

b

c

b

v

vt

t

s

s

c

H

)(Hq

)(Hqbs

**

**

s t t

**

*

*

v v c

*

**

Optimality

b

v

v

t

t

s

sc

)(Hq

c s s

v

v

tt

b

H

)(Hq

bs

*

*

**

s t t

**

*

v v

**

**

*

*

Optimality

)(ˆPfdeg A Maximum Weight of a Perfect Matching

BVv BV

Wv S

wvp

Szvy

\ \

)()(

)()(

)(ˆPfdeg)()( AwAL

B

wA

)()(

Good Characterization

There exist nonsingular matrices Y and Z such that

)(ˆPfdeg)()( AwAL

ZO

OY

DA

AO

ZO

OYA

TT

T

:)(ˆ

)(ˆPfdeg A Maximum Weight of a Perfect Matching in )).(ˆ( AG

satisfies

Combinatorial Relaxation Method by Murota (1990)

Questions

• Polyhedral Description ?

Gyula Pap’s talk

• Applications to Approximate Algorithms

5/3-Approx. Algorithm for Steiner Tree

Prömel & Steger (1998)

How about Metric TSP ?