weighted linear matroid parity - university of toronto · 2012. 6. 28. · satoru iwata (rims,...
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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 ?
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