Extended Formulations for Independence Polytopes of Regular Matroids

Extended Formulations for Independence Polytopes of Regular Matroids Volker Kaibel, Jon Lee, Matthias Walter, Stefan Weltge

Aussois 2015

extended formulation P ⇤ { x | x ⇤ ⇡ (y) , Cy  d } size # inequalities extension complexity xc (P) ⇤ smallest size of any EF for P

M ⇤ (E, I ) matroid independence polytope of M

P (M ) ⇤ conv{ (I ) 2 {0, 1}E | I 2 I }

R������ ’11 There are matroids M such that xc (P (M )) grows exponential.

graphic matroid of G ⇤ (V, E) F ✓ E independent () F is a forest M����� ’91 xc (Psp.forests (G))  O ( |V|· |E| ) Corollary xc (Pforests (G))  O ( |V|· |E| ) Proof: Pforests (G) ⇤ { x 2 RE+ | x  x0 , x0 2 Psp.forests (G) }

cographic matroid of G ⇤ (V, E) dual matroid M ? of graphic matroid bases complements of spanning forests P (M ?) ⇤ { x 2 RE+ | x 

x0 , x0 2 Psp.forests (G) }

Summary For (co)graphic matroids M we have xc (P (M ))  O ( |V|· |E| )  O ( |E| 2 ) .

matrix A 2 Fm⇥E linear matroid MF (A) set of columns independent () linearly independent over F regular matroid for every �eld F there exists some A such that M ⇤ MF (A) examples (co)graphic matroids are regular

R10 is regular 1 0 1 0 0 0

*.0 . R10 ⇤ MR ..0 .0 ,0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

1 1 0 0 1

1 1 1 0 0

0 1 1 1 0

0 0 1 1 1

1 0 +/ 0 // / 1/ 1-

but neither graphic nor cographic

Theorem For regular matroids M we have xc (P (M ))  O ( |E| 2 ) .

Seymour’s decomposition theorem regular matroid

non-leaf nodes are 1-, 2-, or 3-sums of its children

graphic, cographic or R10 number of nodes is linear in |E|

1-sum

I

M1 ⇤ MF2

M1

1

M2 ⇤ MF2

P (M1 xc (P (M1

A

!

1

1

I

M2 ⇤ MF2

I

A

M2 ) ⇤ P (M1 ) Corollary

I

B

B

!

P (M2 )

M2 ))  xc (P (M1 )) + xc (P (M2 ))

!

2-sum

I

M1 ⇤ MF2

M1

I 2 M1

2

a

A

r1

2

M2 ⇤ MF2

M2

spanI1

()

1

!

M2 ⇤ MF2

I r2

I

A

I

bT !

B

abT !

B

I ⇤ I1 [ I2 with I1 2 M1 \ {r1 }, I2 2 M2 \ {r2 } :

! ! I A O abT \ spanI2 ⇤ {O} O O I B

2-sum (cont.) spanI1

a

O

!

< spanI1

! ! I A O abT \ spanI2 ⇤ {O} O O I B I A O O

!

I1 [ {r1 } 2 M1

I

a r1

A

() or ()

or

a

O

!

< spanI2

O abT I B

I2 [ {r2 } 2 M2 1

bT

I r2

B

!

independent sets of a 2-sum M1 2 M2 ⇤ { I \ {r1 , r2 } | I 2 M1

M2 , I \ {r1 , r2 } , ; }

extended formulation P (M1

2

xc (P (M1

M2 ) ⇤ { ⇡ (x) | x 2 P (M1 ) P (M2 ) , xr1 + xr2 ⇤ 1}

Corollary 2 M 2 ))  xc (P (M 1 )) +xc (P (M 2 ))

3-sum

I

a a

A

0 1

cT

I M1

I A

abT

dcT

B

0 1 d d

bT

B

3 M2

⇤ { I \ {p1 , p2 , r1 , r2 , q1 , q2 } | I 2 M1 M2 , I \ {p1 , p2 } , ;, I \ {r1 , r2 } , ;, I \ {q1 , q2 } , ; }

xc (P (M1

Again M ))  xc (P (M1 )) + xc (P (M2 )) 3 2

xc (P (M )) 

X i

O

xc (P (Mi ))  O

✓⇣ X

⇤ O |E|

i 2

|Ei |

⌘ 2◆

✓X i

|Ei | 2

◆

open problems I xc (P (M )) bounded polynomially for F-linear matroids M for some �xed F? I explicit series of matroids M where xc (P (M )) grows exponential

I more methods to design extended formulations (unfortunately, we didn’t need to do this here)