Nonlinear methods in discrete optimization

László Lovász

Eötvös Loránd University, Budapest

lovasz@cs.elte.hu

planar graph

Fáry-Wagner

Every simple planar graph can be drawnin the plane with straight edges

Exercise 1: Prove this.

Rubber bands and planarity

Every 3-connected planar graph can be drawn with straight edges and convex faces.

Tutte (1963)

Rubber bands and planarity

outer face fixed toconvex polygon

edges replaced byrubber bands

2( )i jij E

u uÎ

= -åEEnergy:

Equilibrium:( )

1i j

j N ii

u ud

G 3-connected planar

rubber band embedding is planar

Exercise 2. (a) Let L be a line intersecting the outer polygon P, and let U

be the set of nodes of G that fall on a given (open) side of L. Then U

induces a connected subgraph of G.

(b) There cannot exists a node and a line such that the node and all its

neighbors fall on this line.

(c) Let ab be an edge that is not an edge of P, and let F and F’ be the two

faces incident with ab. Prove that all the other nodes of F fall on one side

of the line through this edge, and all the other nodes of F’ are mapped on

the other side.

(d) Prove the theorem above.

Tutte

Discrete Riemann Mapping Theorem

Coin representation Koebe (1936)

Every planar graph can be represented by touching circles

Can this be obtained from a rubber band representation?

Tutte representation optimal circles

i j i jx x r r- = +

Want:

2( | |)i j i jij E

r r x xÎ

+ - -åMinimize:

( | |) 0i j i jj

ij E

r r x x

Î

+ - - =åOptimum satisfies i:

Rubber bands and strengths

rubber bands havestrengths cij > 0

2( )ij i jij E

c u uÎ

= -åEEnergy:

Equilibrium:( )

( )

ij jj N i

iij

j N i

c u

uc

( ) 0ij i jij E

c u uÎ

- =å

Update strengths:

| |' i jij ij

i j

x xc c

r r

-=

+

The procedure converges to an equilibrium, where

i j i jx x r r- = +

Exercise 3. The edges of a simple planar map are 2-colored

with red and blue. Prove that there is always a node where the

red edges (and so also the blue edges) are consecutive.

There is a node where

“too strong” edges (and

“too weak” edges) are

consecutive.

( ) 2 arctan( )x

tx e dtj- ¥

= ò

A direct optimization proof [Colin de Verdiere]

Variables: ,Vx yÎ Î F

Set

log radii of circles

representing nodes

log radii of circles

inscribed in facets

minimize,

( ) ( )p i ip p ii V p

i p

y x y xj bÎ Î

Î

- - -åF

p

i

ipbFrom any Tutte representation

Polar polytope

: 0polytope,dP P

* { : 1 }: polar polytoped TP y x y x P

Blocking polyhedra Fulkerson 1970

* { : 1 }n TK x x y y K nK convex,ascending

* * *( ) ; facets of vertices ofK K K K

Exercise 4. Let K be the dominant of the convex hull of edgesets of

s-t paths. Prove that the blocker is the dominant of the convex hull of

edge-sets of s-t cuts.

Energy

2 2( ) (0, ) min{| | : }K d K x x K= = ÎE

nK convex, ascending (recessive)

,x K y x y K

*( ) ( ) 1K K =E E

x: shortest vector in K

x*: shortest vector in K*

*x x

Generalized energy

{ }2( , ) min :i iK c c x x K= ÎåE

nK convex, ascending (recessive)

,x K y x y K

1

1 1, * ,...,n

n

c cc c+

æ ö÷ç ÷Î =ç ÷ç ÷çè ø

{ }( , ) min :i iK c c x x K= ÎåL

* *( , ) ( , ) 1K c K c =E E

* *1 ( , ) ( , )K c K c n£ £L L

Exercise 5. Prove these inequalities. Also prove that they are sharp.

x: shortest vector in K

x*: shortest vector in K*

*i i ix Cc x

Example 1.

1 1 2 2 3( ) ( )

3

1 1 2 2 3 3

{( : , , )},

{( : , , )}

( ) energy of rubber bands

iE G E

i

Gj

j

x x x a x a x aK K

y y y b y b y b

K

+ + = - = = =

´ - =

´

= =

=

Í

E

Example 2.

( ) ,

( )

E GK K

K+ =

=

Í

E

s-t flows of value 1 and “everything above”

electrical resistance between nodes s and t

Example 3

Traffic jams (directed)

s t

time to cross e ~ traffic through e = xeN

N cars from s to t

average travel time:2

ex

(xe): flow of value 1 from s to t

Best average travel time = distance of 0 from the directed flow polytope

3

3

3

3

2

2

2

5

4

1

10

10Brooks-Smith-Stone-Tutte 1940

0

3

4

5

67

9

Square tilings I

3

3

3

3

2

2

2

5

4

1

10

10

3

1

4

5

3

9

10

10

9

2

2

2

3

3

Square tilings II

Every triangulation of a quadrilateral can be

represented by a square tiling of a rectangle.

Schramm

3

1

4

5

3

9

10

10

9

2

2

2

3

3

Every triangulation of a quadrilateral can be

represented by a square tiling of a rectangle.

Schramm

If the triangulation is 5-connected, then the

representing squares are non-degeenerate.

K=convex hull of nodesets of u-v paths + +

n

u v

s

tx: shortest vector in K

x*: shortest vector in K*

*x Cx

x gives lengths of edgesof the squares.

Exercise 6. The blocker of K is the dominant of the convex

hull of s-t paths.

Exercise 7. (a) How to get the

position of the center of each square?

(b) Complete the proof.

Unit vector flows

edge dijij" Îv

0ijj

=å v

1ij =v

ij ji=-v v skew symmetric

vector flow

Trivial necessary condition: G is 2-edge-connected.

Conjecture 1. For d=2, every 4-edge-connected graph has

a unit vector flow.

Conjecture 2. For d=3, every 2-edge-connected graph has

a unit vector flow.

Theorem. For d=7, every 2-edge-connected graph has

a unit vector flow.Jain

It suffices to consider 3-edge-connected 3-regular graphs

Exercise 8. Prove conjecture 2 for planar graphs.

[Schramm]

edge skew symmetric (parameter)dijij" Îa

node (vector variabl ) edii" Îx

minimize ij i jij

+ -å a x x

0kj k jij i j

ij jk kj k j

+ -¶+ - = =

¶ + -å å

a x xa x x

x a x x

unit vector flow?

Conjecture 2’.

0ifor which the minimizing x satisfies ij ij i ja$ + - ¹a x x

Conjecture 2’’. Every 3-regular 3-connected graph can be

drawn on the sphere so that every edge is an arc of a large

circle, and at every node, any two edges form 120o.

Exercise 9. Conjectures 2' and 2" are equivalent to Conjecture 2.

Antiblocking polyhedra Fulkerson 1971

* { : 1 }n TK x x y y K

* * *( ) ;K K K K facets of vertices of

nK convex corner

(polarity in the nonnegative orthant)

conv{ :S }TAB( ) stable set inA GG A

: incidence vector of setA A

The stable set polytope

Graph entropy

( )min log : STAB( , ) ( ){ }i i

i V GH p GG x xp

log consti ip x

Körner 1973

( , ) ( ) logn i iH K p H p p p

p: probability distribution on V(G)

( )( ) .99

1lim mi( , ) n log ( [ ])t

t

t

U V GP U

t Gt

H G Up

connected iff distinguishable

Want: encode most of V(G)t by 0-1 words of min length, so that distinguishable words get different codes.

(measure of “complexity” of G)

( , ) ( , ) ( , )H F p H G p H F G p

: ( , ) ( , ) ( )p H G p H G p H p

G

is perfect

Csiszár, Körner, Lovász, Marton, Simonyi

( , ) ( , ) ( )H G p H G p H p