Nonlinear methods in discrete optimization
László Lovász Eötvös Loránd University, Budapest
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 xÎ
+ - - =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 uu
c
( ) 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- ¥
= òA direct optimization proof [Colin de Verdiere]
Variables: ,Vx yÎ Î F
Set
log radii of circlesrepresenting nodes
log radii of circlesinscribed 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 KK
+ ==
ÍE
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: 2ex
(xe): flow of value 1 from s to t
Best average travel time = distance of 0 from the directed flow polytope
33
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 Kx*: 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 hasa unit vector flow.
Conjecture 2. For d=3, every 2-edge-connected graph hasa unit vector flow.
Theorem. For d=7, every 2-edge-connected graph hasa 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 bedrawn 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