![Page 1: Nonlinear methods in discrete optimization László Lovász Eötvös Loránd University, Budapest lovasz@cs.elte.hu](https://reader036.vdocument.in/reader036/viewer/2022062421/56649ce55503460f949b2dbf/html5/thumbnails/1.jpg)
Nonlinear methods in discrete optimization
László Lovász
Eötvös Loránd University, Budapest
![Page 2: Nonlinear methods in discrete optimization László Lovász Eötvös Loránd University, Budapest lovasz@cs.elte.hu](https://reader036.vdocument.in/reader036/viewer/2022062421/56649ce55503460f949b2dbf/html5/thumbnails/2.jpg)
planar graph
Fáry-Wagner
Every simple planar graph can be drawnin the plane with straight edges
Exercise 1: Prove this.
![Page 3: Nonlinear methods in discrete optimization László Lovász Eötvös Loránd University, Budapest lovasz@cs.elte.hu](https://reader036.vdocument.in/reader036/viewer/2022062421/56649ce55503460f949b2dbf/html5/thumbnails/3.jpg)
Rubber bands and planarity
Every 3-connected planar graph can be drawn with straight edges and convex faces.
Tutte (1963)
![Page 4: Nonlinear methods in discrete optimization László Lovász Eötvös Loránd University, Budapest lovasz@cs.elte.hu](https://reader036.vdocument.in/reader036/viewer/2022062421/56649ce55503460f949b2dbf/html5/thumbnails/4.jpg)
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
![Page 5: Nonlinear methods in discrete optimization László Lovász Eötvös Loránd University, Budapest lovasz@cs.elte.hu](https://reader036.vdocument.in/reader036/viewer/2022062421/56649ce55503460f949b2dbf/html5/thumbnails/5.jpg)
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
![Page 6: Nonlinear methods in discrete optimization László Lovász Eötvös Loránd University, Budapest lovasz@cs.elte.hu](https://reader036.vdocument.in/reader036/viewer/2022062421/56649ce55503460f949b2dbf/html5/thumbnails/6.jpg)
Discrete Riemann Mapping Theorem
Coin representation Koebe (1936)
Every planar graph can be represented by touching circles
![Page 7: Nonlinear methods in discrete optimization László Lovász Eötvös Loránd University, Budapest lovasz@cs.elte.hu](https://reader036.vdocument.in/reader036/viewer/2022062421/56649ce55503460f949b2dbf/html5/thumbnails/7.jpg)
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:
![Page 8: Nonlinear methods in discrete optimization László Lovász Eötvös Loránd University, Budapest lovasz@cs.elte.hu](https://reader036.vdocument.in/reader036/viewer/2022062421/56649ce55503460f949b2dbf/html5/thumbnails/8.jpg)
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
![Page 9: Nonlinear methods in discrete optimization László Lovász Eötvös Loránd University, Budapest lovasz@cs.elte.hu](https://reader036.vdocument.in/reader036/viewer/2022062421/56649ce55503460f949b2dbf/html5/thumbnails/9.jpg)
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.
![Page 10: Nonlinear methods in discrete optimization László Lovász Eötvös Loránd University, Budapest lovasz@cs.elte.hu](https://reader036.vdocument.in/reader036/viewer/2022062421/56649ce55503460f949b2dbf/html5/thumbnails/10.jpg)
There is a node where
“too strong” edges (and
“too weak” edges) are
consecutive.
![Page 11: Nonlinear methods in discrete optimization László Lovász Eötvös Loránd University, Budapest lovasz@cs.elte.hu](https://reader036.vdocument.in/reader036/viewer/2022062421/56649ce55503460f949b2dbf/html5/thumbnails/11.jpg)
( ) 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
![Page 12: Nonlinear methods in discrete optimization László Lovász Eötvös Loránd University, Budapest lovasz@cs.elte.hu](https://reader036.vdocument.in/reader036/viewer/2022062421/56649ce55503460f949b2dbf/html5/thumbnails/12.jpg)
Polar polytope
: 0polytope,dP P
* { : 1 }: polar polytoped TP y x y x P
![Page 13: Nonlinear methods in discrete optimization László Lovász Eötvös Loránd University, Budapest lovasz@cs.elte.hu](https://reader036.vdocument.in/reader036/viewer/2022062421/56649ce55503460f949b2dbf/html5/thumbnails/13.jpg)
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.
![Page 14: Nonlinear methods in discrete optimization László Lovász Eötvös Loránd University, Budapest lovasz@cs.elte.hu](https://reader036.vdocument.in/reader036/viewer/2022062421/56649ce55503460f949b2dbf/html5/thumbnails/14.jpg)
Energy
2 2( ) (0, ) min{| | : }K d K x x K= = ÎE
nK convex, ascending (recessive)
,x K y x y K
![Page 15: Nonlinear methods in discrete optimization László Lovász Eötvös Loránd University, Budapest lovasz@cs.elte.hu](https://reader036.vdocument.in/reader036/viewer/2022062421/56649ce55503460f949b2dbf/html5/thumbnails/15.jpg)
*( ) ( ) 1K K =E E
x: shortest vector in K
x*: shortest vector in K*
*x x
![Page 16: Nonlinear methods in discrete optimization László Lovász Eötvös Loránd University, Budapest lovasz@cs.elte.hu](https://reader036.vdocument.in/reader036/viewer/2022062421/56649ce55503460f949b2dbf/html5/thumbnails/16.jpg)
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
![Page 17: Nonlinear methods in discrete optimization László Lovász Eötvös Loránd University, Budapest lovasz@cs.elte.hu](https://reader036.vdocument.in/reader036/viewer/2022062421/56649ce55503460f949b2dbf/html5/thumbnails/17.jpg)
* *( , ) ( , ) 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
![Page 18: Nonlinear methods in discrete optimization László Lovász Eötvös Loránd University, Budapest lovasz@cs.elte.hu](https://reader036.vdocument.in/reader036/viewer/2022062421/56649ce55503460f949b2dbf/html5/thumbnails/18.jpg)
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
![Page 19: Nonlinear methods in discrete optimization László Lovász Eötvös Loránd University, Budapest lovasz@cs.elte.hu](https://reader036.vdocument.in/reader036/viewer/2022062421/56649ce55503460f949b2dbf/html5/thumbnails/19.jpg)
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
![Page 20: Nonlinear methods in discrete optimization László Lovász Eötvös Loránd University, Budapest lovasz@cs.elte.hu](https://reader036.vdocument.in/reader036/viewer/2022062421/56649ce55503460f949b2dbf/html5/thumbnails/20.jpg)
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
![Page 21: Nonlinear methods in discrete optimization László Lovász Eötvös Loránd University, Budapest lovasz@cs.elte.hu](https://reader036.vdocument.in/reader036/viewer/2022062421/56649ce55503460f949b2dbf/html5/thumbnails/21.jpg)
3
3
3
3
2
2
2
5
4
1
10
10
![Page 22: Nonlinear methods in discrete optimization László Lovász Eötvös Loránd University, Budapest lovasz@cs.elte.hu](https://reader036.vdocument.in/reader036/viewer/2022062421/56649ce55503460f949b2dbf/html5/thumbnails/22.jpg)
3
1
4
5
3
9
10
10
9
2
2
2
3
3
Square tilings II
![Page 23: Nonlinear methods in discrete optimization László Lovász Eötvös Loránd University, Budapest lovasz@cs.elte.hu](https://reader036.vdocument.in/reader036/viewer/2022062421/56649ce55503460f949b2dbf/html5/thumbnails/23.jpg)
Every triangulation of a quadrilateral can be
represented by a square tiling of a rectangle.
Schramm
![Page 24: Nonlinear methods in discrete optimization László Lovász Eötvös Loránd University, Budapest lovasz@cs.elte.hu](https://reader036.vdocument.in/reader036/viewer/2022062421/56649ce55503460f949b2dbf/html5/thumbnails/24.jpg)
3
1
4
5
3
9
10
10
9
2
2
2
3
3
![Page 25: Nonlinear methods in discrete optimization László Lovász Eötvös Loránd University, Budapest lovasz@cs.elte.hu](https://reader036.vdocument.in/reader036/viewer/2022062421/56649ce55503460f949b2dbf/html5/thumbnails/25.jpg)
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.
![Page 26: Nonlinear methods in discrete optimization László Lovász Eötvös Loránd University, Budapest lovasz@cs.elte.hu](https://reader036.vdocument.in/reader036/viewer/2022062421/56649ce55503460f949b2dbf/html5/thumbnails/26.jpg)
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.
![Page 27: Nonlinear methods in discrete optimization László Lovász Eötvös Loránd University, Budapest lovasz@cs.elte.hu](https://reader036.vdocument.in/reader036/viewer/2022062421/56649ce55503460f949b2dbf/html5/thumbnails/27.jpg)
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.
![Page 28: Nonlinear methods in discrete optimization László Lovász Eötvös Loránd University, Budapest lovasz@cs.elte.hu](https://reader036.vdocument.in/reader036/viewer/2022062421/56649ce55503460f949b2dbf/html5/thumbnails/28.jpg)
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.
![Page 29: Nonlinear methods in discrete optimization László Lovász Eötvös Loránd University, Budapest lovasz@cs.elte.hu](https://reader036.vdocument.in/reader036/viewer/2022062421/56649ce55503460f949b2dbf/html5/thumbnails/29.jpg)
[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?
![Page 30: Nonlinear methods in discrete optimization László Lovász Eötvös Loránd University, Budapest lovasz@cs.elte.hu](https://reader036.vdocument.in/reader036/viewer/2022062421/56649ce55503460f949b2dbf/html5/thumbnails/30.jpg)
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.
![Page 31: Nonlinear methods in discrete optimization László Lovász Eötvös Loránd University, Budapest lovasz@cs.elte.hu](https://reader036.vdocument.in/reader036/viewer/2022062421/56649ce55503460f949b2dbf/html5/thumbnails/31.jpg)
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)
![Page 32: Nonlinear methods in discrete optimization László Lovász Eötvös Loránd University, Budapest lovasz@cs.elte.hu](https://reader036.vdocument.in/reader036/viewer/2022062421/56649ce55503460f949b2dbf/html5/thumbnails/32.jpg)
conv{ :S }TAB( ) stable set inA GG A
: incidence vector of setA A
The stable set polytope
![Page 33: Nonlinear methods in discrete optimization László Lovász Eötvös Loránd University, Budapest lovasz@cs.elte.hu](https://reader036.vdocument.in/reader036/viewer/2022062421/56649ce55503460f949b2dbf/html5/thumbnails/33.jpg)
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)
![Page 34: Nonlinear methods in discrete optimization László Lovász Eötvös Loránd University, Budapest lovasz@cs.elte.hu](https://reader036.vdocument.in/reader036/viewer/2022062421/56649ce55503460f949b2dbf/html5/thumbnails/34.jpg)
( )( ) .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)
![Page 35: Nonlinear methods in discrete optimization László Lovász Eötvös Loránd University, Budapest lovasz@cs.elte.hu](https://reader036.vdocument.in/reader036/viewer/2022062421/56649ce55503460f949b2dbf/html5/thumbnails/35.jpg)
( , ) ( , ) ( , )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
![Page 36: Nonlinear methods in discrete optimization László Lovász Eötvös Loránd University, Budapest lovasz@cs.elte.hu](https://reader036.vdocument.in/reader036/viewer/2022062421/56649ce55503460f949b2dbf/html5/thumbnails/36.jpg)