Nonlinear methods in discrete optimization László Lovász Eötvös Loránd University, Budapest

planar graph Fáry-Wagner Every simple planar graph can be drawn in 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 to convex polygon edges replaced by rubber bands Energy: Equilibrium:

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 Want: Minimize: Optimum satisfies  i:

Rubber bands and strengths rubber bands have strengths c ij > 0 Energy: Equilibrium:

Update strengths: The procedure converges to an equilibrium, where 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.

A direct optimization proof [Colin de Verdiere] Variables: Set log radii of circles representing nodes log radii of circles inscribed in facets minimize p i From any Tutte representation

Polar polytope

Blocking polyhedra Fulkerson 1970 convex, ascending 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 convex, ascending (recessive)

x : shortest vector in K x *: shortest vector in K *

Generalized energy convex, ascending (recessive)

Exercise 5. Prove these inequalities. Also prove that they are sharp. x : shortest vector in K x *: shortest vector in K *

Example 1. Example 2. 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 = x e N N cars from s to t average travel time: (x e ): flow of value 1 from s to t Best average travel time = distance of 0 from the directed flow polytope

Brooks-Smith-Stone-Tutte Square tilings I

Square tilings II

Every triangulation of a quadrilateral can be represented by a square tiling of a rectangle. Schramm

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 t x : shortest vector in K x *: shortest vector in K * x gives lengths of edges of 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 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] unit vector flow?

Conjecture 2’. 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 120 o. Exercise 9. Conjectures 2' and 2" are equivalent to Conjecture 2.

Antiblocking polyhedra Fulkerson 1971 convex corner (polarity in the nonnegative orthant)

The stable set polytope

Graph entropy Körner 1973 p : probability distribution on V(G)

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 )

Csiszár, Körner, Lovász, Marton, Simonyi