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Nonlinear methods in discrete optimization L á szl ó Lov á sz Eötvös Loránd University, Budapest lovasz@cs.elte.hu. Every simple planar graph can be drawn in the plane with straight edges. Fáry-Wagner. planar graph. Exercise 1 : Prove this.
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Nonlinear methods in discrete optimization László Lovász Eötvös Loránd University, Budapest lovasz@cs.elte.hu
Every simple planar graph can be drawn in the plane with straight edges Fáry-Wagner planar graph Exercise 1: Prove this.
Every 3-connected planar graph can be drawn with straight edges and convex faces. Rubber bands and planarity Tutte (1963)
outer face fixed to convex polygon edges replaced by rubber bands Energy: Equilibrium: Rubber bands and planarity
G 3-connected planar rubber band embedding is planar Tutte 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.
Coin representation Koebe (1936) Every planar graph can be represented by touching circles Discrete Riemann Mapping Theorem
Want: Minimize: Optimum satisfies i: Can this be obtained from a rubber band representation? Tutte representation optimal circles
Energy: Equilibrium: Rubber bands and strengths rubber bands have strengths cij > 0
The procedure converges to an equilibrium, where Update strengths: 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.
p i From any Tutte representation log radii of circles representing nodes Variables: log radii of circles inscribed in facets minimize A direct optimization proof [Colin de Verdiere] Set
convex, ascending Blocking polyhedra Fulkerson 1970 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.
x: shortest vector in K x*: shortest vector in K*
convex, ascending (recessive) Generalized energy
x: shortest vector in K x*: shortest vector in K* Exercise 5. Prove these inequalities. Also prove that they are sharp.
Example 2. s-t flows of value 1 and “everything above” electrical resistance between nodes s and t Example 1.
Traffic jams (directed) time to cross e ~ traffic through e = xeN s t average travel time: Example 3 N cars from sto t (xe): flow of value 1 from s to t Best average travel time = distance of 0 from the directed flow polytope
10 0 3 3 4 1 3 2 3 4 5 2 6 5 7 3 2 9 10 Square tilings I Brooks-Smith-Stone-Tutte 1940
10 3 3 4 1 2 3 2 5 3 2 10
10 9 9 10 Square tilings II 3 3 4 2 3 1 2 5 3 2
Every triangulation of a quadrilateral can be represented by a square tiling of a rectangle. Schramm
10 9 9 10 3 3 4 2 3 1 2 5 3 2
If the triangulation is 5-connected, then the representing squares are non-degeenerate. Every triangulation of a quadrilateral can be represented by a square tiling of a rectangle. Schramm
t x: shortest vector in K x*: shortest vector in K* K=convex hull of nodesets of u-v paths + +n u v 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. x gives lengths of edges of the squares. s
skew symmetric vector flow Unit vector flows Trivial necessary condition: G is 2-edge-connected.
Theorem. For d=7, every 2-edge-connected graph has a unit vector flow. Jain 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. It suffices to consider 3-edge-connected 3-regular graphs Exercise 8. Prove conjecture 2 for planar graphs.
unit vector flow? [Schramm]
Conjecture 2’. Exercise 9. Conjectures 2' and 2" are equivalent to 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 120o.
convex corner Antiblocking polyhedra Fulkerson 1971 (polarity in the nonnegative orthant)
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)
