Nonlinear methods in discrete optimization

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Nonlinear methods in discrete optimization László
Lovász Eötvös Loránd University, Budapest
lovasz_at_cs.elte.hu
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planar graph
Exercise 1 Prove this.
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Rubber bands and planarity
Tutte (1963)
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Rubber bands and planarity
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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.
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Coin representation Koebe (1936)
Every planar graph can be represented by touching
circles
Discrete Riemann Mapping Theorem
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Can this be obtained from a rubber band
representation?
Tutte representation ? optimal circles
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Rubber bands and strengths
rubber bands have strengths cij gt 0
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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.
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There is a node where too strong edges
(and too weak edges) are consecutive.
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A direct optimization proof Colin de Verdiere
Set
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Polar polytope
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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.
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Energy
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Generalized energy
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Exercise 5. Prove these inequalities. Also prove
that they are sharp.
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Example 1.
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Example 3
N cars from s to t
(xe) flow of value 1 from s to t
Best average travel time distance of 0 from
the directed flow polytope
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Square tilings I
Brooks-Smith-Stone-Tutte 1940
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10
3
3
4
1
2
3
2
5
3
2
10
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Square tilings II
3
3
4
2
3
1
2
5
3
2
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Every triangulation of a quadrilateral can be
represented by a square tiling of a rectangle.
Schramm
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3
3
4
2
3
1
2
5
3
2
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Every triangulation of a quadrilateral can be
represented by a square tiling of a rectangle.
Schramm
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t
Kconvex 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
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Unit vector flows
Trivial necessary condition G is
2-edge-connected.
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Conjecture 1. For d2, every 4-edge-connected
graph has a unit vector flow.
Conjecture 2. For d3, 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.
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Schramm
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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.
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Antiblocking polyhedra
Fulkerson 1971
(polarity in the nonnegative orthant)
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The stable set polytope
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Graph entropy
Körner 1973
p probability distribution on V(G)
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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)
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