Hierarchical Decompositions for Congestion Minimization in Networks Harald Räcke 1

Approximation Tree Approximation

Approximating a Graph by a Tree Task: Given an (undirected) graph G. Compute a decomposition tree that is similar. decomposition tree:  bijection between leaf nodes of the tree and graph nodes embedding  each internal node is mapped to some graph node  tree edges are mapped to path in the graph between the corresponding endpoints 3 abcdefghij a g b e i j f h d c

Similarity 4

Unfortunately this does not work: Any tree will distort at least one edge/pair by a lot in a cycle. Solution: Convex combination of trees probability distribution over trees. What is the expected factor by which the length of an edge increases when choosing a random tree? Distance-based Decompositions 5

6

Cut-based Decompositions 7

Part 1. Cut-based Decompositions 8 a g b e i j f h d c abcdefghij ce b h

Part 2. Cut-based Decompositions 9 abcdefghij ce i h a g b e i j f h d c

Part 1: simply make capacities in the tree large enough!!! Cut-based Decompositions 10 a g b e i j f h d c abcdefghij ce b h

Part 1: simply make capacities in the tree large enough!!! Cut-based Decompositions 11 a g b e i j f h d c abcdefghij ce b h

Part 2: Not possible. The high capacity tree edge is mapped to a single path. Cut-based Decompositions 12 a g b e i j f h d c abcdefghij ce b h

Cut-based Decompositions 13

Cut-based Decompositions 14

Cut-based Decompositions 15

Minimum Bisection black vertices white vertices

Minimum Bisection Minimum Bisection on leaf nodes of the tree: Color each leaf node either black or white such that each color is used the same number of times. Minimize total capacity of edges that you need to take out in order to disconnect black from white vertices.

Minimum Bisection Algorithm: For each tree in the support of the convex combination compute a minimum leaf bisection. Output the smallest bisection found. 18 tree is better network T* is optimal

Advantages of having a single tree Some applications seem to require a single tree. E.g. all-or-nothing multicommodity flow. Oblivious routing can be done with small routing tables (poly-logarithmic per edge). See Matthews talk…

Hierarchical Routing Scheme

intermediate clusters.

Questions Embedding: How are intermediate nodes chosen? How are routing paths between intermediate nodes chosen? Decomposition: How is the partitioning done?

Choosing Intermediate Nodes

Estimating Traffic

Choosing Routing Paths

Select routing paths according to this solution!

Observation

Goal

A Bad-Case Example

Precondition

Decomposition Theorem

The Algorithm Capacity of edges bet- ween different clusters decreases.

Further Results/Open Questions