Graph Decompositions, Metric Geometry, and Approximation Algorithms

Yuval Rabani

Technion

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In recent years, deep results in geometric functional analysis, and
specifically in the non-linear theory of Banach spaces, have been
successfully applied to solve fundamental problems in several areas of
the theory of combinatorial algorithms.  In a somewhat more surprising
twist of events, algorithmic insight and tools have been used to
revive the non-linear theory of Banach spaces, settling outstanding
open questions and generating new research directions.

I will discuss one aspect of this amazingly fruitful endeavor that
relates intriguing NP-hard problems in combinatorial optimization to
questions on Lipschitz maps of finite metric spaces. I will discuss
partitions of graphs into clusters that satisfy certain boundary
conditions, a concept underlying both sides of this relationship. I
will overview between two and four partition schemes, each with its
own set of applications and substantial open problems.