Existential Polytime and Polyhedral Combinatorics
A minicourse by
in Belgrade, June 4 - 7, 2015.
Mathematical Institute of the Serbian Academy of Sciences and Arts
For thousands of years, and even now in group theoretic geometry, the
beautiful symmetries of a handful of polyhedra with a handful of facets
have been at the center of refined mathematics. Since the advent of
Turing's computers and operations research, beauty has been found in
polyhedra regardless of symmetry, with facets as numerous as the stars.
Linear-algebra theory is being nudged by great systems of linear
inequalities as inputs.`Polyhedra' means
their solution sets. `Existential polytime', i.e. NP, means certifiable in
polynomial time when true.
The course will explore some of the polyhedra which have edged aside the
Topics of the various days are related
but presentations will be independent with some reference to each other.
Presentation each day will be in the morning, 10am - 12am, with elaboration
and tutorial during the refreshment after the lecture.
Combinatorial structure of paths, network flows, optimum
assignment, stable marriage.
Routes for Chinese postmen, traveling salesmen, and itinerant
preachers, optimum systems of trees and branchings.
Submodular set functions and polymatroids.
The greedy algorithm. Combinatorics of (linear) dependence.
Duality and contractions. Exact matrix implementations.
Matroid partitioning and Lehman's game. Polymatroid intersections.
The ellipsoid method for exponentially large linear programs.
Some Pure Math of Classical Game Theory.
Bimatrix games and beautiful-mind Nash equilibria; Room partitioning and
N-person cooperative games; set functions and cores. Algorithms and
Some of the Resources. Optima interview
(Discrete) Optimization Stories
Books on Combinatorial Optimization including
Schrijver, Combinatorial Optimization: Polyhedra and Efficiency
Jack Edmonds, 3^4, the teacher, is the Dumbledore, Gandalf, Merlin, Obi-Wan
Kenobi, of the subject.
Muggles are welcome.
Please register at no cost: