Discrete mathematics:

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Discrete mathematics the last and next decade
László Lovász Microsoft Research One Microsoft
Way, Redmond, WA 98052 lovasz_at_microsoft.com
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Higlights of the 90s
Approximation algorithms positive and
negative results
Discrete probability Markov chains, high
concentration, nibble methods, phase
transitions
Pseudorandom number generators from art
to science theory and constructions
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Approximation algorithms The Max Cut Problem
NP-hard
Approximations?
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Easy with 50 error Erdos
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???
Arora-Lund-Motwani- Sudan-Szegedy 92 Hastad
NP-hard with 6 error
(Interactive proof systems, PCP)
Polynomial with 12 error
Goemans-Williamson 93
(semidefinite optimization)
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Discrete probability random
structures randomized algorithms
algorithms on random input
statistical mechanics phase transitions
high concentration
pseudorandom numbers
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Algorithms and probability
Randomized algorithms (making coin flips)
important applications
(primality testing,
integration, optimization,
volume computation, simulation)
difficult to analyze
Algorithms with stochastic input
even more important applications
even more difficult to analyze
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Difficulty after a few iterations, complicated
function of the original random
variables arise.
New methods in probability Strong
concentration (Talagrand)
Laws of Large Numbers sums of independent
random variables is strongly concentrated
General strong concentration very general
smooth functions of independent
random variables are strongly concentrated
Nible, martingales, rapidly mixing Markov chains,
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Example
(was open for 30 years)
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First idea use algebraic construction (conics,)
gives only about q
Solution
Rödl nibble strong concentration results
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Driving forces for the next decade
New areas of applications
The study of very large structures
More tools from classical areas in mathematics
More applications in classical areas?!
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New areas of application
Biology genetic code population
dynamics protein folding
Physics elementary particles, quarks, etc.
(Feynman graphs)
statistical mechanics
(graph theory, discrete probability)
Economics indivisibilities
(integer programming, game theory)
Computing algorithms, complexity, databases,
networks, VLSI, ...
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Very large structures

• internet
• VLSI
• databases

How to model these?
non-constant but stable partly random

• genetic code
• brain
• animal
• ecosystem

• -economy
• society

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Very large structures how to model them?
Graph minors
Robertson, Seymour, Thomas
If a graph does not contain a given minor, then
it is essentially a
1-dimensional structure of 2-dimensional
pieces.
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Very large structures how to model them?
Regularity Lemma
Szeméredi
The nodes of every graph can be partitioned into
a bounded number of essentially equal parts so
that almost all bipartite graphs between 2
parts are essentially random (with different
densities).
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Very large structures

• -internet
• VLSI
• databases
• genetic code
• brain
• animal
• ecosystem
• economy
• society

How to model these?
How to handle them algorithmically?
heuristics/approximation
algorithms
linear time algorithms
sublinear time algorithms (sampling)
A complexity theory of linear time?
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More and more tools from classical math
Linear algebra eigenvalues
semidefinite optimization
higher incidence matrices
homology theory
Geometry geometric representations of
graphs convexity
Analysis generating functions
Fourier analysis, quantum
computing
Number theory cryptography
Topology, group theory, algebraic
geometry, special functions, differential
equations,
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Example 1 Geometric representations of graphs
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Coin representation
Koebe (1936)
Every planar graph can be represented by touching
circles
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Polyhedral version
Every 3-connected planar graph is the skeleton
of a convex polytope such that every edge
touches the unit sphere
Andreev
Cage Represention
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From polyhedra to circles
horizon
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From polyhedra to representation of the dual
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Cage representation ? Riemann Mapping Theorem
? Koebe
? Sullivan
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The Colin de Verdière number
G connected graph Roughly ?(G) multiplicity
of second largest eigenvalue of
adjacency matrix

Largest has multiplicity 1.
But maximize over weighting the edges and
diagonal entries
(But non-degeneracy condition on weightings)
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Colin de Verdière, using pdes Van der Holst,
elementary proof
µ(G)?3 ? G is a planar
3 if G is 3-connected
may assume second largest eigenvalue is 0
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L-Schrijver
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Cage representation ? Riemann Mapping Theorem
? Koebe
? Sullivan
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Example 2 volume computation
, convex
Given
Want volume of K
Not possible in polynomial time, even if encn.
Possible in randomized polynomial time, for
arbitrarily small e.
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Classical probability use eigenvalue gap
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History earlier highlights 60 polyhedral
combinatorics, polynomial time, random
graphs, extremal graph theory, matroids 70
4-Color Theorem, NP-completeness,
hypergraph theory, Szemerédi Lemma 80 graph
minor theory, cryptography
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• Highlights if the last 4 decades
• New applications
• physics, biology, computing, economics
• 3. Main trends in discrete math
• -Very large structures
• -More and more applications of methods from
• classical math
• -Discrete probability

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Optimization discrete ? linear ? semidefinite
? ?