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Discrete mathematics: the last and next decade

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Discrete mathematics:

the last and next decade

László Lovász

Microsoft Research

One Microsoft Way, Redmond, WA 98052

Higlights of the 90’s:

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

maximize

Approximation algorithms:

The Max Cut Problem

NP-hard

…Approximations?

Easy with 50% error Erdős~’65:

???

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)

Discrete probability

random structures

randomized algorithms

algorithms on random input

statistical mechanics

phase transitions

high concentration

pseudorandom numbers

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

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,…

qpolylog(q)

Want:

such that:

Few vectors

- any 3 linearly independent

- every vector is a linear combination of 2

Every finite projective plane of order q

has a complete arc of size qpolylog(q).

Kim-Vu

Example

(was open for 30 years)

at random

Second idea: choose

?????

First idea: use algebraic construction (conics,…)

gives only about q

Solution:

Rödl nibble + strong concentration results

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?!

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, ...

Very large structures

- internet
- VLSI
- databases

How to model these?

non-constant but stable

partly random

- genetic code
- brain
- animal
- ecosystem

- -economy
- society

up to a bounded number

of additional nodes

tree-decomposition

embeddable in a fixed surface

except for “fringes”

of bounded depth

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.

given >0 and k>1,

the number of parts is

between k and f(k, )

difference at most 1

with k2 exceptions

for subsets X,Y of the two parts,

# of edges between X and Y

is p|X||Y| n2

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).

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?

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,…

3-connected planar graph

Every 3-connected planar graph

is the skeleton of a polytope.

Steinitz

Example 1: Geometric representations of graphs

Coin representation

Koebe (1936)

Every planar graph can be represented by touching circles

Polyhedral version

Every 3-connected planar graph

is the skeleton of a convex polytope

such that every edge

touches the unit sphere

Andre’ev

“Cage Represention”

From polyhedra to circles

horizon

From polyhedra to representation of the dual

Cage representation Riemann Mapping Theorem

Koebe

Sullivan

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)

Representation of G in R3

basis of nullspace of M

Colin de Verdière, using pde’s

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

G 3-connected

planar

nullspace representation gives

planar embedding in S2

The vectors can be rescaled so that

we get a Steinitz representation.

LL

L-Schrijver

Nullspace representation

from the CdV matrix

eigenfunctions of the

Laplacian

~

Cage representation Riemann Mapping Theorem

Koebe

Sullivan

by a membership oracle;

with relative error ε

Example 2: volume computation

, convex

Given:

Want: volume of K

Not possible in polynomial time, even if ε=ncn.

Possible in randomized polynomial time,

for arbitrarily small ε.

Complexity:

For self-reducible problems,

counting sampling

(Jerrum-Valiant-Vazirani)

Enough to sample

from convex bodies

Algorithmic results:

Use rapidly mixing

Markov chains

(Broder; Jerrum-Sinclair)

Enough to estimate

the mixing rate

of random walk

on lattice in K

Dyer

Frieze

Kannan

1989

Graph theory (expanders):

use conductance to

estimate eigenvalue gap

Alon, Jerrum-Sinclair

Enough to prove

isoperimetric inequality

for subsets of K

Differential geometry:

Isoperimetric inequality

Classical probability:

use eigenvalue gap

Use conductance to

estimate mixing rate

Jerrum-Sinclair

Enough to prove

isoperimetric inequality

for subsets of K

Differential geometry:

properties of minimal

cutting surface

Isoperimetric inequality

Differential equations:

bounds on Poincaré

constant

Paine-Weinberger

bisection method,

improved

isoperimetric inequality

LL-Simonovits 1990

Log-concave functions:

reduction to integration

Applegate-Kannan 1992

Brunn-Minkowski Thm:

Ball walk

LL 1992

Log-concave functions:

reduction to integration

Applegate-Kannan 1992

Convex geometry:

Ball walk

LL 1992

Statistics:

Better error handling

Dyer-Frieze 1993

Optimization:

Better prepocessing

LL-Simonovits 1995

Functional analysis:

isotropic position of

convex bodies

achieving

isotropic position

Kannan-LL-Simonovits 1998

Geometry:

projective (Hilbert)

distance

affin invariant

isoperimetric inequality

analysis if hit-and-run walk

LL 1999

Differential equations:

log-Sobolev inequality

elimination of

“start penalty” for

lattice walk

Frieze-Kannan 1999

log-Cheeger inequality

elimination of

“start penalty” for

ball walk

Kannan-LL 1999

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

- 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

Optimization:

discrete linear semidefinite ?