Discrete mathematics:the last and next decade

László Lovász

Microsoft Research

One Microsoft Way, Redmond, WA 98052

lovasz@microsoft.com

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

Approximation algorithms:The Max Cut Problem

maximize

NP-hard

…Approximations?

Easy with 50% error Erdős ~’65:

Arora-Lund-Motwani-Sudan-Szegedy ’92:Hastad

Polynomial with 12% error Goemans-Williamson ’93:

???

NP-hard with 6% error

(Interactive proof systems, PCP)

(semidefinite optimization)

Discrete probability

random structures

randomized algorithms

algorithms on random input

statistical mechanics

phase transitions

high concentration

pseudorandom numbers

Randomized algorithms (making coin flips):

Algorithms and probability

Algorithms with stochastic input:

difficult to analyze

even more difficult to analyze

important applications (primality testing, integration, optimization, volume computation, simulation)

even more important applications

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 concentratedGeneral strong concentration: very general “smooth” functions of independent random variables are strongly concentrated

Nible, martingales, rapidly mixing Markov chains,…

Example

1 2 33, , ,. ( ).. Ga Fa qa Want: such that:

- any 3 linearly independent

- every vector is a linear combination of 2

Few vectors

q polylog(q)

(was open for 30 years)

Every finite projective plane of order qhas a complete arc of size q polylog(q).

Kim-Vu

Second idea: choose 1 2 3, , ,...a a a at random

?????

Solution: Rödl nibble + strong concentration results

First idea: use algebraic construction (conics,…)

gives only about q

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

-genetic code

-brain

-animal

-ecosystem

-economy

-society

How to model these?

non-constant but stablepartly random

-internet

-VLSI

-databases

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.

up to a bounded number of additional nodes

tree-decomposition

embeddable in a fixed surfaceexcept for “fringes” of bounded depth

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 partsso that almost all bipartite graphs between 2 partsare essentially random(with different densities).

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

How to model these?

How to handle themalgorithmically?

heuristics/approximation algorithms

-internet

-VLSI

-databases

-genetic code -brain

-animal

-ecosystem

-economy

-society

A complexity theory of linear time?

Very large structures

linear time algorithms

sublinear time algorithms (sampling)

Linear algebra : eigenvalues semidefinite optimization higher incidence matrices homology theory

More and more tools from classical math

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,…

Steinitz

Every 3-connected planar graphis the skeleton of a polytope.

3-connected planar graph

Example 1: Geometric representations of graphs

Coin representation

Every planar graph can be represented by touching circles

Koebe (1936)

Polyhedral version

Andre’ev

Every 3-connected planar graph is the skeleton of a convex polytope

such that every edge touches the unit sphere

“Cage Represention”

From polyhedra to circles

horizon

From polyhedra to representation of the dual

Cage representation Riemann Mapping Theorem

Sullivan

Koebe

The Colin de Verdière number

G: connected graph

Roughly: (G) = multiplicity of second largest eigenvalue

of adjacency matrix

(But: non-degeneracy condition on weightings)

Largest has multiplicity 1.

But: maximize over weighting the edges and diagonal entries

μ(G)3 G is a planar Colin de Verdière, using pde’sVan der Holst, elementary proof

=3 if G is 3-connected

1

2

n

uu

u

Representation of G in 3

0ij jj

M u

basis of nullspace of M

11 21 31

12 22 3

1 2 3

2

12 22 32

:

x x xx x x

x

x x

x x

x

may assume second largest eigenvalue is 0

G 3-connectedplanar

nullspace representation gives

planar embedding in 2

L-Schrijver

The vectors can be rescaled so that we get a Steinitz representation. LL

Cage representation Riemann Mapping Theorem

Sullivan

Koebe

Nullspace representationfrom the CdV matrix ~

eigenfunctions of theLaplacian

Example 2: volume computation

nK Given: , convex

Want: volume of K

by a membership oracle;2(0,1) (0, )B K B n

with relative error ε

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 samplefrom 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

Graph theory (expanders):use conductance toestimate eigenvalue gapAlon, Jerrum-Sinclair

Enough to proveisoperimetric inequalityfor subsets of K

Differential geometry: Isoperimetric inequality

DyerFriezeKannan1989

* 27( )O n

Classical probability:use eigenvalue gap

Use conductance toestimate mixing rateJerrum-Sinclair

Enough to proveisoperimetric inequalityfor subsets of K

Differential geometry:properties of minimalcutting surface

Isoperimetric inequality

Differential equations:bounds on PoincaréconstantPaine-Weinberger

bisection method,improvedisoperimetric inequalityLL-Simonovits 1990

* 16( )O nLog-concave functions: reduction to integration

Applegate-Kannan 1992* 10( )O n

Brunn-Minkowski Thm: Ball walkLL 1992

* 10( )O n

Log-concave functions: reduction to integrationApplegate-Kannan 1992

* 10( )O n

Convex geometry: Ball walkLL 1992

* 10( )O n

Statistics: Better error handlingDyer-Frieze 1993

* 8( )O n

Optimization: Better prepocessingLL-Simonovits 1995

* 7( )O n

achieving isotropic positionKannan-LL-Simonovits 1998

* 5( )O nFunctional analysis:isotropic position ofconvex bodies

Geometry:projective (Hilbert)distance

affin invariant isoperimetric inequalityanalysis if hit-and-run walkLL 1999

* 5( )O n

Differential equations:log-Sobolev inequality

elimination of “start penalty” forlattice walkFrieze-Kannan 1999

log-Cheeger inequality elimination of “start penalty” forball walkKannan-LL 1999

* 5( )O n

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

1. Highlights if the last 4 decades

2. 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 ?