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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
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 ?