discrete mathematics: the last and next decade lászló lovász microsoft research one microsoft...
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Approximation algorithms: The Max Cut Problem maximize NP-hard …Approximations?TRANSCRIPT
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Discrete mathematics:the last and next decade
László Lovász
Microsoft Research
One Microsoft Way, Redmond, WA 98052
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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
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Approximation algorithms:The Max Cut Problem
maximize
NP-hard
…Approximations?
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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)
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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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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
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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 concentratedGeneral strong concentration: very general “smooth” functions of independent random variables are strongly concentrated
Nible, martingales, rapidly mixing Markov chains,…
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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
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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
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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
-genetic code
-brain
-animal
-ecosystem
-economy
-society
How to model these?
non-constant but stablepartly random
-internet
-VLSI
-databases
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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.
up to a bounded number of additional nodes
tree-decomposition
embeddable in a fixed surfaceexcept for “fringes” of bounded depth
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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 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
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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)
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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,…
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Steinitz
Every 3-connected planar graphis the skeleton of a polytope.
3-connected planar graph
Example 1: Geometric representations of graphs
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Coin representation
Every planar graph can be represented by touching circles
Koebe (1936)
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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”
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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
Sullivan
Koebe
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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
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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
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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
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Cage representation Riemann Mapping Theorem
Sullivan
Koebe
Nullspace representationfrom the CdV matrix ~
eigenfunctions of theLaplacian
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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.
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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
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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
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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
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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
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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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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
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Optimization:
discrete linear semidefinite ?