Discrete mathematics: the last and next decade László Lovász Microsoft Research One Microsoft Way, Redmond, WA

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 concentrated General strong concentration: very general “smooth” functions of independent random variables are strongly concentrated Nible, martingales, rapidly mixing Markov chains,…

Example 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 q has a complete arc of size  q polylog ( q ). Kim-Vu

Second idea: choose 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 stable partly random -internet -VLSI -databases

Very large structures: how to model them? Graph minorsRobertson, 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 surface except for “fringes” of bounded depth

Very large structures: how to model them? Regularity LemmaSzeméredi 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). given  >0 and k>1, the number of parts is between k and f(k,  ) difference at most 1 with  k 2 exceptions for subsets X,Y of the two parts, # of edges between X and Y is p|X||Y|   n 2

How to model these? How to handle them algorithmically? 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 graph is 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’s Van der Holst, elementary proof = 3 if G is 3 -connected Representation of G in R 3 basis of nullspace of M may assume second largest eigenvalue is 0

G 3-connected planar  nullspace representation gives planar embedding in S 2 L-Schrijver The vectors can be rescaled so that we get a Steinitz representation. LL

Cage representation  Riemann Mapping Theorem  Sullivan  Koebe Nullspace representation from the CdV matrix ~ eigenfunctions of the Laplacian

Example 2: volume computation Given:, convex Want: volume of K by a membership oracle; with relative error ε Not possible in polynomial time, even if ε=n cn. 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 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 Dyer Frieze Kannan 1989 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 achieving isotropic position Kannan-LL-Simonovits 1998 Functional analysis: isotropic position of convex bodies

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 inequalityelimination 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

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  ?