Hyperfinite graphs and combinatorial optimization László Lovász Hungarian Academy of Sciences and Eötvös Loránd University, Budapest January 2018

Don in Epidauros 1995 January 2018

Bounded degree ( D) Borel graph Graphing, definition Bounded degree ( D) Borel graph on standard probability space V (say [0,1]) with “double counting” condition: Extends to measure  on Borel subsets of V2. „edge measure” January 2018

„The notion of an algorithm is basic to Why graphings? „The notion of an algorithm is basic to all of computer programming, so we should begin with a careful analysis of this concept.” Donald Knuth January 2018

Find a perfect matching in G. Why graphings? Find a perfect matching in G. distributed algorithm: Nguyen and Onak January 2018

distributed algorithms Why graphings? distributed algorithms local algorithms (on bounded degree graphs) January 2018

distributed algorithms Why graphings? distributed algorithms local algorithms (on bounded degree graphs) property testing (on bounded degree graphs) January 2018

distributed algorithms Why graphings? distributed algorithms local algorithms (on bounded degree graphs) property testing (on bounded degree graphs) local algorithms on graphings The infinite is a good approximation of the large finite. January 2018

Why graphings? Ergodic theory Finitely generated groups Limits of convergent graphs sequences with bounded degree January 2018

Graphing, examples January 2018

Graphing, examples unit circumference  irrational components: 2-way infinite paths January 2018

, irrational, lin. indep 1x1 torus , irrational Graphing, examples unit circumference , irrational, lin. indep   1x1 torus , irrational components: grids   components: grids January 2018

Penrose tilings rhombic icosahedron de Bruijn - Bárász January 2018

Local equivalence, definition xV  Gx: component of x uniform random x  Gx: random connected rooted (countable) graph unimodular random network Benjamini - Schramm January 2018

Local equivalence, definition xV  Gx: component of x G1, G2 locally equivalent: for random uniform xV, distributions of (G1)x and (G2)x are the same January 2018

Local isomorphism, definition  : V(G1)  V(G2) local isomorphism: measure preserving and (x) isomorphism between (G1)x and (G2)(x) Existence of local isomorphism proves local equivalence. January 2018

Local isomorphism, example   (x,y)  x+y mod 1    components: grids components: grids January 2018

G1 and G2 are locally equivalent  Local equivalence G1 and G2 are locally equivalent  G and local isomorphisms GG1, GG2. G1 G2 G January 2018

Graph partition problem k-edge-separator: TE(G), component of G-T has  k nodes Graphing G hyperfinite: sepk(G)0 (k) January 2018

Hyperfinite graphings, examples   Diophantine approximation   January 2018

Hyperfinite graph families Family G of finite graphs is hyperfinite: Hyperfinite: paths, trees, planar graphs, every non-trivial minor-closed property Non-hyperfinite: expanders January 2018

Every graph property is testable in any family of hyperfinite graphs. Hyperfinite graph families Every graph property is testable in any family of hyperfinite graphs. Many graph properties are polynomial time testable for graphs with bounded tree-width. Newman – Sohler (Benjamini-Schramm-Shapira, Elek) January 2018

{Gn} is hyperfinite  G is hyperfinite Hyperfinite graph sequences If Gn  G locally, then {Gn} is hyperfinite  G is hyperfinite Schramm (Benjamini-Shapira-Schramm) January 2018

? Hyperfinite graphings If G1 and G2 are locally equivalent, then G1 is hyperfinite  G2 is hyperfinite G1 G2 G ? January 2018

Local isomorphism forward Diophantine approximation  (x,y)  x+y mod 1     January 2018

Pushing forward and pulling back measure preserving subset linear relaxation measure January 2018

Fractional graph partition problem T: optimal k-edge-separator January 2018

Fractional graph partition problem probability distribution „marginal” uniform expected expansion January 2018

Fractional graph/graphing partition problem Define Can be defined for graphings  probability distribution on Rk with uniform marginal no dependence on k January 2018

Hyperfinite graphings If G1 and G2 are locally equivalent, then G1 is hyperfinite  G2 is hyperfinite G1 G2 G January 2018

Algorithm: For j=1,2,..., select Y1,Y2,...k so that Proof sketch Algorithm: For j=1,2,..., select Y1,Y2,...k so that Yj is the minimizer of Output: X=Y1 Y2 ... On a graphing: no uncountable sequence of steps! Phases... January 2018

Fractional separation duality If G1 and G2 are locally equivalent, then G1 G2 easy ? Duality! January 2018

Fractional separation duality Hahn-Banach + Riesz Representation January 2018

Fractional separation duality January 2018

Thanks, that’s all! January 2018