Hyperfinite graphs and combinatorial optimization László Lovász Hungarian Academy of Sciences and Eötvös Loránd University, Budapest August 2017

Graphing, definition Bounded degree ( D) Borel graph on V=[0,1] with “measure-preserving” condition: Extends to measure  on Borel subsets of [0,1]2 „edge measure” August 2017

Graphing, examples unit circumference  irrational components: 2-way infinite paths August 2017

Graphing, examples unit circumference 1x1 torus , irrational components: grids   components: grids August 2017

Graphing, examples ... ... ... ... 1 ... 1 ... connected to 1 ... 1 ... connected to August 2017

Local equivalence, definition xV  Gx: component of x uniform random x  Gx: random connected rooted (countable) graph = unimodular random network G1, G2 locally equivalent: distributions of (G1)x and (G2)x are the same August 2017

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. August 2017

Local isomorphism, example   (x,y)  x+y mod 1    components: grids components: grids August 2017

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 August 2017

Graph partition problem k-edge-separator: TE(G), component of G-T has  k nodes August 2017

Hyperfinite (amenable) graphings Graphing G hyperfinite: sepk(G)0 (k)  k nodes (T) small August 2017

Hyperfinite graphings, examples 1 ... not hyperfinite     August 2017

? Hyperfinite graphings If G1 and G2 are locally equivalent, then G1 is hyperfinite  G2 is hyperfinite G1 G2 G ? August 2017

Local isomorphism forward  (x,y)  x+y mod 1     August 2017

Fractional graph partition problem T: optimal k-edge-separator August 2017

Fractional graph partition problem probability distribution „marginal” uniform expected expansion August 2017

Fractional graph partition problem Define Can be defined for graphings  probability distribution on Rk with uniform marginal no dependence on k August 2017

Hyperfinite graphings If G1 and G2 are locally equivalent, then G1 is hyperfinite  G2 is hyperfinite G1 G2 G August 2017

Hyperfinite graphings    August 2017

Algorithm: For j=1,2,..., select Y1,Y2,... so that Proof sketch Algorithm: For j=1,2,..., select Y1,Y2,... so that Yj is the minimizer of Output: X=Y1 Y2 ... On a graphing: no uncountable sequence of steps! Phases... August 2017

Fractional separation If G1 and G2 are locally equivalent, then G1 G2 easy ? Duality! August 2017

Fractional separation, duality in finite case Linear program: Dual program: August 2017

Fractional separation, duality in infinite case August 2017

Fractional separation, duality in infinite case Duality - Hahn-Banach + Riesz Representation Compactification of graphings August 2017

(I,A): standard Borel space : probability measure Compact graphings Graphing: (I,A,,E) (I,A): standard Borel space : probability measure E: symmetric Borel subset of IxI with measure-preserving condition August 2017

Compact graphing: (J,A,,E) J: compact metric space A: Borel sets of J Compact graphings Compact graphing: (J,A,,E) J: compact metric space A: Borel sets of J : probability measure E: symmetric Borel subset of IxI with measure-preserving condition, and August 2017

Compact graphings     not compact compact August 2017

Every graphing can be obtained from a Compact graphings Every graphing can be obtained from a compact graphing by deleting components of total measure zero. August 2017

Fractional separation, duality in infinite case August 2017

co-NP characterization of hyperfiniteness Graphing G is not hyperfinite  August 2017

Fractional separation If G1 and G2 are locally equivalent, then sep*(G1) = sep*(G2) G1 G2 easy ? August 2017

Pushing forward and pulling back measure preserving subset linear relaxation measure function August 2017

Hyperfinite (amenable) graph families Family G of graphs is hyperfinite: Hyperfinite: paths, trees, planar graphs, every non-trivial minor-closed property Non-hyperfinite: expanders August 2017

Hyperfinite graph sequences Every locally convergent hyperfinite graph sequence is locally-globally convergent. Elek Hatami – L – Szegedy Every property of hyperfinite graphs is testable. Newman – Sohler (Benjamini-Schramm-Shapira, Elek) August 2017

Hyperfinite (amenable) graph families O(1) nodes o(n) edges August 2017

{Gn} is hyperfinite  G is hyperfinite If Gn  G locally, then {Gn} is hyperfinite  G is hyperfinite Schramm Benjamini-Shapira-Schramm August 2017

Thanks, that’s all! August 2017

Combopt generalization (V,H): hypergraph on n vertices, x {x}H w: HR+, w({x})  1 August 2017

Combopt generalization August 2017