Overview Resolutions Density Decomposing Flatening Model Checking Structural Sparsity Limits Structural Sparsity Jaroslav Nešetřil Patrice Ossona de Mendez Charles University CAMS, CNRS/EHESS LIA STRUCO Praha, Czech Republic Paris, France � 2 2 � � 2 � 2 — Melbourne — 2
Overview Resolutions Density Decomposing Flatening Model Checking Structural Sparsity Limits The Dense–Sparse Dichotomy
Overview Resolutions Density Decomposing Flatening Model Checking Structural Sparsity Limits Overview CC BY http://freeaussiestock.com/free/Northern_Territory/slides/desert_plain.htm, CC BY-SA Steven Gerner, CC BY Olga Berrios
Overview Resolutions Density Decomposing Flatening Model Checking Structural Sparsity Limits What is a Sparse Structure?
Overview Resolutions Density Decomposing Flatening Model Checking Structural Sparsity Limits Class Resolutions
� Overview Resolutions Density Decomposing Flatening Model Checking Structural Sparsity Limits Topological resolution of a class C Shallow topological minors at depth t : H G ≤ 2 t C � ▽ t = { H : some ≤ 2 t -subdivision of H is a subgraph of some G ∈ C} . Topological resolution : C ⊆ C � ▽ 0 ⊆ C � ▽ 1 ⊆ . . . ⊆ C � ▽ t ⊆ . . . ⊆ C � ▽ ∞ time
Overview Resolutions Density Decomposing Flatening Model Checking Structural Sparsity Limits The Somewhere dense — Nowhere dense dichotomy A class C is somewhere dense if there exists τ such that C � ▽ τ contains all graphs. ( ∃ τ ) ω ( C � ⇐ ⇒ ▽ τ ) = ∞ . A class C is nowhere dense otherwise. ( ∀ τ ) ω ( C � ⇐ ⇒ ▽ τ ) < ∞ .
Overview Resolutions Density Decomposing Flatening Model Checking Structural Sparsity Limits Every kind of shallow minors Topological minor Minor Immersion ≤ t ≤ 2 t ≤ 2 t ≤ s + 1
Overview Resolutions Density Decomposing Flatening Model Checking Structural Sparsity Limits Class Taxonomy d χ ω Minors Topological Bounded Nowhere minors expansion dense Immersions Definition
Overview Resolutions Density Decomposing Flatening Model Checking Structural Sparsity Limits Class Taxonomy d χ ω Bounded Bounded Nowhere Minors expansion expansion dense Topological Bounded Bounded Nowhere minors expansion expansion dense Bounded Bounded Nowhere Immersions expansion expansion dense Theorem (Nešetřil, POM 2012)
Overview Resolutions Density Decomposing Flatening Model Checking Structural Sparsity Limits The Nowhere Dense World Almost wide Nowhere dense Excluded Bounded Locally bounded topological minor expansion expansion Locally excluded Excluded minor minor Locally bounded Bounded genus tree-width Bounded degree Planar More. . .
Overview Resolutions Density Decomposing Flatening Model Checking Structural Sparsity Limits Density
Overview Resolutions Density Decomposing Flatening Model Checking Structural Sparsity Limits What is unavoidable in dense graphs? Theorem (Erdős, Simonovits, Stone 1966) � � � n � 1 + o ( n 2 ) . ex( n, H ) = 1 − χ ( H ) − 1 2 Theorem (Bukh, Jiang 2016) � k log k n 1+ 1 k + O ( n ) . ex( n, C 2 k ) ≤ 80 Theorem (Jiang 2010) ) = O ( n 1+ 10 ex( n, K ( ≤ p ) p ) . t
� � Overview Resolutions Density Decomposing Flatening Model Checking Structural Sparsity Limits Concentration Theorem (Jiang 2010) ) = O ( n 1+ 10 ex( n, K ( ≤ p ) p ) . t ▽ 10 t C ⊆ C � ▽ 0 ⊆ . . . ⊆ C � ▽ t ⊆ . . . ⊆ C � ⊆ . . . ⊆ C � ▽ ∞ ǫ � G � > C t | G | 1+ ǫ K t � G � = number of edges | G | = number of vertices Hence: log � G � log � G � lim sup log | G | > 1 + ǫ = ⇒ lim sup log | G | = 2 . ▽ 10 t G ∈C � G ∈C � ▽ t ǫ
Overview Resolutions Density Decomposing Flatening Model Checking Structural Sparsity Limits Classification by logarithmic density Theorem (Class trichotomy — Nešetřil and POM) Let C be an infinite class of graphs. Then log � G � log | G | ∈ {−∞ , 0 , 1 , 2 } . sup lim sup t G ∈C � ▽ t • bounded size class ⇐ ⇒ −∞ or 0 ; • nowhere dense class ⇐ ⇒ −∞ , 0 or 1 ; • somewhere dense class ⇐ ⇒ 2 .
Overview Resolutions Density Decomposing Flatening Model Checking Structural Sparsity Limits Decomposing
Overview Resolutions Density Decomposing Flatening Model Checking Structural Sparsity Limits Tree-depth Definition The tree-depth td( G ) of a graph G is the minimum height of a rooted forest Y s.t. G ⊆ Closure( Y ) . td( P n ) = log 2 ( n + 1)
Overview Resolutions Density Decomposing Flatening Model Checking Structural Sparsity Limits Low tree-depth decompositions χ p ( G ) is the minimum of colors such that every subset I of ≤ p colors induces a subgraph G I so that td( G I ) ≤ | I | . Theorem (Nešetřil and POM; 2006, 2010) ∀ p, sup χ p ( G ) < ∞ ⇐ ⇒ C has bounded expansion. G ∈C log χ p ( G ) ∀ p, lim sup = 0 ⇐ ⇒ C is nowhere dense. log | G | G ∈C (extends DeVos, Ding, Oporowski, Sanders, Reed, Seymour, Vertigan on low tree-width decomposition of proper minor closed classes, 2004)
� � Overview Resolutions Density Decomposing Flatening Model Checking Structural Sparsity Limits Logarithmic density (again) Theorem (Nešetřil and POM) Somewhere dense = | F | log(# F ⊆ G ) ∀ F : sup lim sup log | G | t G ∈C � ▽ t ∈{−∞ , 0 , 1 ,...α ( F ) } Nowhere dense Remark Proof based on Low Tree-Depth Decompositions and regularity properties of bounded height trees. Details
Overview Resolutions Density Decomposing Flatening Model Checking Structural Sparsity Limits Flatening
Overview Resolutions Density Decomposing Flatening Model Checking Structural Sparsity Limits Quasi-wide classes A class C of graphs is quasi-wide if ∀ d ∃ s ∀ m ∃ N : ∀ G ∈ C , | G | ≥ N , ∃ S, A ⊆ V ( G ) with • | S | ≤ s, | A | ≥ m , • ∀ x � = y ∈ A \ S, dist G − S ( x, y ) > d . − →
Overview Resolutions Density Decomposing Flatening Model Checking Structural Sparsity Limits Quasi-wide classes A class C of graphs is quasi-wide if ∀ d ∃ s ∀ m ∃ N : ∀ G ∈ C , | G | ≥ N , ∃ S, A ⊆ V ( G ) with • | S | ≤ s, | A | ≥ m , • ∀ x � = y ∈ A \ S, dist G − S ( x, y ) > d . Theorem (Nešetril and POM) A hereditary class of graphs is quasi-wide if and only if it is nowhere dense.
Overview Resolutions Density Decomposing Flatening Model Checking Structural Sparsity Limits r -neighbourhood covers 2 r r
Overview Resolutions Density Decomposing Flatening Model Checking Structural Sparsity Limits r -neighbourhood covers Theorem (Grohe, Kreutzer, Siebertz 2013) For every C nowhere dense (resp. bounded expansion) class of graphs there is f s.t. ∀ r ∈ N , ǫ > 0 , and G ∈ C with | G | ≥ f ( r, ǫ ) there exists a family X of subgraphs of G s.t. • the maximum radius of H ∈ X is ≤ 2 r ; • every v ∈ G has all its r -neighborhood in some H ∈ X ; • every v ∈ G belongs to at most | G | ǫ (resp. K ( C , r, ǫ ) ) subgraphs in X . Remark Actually a characterization of nowhere dense and bounded expansion monotone classes.
Overview Resolutions Density Decomposing Flatening Model Checking Structural Sparsity Limits Model Checking
Overview Resolutions Density Decomposing Flatening Model Checking Structural Sparsity Limits Model checking Theorem (Dvořák, Kráľ, Thomas 2010) For every class C with bounded expansion, every property of graphs definable in first-order logic can be decided in time O ( n ) on C . Theorem (Kazana, Segoufin 2013) For every class C with bounded expansion, every first-order definable subset can be enumerated in lexicographic order in constant time between consecutive outputs and linear time preprocessing time.
Overview Resolutions Density Decomposing Flatening Model Checking Structural Sparsity Limits Model checking Theorem (Grohe, Kreutzer, Siebertz 2014) For every nowhere dense class C and every ǫ > 0 , every property of graphs definable in first-order logic can be decided in time O ( n 1+ ǫ ) on C . Theorem (Dvořák, Kráľ, Thomas 2010; Kreutzer 2011) if a monotone class C is somewhere dense, then deciding first- order properties of graphs in C is not fixed-parameter tractable (unless FPT = W[1] . Remark Hence a characterization of nowhere dense/somewhere dense dichotomy in terms of algorithmic complexity.
Overview Resolutions Density Decomposing Flatening Model Checking Structural Sparsity Limits Excluded Structures
Overview Resolutions Density Decomposing Flatening Model Checking Structural Sparsity Limits Nowhere dense topological minor At each depth, an excluded minor immersion
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