On metric embeddings, shortest path decompositions and face cover of planar graphs Arnold Filtser Ben-Gurion University Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 1 / 34 This talk is based on the following papers: Metric Embedding via Shortest Path Decompositions Ittai Abraham, Arnold Filtser, Anupam Gupta, Ofer Neiman. A face cover perspective to `1 embeddings of planar graphs Arnold Filtser Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 2 / 34 Theorem (Bourgain 85) Every n-point metric (X ; dX ) is embeddable into Euclidean space with distortion O(log n). Tight. Metric Embeddings (X; dX) (Rd; k·k ) f : X ! Rd p Embedding f ∶ X → Rd has distortion t if for all x; y ∈ X dX (x; y) ≤ Yf (x) − f (y)Yp ≤ t ⋅ dX (x; y) Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 3 / 34 Tight. Metric Embeddings (X; dX) (Rd; k·k ) f : X ! Rd p Embedding f ∶ X → Rd has distortion t if for all x; y ∈ X dX (x; y) ≤ Yf (x) − f (y)Yp ≤ t ⋅ dX (x; y) Theorem (Bourgain 85) Every n-point metric (X ; dX ) is embeddable into Euclidean space with distortion O(log n). Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 3 / 34 Metric Embeddings (X; dX) (Rd; k·k ) f : X ! Rd p Embedding f ∶ X → Rd has distortion t if for all x; y ∈ X dX (x; y) ≤ Yf (x) − f (y)Yp ≤ t ⋅ dX (x; y) Theorem (Bourgain 85) Every n-point metric (X ; dX ) is embeddable into Euclidean space with distortion O(log n). Tight. Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 3 / 34 NP-hard. Sparsest Cut Problem G = (V ; E; w) is a weighted graph. The sparsity S ⊆ V : ¯ wE(S; S) S φ( ) = ¯ min SSS; SSS Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 4 / 34 NP-hard. Sparsest Cut Problem G = (V ; E; w) is a weighted graph. The sparsity S ⊆ V : ¯ wE(S; S) S φ( ) = ¯ min SSS; SSS S 1 1 3 15 2 φ(S) = 8 2 3 2 1 S¯ Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 4 / 34 NP-hard. Sparsest Cut Problem G = (V ; E; w) is a weighted graph. The sparsity S ⊆ V : ¯ wE(S; S) S φ( ) = ¯ min SSS; SSS S 1 1 3 15 2 φ(S) = 8 2 3 2 1 S¯ Sparsest Cut Problem Find the cut with minimum sparsity minS⊊V φ(S). Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 4 / 34 Sparsest Cut Problem G = (V ; E; w) is a weighted graph. The sparsity S ⊆ V : ¯ wE(S; S) S φ( ) = ¯ min SSS; SSS S 1 1 3 15 2 φ(S) = 8 2 3 2 1 S¯ Sparsest Cut Problem Find the cut with minimum sparsity minS⊊V φ(S). NP-hard. Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 4 / 34 Sparsest Cut Problem G = (V ; E; w) is a weighted graph. The sparsity S ⊆ V : ¯ wE(S; S) S φ( ) = ¯ min SSS; SSS Sparsest Cut Problem Find the cut with minimum sparsity minS⊊V φ(S). NP-hard. d G embeds into `1 = (R ; Y ⋅ Y1) t-approximation for with distortion t. ⇒ sparsest cut. Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 4 / 34 Planar. Excluding a fixed minor. Bounded treewidth. Bounded Pathwidth. Try special graph families: Special Graph Families Bourgain's log n distortion: best possible for general metric spaces. Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 5 / 34 Planar. Excluding a fixed minor. Bounded treewidth. Bounded Pathwidth. Special Graph Families Bourgain's log n distortion: best possible for general metric spaces. Try special graph families: Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 5 / 34 Excluding a fixed minor. Bounded treewidth. Bounded Pathwidth. Special Graph Families Bourgain's log n distortion: best possible for general metric spaces. Try special graph families: Planar. Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 5 / 34 Bounded treewidth. Bounded Pathwidth. Special Graph Families Bourgain's log n distortion: best possible for general metric spaces. Try special graph families: Planar. Excluding a fixed minor. Excluded as a minor Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 5 / 34 Bounded Pathwidth. Special Graph Families Bourgain's log n distortion: best possible for general metric spaces. Try special graph families: Planar. Excluding a fixed minor. Bounded treewidth. 3 1 1 3 4 5 7 5 9 1 1 3 4 6 10 6 8 1 1 1 2 9 11 2 3 4 2 3 4 2 2 7 11 2 2 10 12 3 4 8 12 4 Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 5 / 34 Special Graph Families Bourgain's log n distortion: best possible for general metric spaces. Try special graph families: Planar. 3 Excluding a fixed minor. 5 7 Bounded treewidth. Bounded Pathwidth. 6 8 1 2 9 11 10 12 4 1 1 1 1 1 1 1 1 2 2 2 2 3 3 4 4 3 3 4 4 5 6 9 9 7 8 11 12 Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 5 / 34 G has an SPD of depth k if after removing some shortest path P, every connected component in GP has an SPD of depth k − 1. Shortest Path Decompositions Definition (Depth of a Shortest Path Decompositions) Every (weighted) path graph has an SPD of depth 1. Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 6 / 34 Shortest Path Decompositions Definition (Depth of a Shortest Path Decompositions) Every (weighted) path graph has an SPD of depth 1. G has an SPD of depth k if after removing some shortest path P, every connected component in GP has an SPD of depth k − 1. Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 6 / 34 Shortest Path Decompositions Definition (Depth of a Shortest Path Decompositions) Every (weighted) path graph has an SPD of depth 1. G has an SPD of depth k if after removing some shortest path P, every connected component in GP has an SPD of depth k − 1. The diamond graph D4 and its SPD . The SPD depth is 4. Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 6 / 34 Shortest Path Decompositions Definition (Depth of a Shortest Path Decompositions) Every (weighted) path graph has an SPD of depth 1. G has an SPD of depth k if after removing some shortest path P, every connected component in GP has an SPD of depth k − 1. Planar. G Planar SPDdepth O(log n). (Cycle separator). ⇒ Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 6 / 34 Shortest Path Decompositions Definition (Depth of a Shortest Path Decompositions) Every (weighted) path graph has an SPD of depth 1. G has an SPD of depth k if after removing some shortest path P, every connected component in GP has an SPD of depth k − 1. Planar. G Planar SPDdepth O(log n). (Cycle separator). Minor-free. G excludes H as a minor ⇒ SPDdepth O(g(H) ⋅ log n). ⇒ Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 6 / 34 Shortest Path Decompositions Definition (Depth of a Shortest Path Decompositions) Every (weighted) path graph has an SPD of depth 1. G has an SPD of depth k if after removing some shortest path P, every connected component in GP has an SPD of depth k − 1. Planar. G Planar SPDdepth O(log n). (Cycle separator). Minor-free. G excludes H as a minor ⇒ SPDdepth O(g(H) ⋅ log n). ⇒ Treewidth. G has treewidth-k SPDdepth O(k log n). ⇒ Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 6 / 34 Shortest Path Decompositions Definition (Depth of a Shortest Path Decompositions) Every (weighted) path graph has an SPD of depth 1. G has an SPD of depth k if after removing some shortest path P, every connected component in GP has an SPD of depth k − 1. Planar. G Planar SPDdepth O(log n). (Cycle separator). Minor-free. G excludes H as a minor ⇒ SPDdepth O(g(H) ⋅ log n). ⇒ Treewidth. G has treewidth-k SPDdepth O(k log n). Pathwidth. G has pathwidth-k SPDdepth k + 1. ⇒ 1 1 1 1 1 1 1 1 2 2 ⇒ 2 2 3 3 4 4 3 3 4 4 5 6 9 10 7 8 11 12 Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 6 / 34 Shortest Path Decompositions There is a graph with SPDdepth 2, which contain Kn as a minor. n n 1 1 1 1 1 1 1 1 n Shortest Path Decompositions There is a graph with SPDdepth 2, which contain Kn as a minor. n n 1 1 1 1 1 1 1 1 n Shortest Path Decompositions There is a unweighted graph with SPDdepth 3, containing Kn. n 1 1 1 1 1 1 1 1 Main Result Theorem (Embeddings by SPDdepth [Abraham, F, Gupta, Neiman 18]) Let G = (V ; E) be a weighted graph with SPDdepth k.