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Polynomial Bounds for the Grid-Minor Theorem Julia Chuzhoy Toyota Technological Institute at Chicago Chandra Chekuri University of Illinois at Urbana-Champaign
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Grid Minor Theorem (Excluded Grid Theorem) [Robertson, Seymour ‘86] Graph Minor Theory [Robertson – Seymour] – Wagner’s conjecture: any infinite sequence of finite graphs contains two graphs G,G’ where G is a minor of G’ – Grid-Minor Theorem: if the treewidth of G is large, then G contains a large grid minor
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Grid Minor Theorem (Excluded Grid Theorem) [Robertson, Seymour ‘86] Graph Minor Theory [Robertson – Seymour] – Wagner’s conjecture: any infinite sequence of finite graphs contains two graphs G,G’ where G is a minor of G’ – Grid-Minor Theorem: if the treewidth of G is large, then G contains a large grid minor
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Treewidth Trees General Graphs
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Tree Decomposition d e b c a f g h Example from Bodlaender’s talk
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g h g a f a f c c d e Tree Decomposition d e b c a f g h b c a Example from Bodlaender’s talk
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c d e Tree Decomposition d e b c a f g h a f g h c g a f b c a Example from Bodlaender’s talk
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c d e Tree Decomposition d e b c a f g h a f g h c g a f b c a Example from Bodlaender’s talk
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c d e Tree Decomposition d e b c a f g h a f g h c g a f b c a Example from Bodlaender’s talk
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c d e Tree Decomposition d e b c a f g h a f g h c g a f b c a Example from Bodlaender’s talk
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c d e Tree Decomposition d e b c a f g h a f g h c g a f b c a Example from Bodlaender’s talk
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c d e Tree Decomposition d e b c a f g h a f g h c g a f b c a Example from Bodlaender’s talk Decomposition width = max # of vertices in a bag -1 Treewidth: min width of any decomposition
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Treewidth of Some Graphs Tree: 1 Cycle: 2 (√n×√n)-grid: √n n-vertex expander: Ω(n)
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Well-Linkedness
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A set T of vertices is well-linked in G iff for any two equal- sized subsets A,B of T, we can connect A to B with |A| disjoint paths.
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Treewidth and Well-Linkedness Thm. Let k be the maximum size of any well- linked set of vertices in G. Then: k≤treewidth(G)≤4k.
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Treewidth Trees Small-Treewidth Graphs Large-Treewidth Graphs
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Grid-Minor Theorem [Robertson, Seymour] If the treewidth of G is large, then it contains a large grid minor.
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Grid-Minor Theorem [Robertson, Seymour] If the treewidth of G is large, then it contains a large grid minor. We can obtain the grid from G by a sequence of edge-deletion and edge-contraction operations a size-4 grid
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Minors by Embedding
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Grid-Minor Theorem [Robertson, Seymour] If the treewidth of G is large, then it contains a large grid minor, so: G contains many disjoint cycles G contains many disjoint cycles of length 0 mod m G contains a convenient routing structure The size of the vertex cover in G is large …
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Applications Fixed parameter tractability Erdos-Posa type results Graph minor theory …
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If the treewidth of G is large, then it contains a large grid minor. Grid-Minor Theorem
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If the treewidth of G is k, then it contains a grid minor of size f(k). Easy to see that [Robertson, Seymour ‘94]: Conjecture [Robertson, Seymour ‘94]: How large is f(k)?
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Grid-Minor Theorem If the treewidth of G is k, then it contains a grid minor of size f(k). [Robertson, Seymour, Thomas ‘89] : [Diestel, Gorbunov, Jensen, Thomassen ‘99] – simpler proof [Kawarabayashi, Kobayashi ‘12], [Leaf, Seymour ‘12]: This talk:
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Grid-Minor Theorem If the treewidth of G is k, then it contains a grid minor of size f(k). In some families of graphs f(k)=Ω(k) – Planar graphs [Robertson, Seymour, Thomas ‘94] – Bounded genus graphs [Demaine, Fomin, Hajiaghayi, Thilikos ‘05] – Graphs excluding a fixed minor [Demaine, Hajiaghayi ‘08]
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Path-of-Sets System
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A Path-of-Sets System C1C1 C2C2 C3C3 …ChCh Each C i is a connected cluster The clusters are disjoint Every consecutive pair of clusters connected by h paths All blue paths are disjoint from each other and internally disjoint from the clusters …
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A Path-of-Sets System C1C1 C2C2 C3C3 …ChCh Each C i is a connected cluster The clusters are disjoint Every consecutive pair of clusters connected by h paths All blue paths are disjoint from each other and internally disjoint from the clusters h …
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A Path-of-Sets System C1C1 C2C2 C3C3 …ChCh … CiCi Interface vertex The interface vertices are well-linked inside C i
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A Path-of-Sets System C1C1 C2C2 C3C3 …ChCh … CiCi The interface vertices are well-linked inside C i
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A Path-of-Sets System C1C1 C2C2 C3C3 …ChCh CiCi The interface vertices are well-linked inside C i
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A Path-of-Sets System C1C1 C2C2 C3C3 …ChCh The interface vertices are well-linked inside C i CiCi
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A Path-of-Sets System C1C1 C2C2 C3C3 …ChCh The interface vertices are well-linked inside C i CiCi
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A Path-of-Sets System C1C1 C2C2 C3C3 …ChCh h … Thm [Leaf, Seymour ‘12]: Given a path-of-sets system, we can efficiently find a grid minor of size Ω(√h). Corollary: enough to find a path-of-sets system with h=poly(k), where k is the treewidth.
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From Path-of-Sets System to Grid Minor
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Building the Grid
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C1C1 C1C1 C2C2 C2C2 C3C3 C3C3 C4C4 C4C4 C1C1 C2C2 C3C3 …ChCh P1P1 P2P2 P3P3 PhPh … …
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Direct vs Indirect Path Direct path Indirect path
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Building the Grid C1C1 C2C2 C3C3 …ChCh C1C1 C1C1 C2C2 C2C2 C3C3 C3C3 C4C4 C4C4 For each C i, we’ll be looking for a direct path connecting some consecutive pair of horizontal paths C1C1 C2C2 C3C3 …ChCh P1P1 P2P2 P3P3 PhPh … …
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Routing Inside Clusters CiCi P1P1 P2P2 P3P3 P4P4
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CiCi P1P1 P2P2 P3P3 P4P4 P1P1 P2P2 P3P3 P4P4 Path graph H i for C i
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Routing Inside Clusters CiCi P1P1 P2P2 P3P3 P4P4 P1P1 P2P2 P3P3 P4P4 Path graph H i for C i
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Routing Inside Clusters CiCi P1P1 P2P2 P3P3 P4P4 P1P1 P2P2 P3P3 P4P4 Path graph H i for C i
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Routing Inside Clusters P1P1 P2P2 P3P3 P4P4 Good scenario: The path graph for all C i contains the same path P1P1 P2P2 P3P3 P4P4 “Bad” scenario:
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Routing Inside Clusters CiCi P1P1 P2P2 P3P3 P4P4 P1P1 P2P2 P3P3 P4P4 Path graph H i for C i
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Inside the Super-Clusters Thm: for any n-vertex graph G, Either there is a tree in G with Ω(√n) leaves Or there is a 2-path in G of length Ω(√n)
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Inside the Super-Clusters Thm: for any n-vertex graph G, Either there is a tree in G with Ω(√n) leaves Or there is a 2-path in G of length Ω(√n)
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Routing Inside Clusters CiCi P1P1 P2P2 P3P3 P4P4 P1P1 P2P2 P3P3 P4P4 Path graph H i for C i Cluster C i is good if H i has a tree with √h leaves. Assume all clusters are good.
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Routing Inside Clusters CiCi P1P1 P2P2 P3P3 P4P4 P1P1 P2P2 P3P3 P4P4 Path graph H i for C i
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Routing Inside Clusters CiCi P1P1 P2P2 P3P3 P4P4 P1P1 P2P2 P3P3 P4P4 Path graph H i for C i
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Routing Inside Clusters CiCi P1P1 P2P2 P3P3 P4P4 P1P1 P2P2 P3P3 P4P4 Path graph H i for C i We say that C i chooses the paths corresponding to the leaves of the tree.
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Routing Inside Clusters …
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…
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…
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If r is large enough, then some choice of √h will repeat h times. … r
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Routing Inside Clusters
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Re-connect the paths via even-indexed clusters, so all odd-indexed clusters choose the same paths!
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Completing the Proof Super- cluster
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Completing the Proof For each super-cluster S i : Either build a large grid minor inside S i Or show that S i is a good cluster
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Inside the Super-Clusters P1P1 P2P2 P3P3 P4P4 P1P1 P2P2 P3P3 P4P4 P1P1 P2P2 P3P3 P4P4 P1P1 P2P2 P3P3 P4P4 P1P1 P2P2 P3P3 P4P4 P1P1 P2P2 P3P3 P4P4 H: path-graph for the super- cluster
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Inside the Super-Clusters P1P1 P2P2 P3P3 P4P4 P1P1 P2P2 P3P3 P4P4 H: path-graph for the super- cluster Either H contains a tree with many leaves Or it contains a long 2-path Can build a grid-minor directly Either H contains a tree with many leaves Or it contains a long 2-path Can build a grid-minor directly P1P1 P2P2 P3P3 P4P4 H1H1 P1P1 P2P2 P3P3 P4P4 H2H2 P1P1 P2P2 P3P3 P4P4 H4H4 P1P1 P2P2 P3P3 P4P4 H3H3
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Inside the Super-Clusters P1P1 P2P2 P3P3 P4P4 H: path-graph for the super- cluster P1P1 P2P2 P3P3 P4P4 H1H1 P1P1 P2P2 P3P3 P4P4 H2H2 P1P1 P2P2 P3P3 P4P4 H4H4 P1P1 P2P2 P3P3 P4P4 H3H3 v1v1 v2v2 v3v3 v4v4 …v √h Want to show: this path appears in all H i ’s Will show: large sub-path appears in half the H i ’s Want to show: this path appears in all H i ’s Will show: large sub-path appears in half the H i ’s
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Inside the Super-Clusters P1P1 P2P2 P3P3 P4P4 H: path-graph for the super- cluster P1P1 P2P2 P3P3 P4P4 H1H1 v1v1 v2v2 v3v3 v4v4 …v √h
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Inside the Super-Clusters P1P1 P2P2 P3P3 P4P4 H: path-graph for the super- cluster P1P1 P2P2 P3P3 P4P4 H1H1 v1v1 v2v2 v3v3 v4v4 …v √h
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Inside the Super-Clusters P1P1 P2P2 P3P3 P4P4 H: path-graph for the super- cluster P1P1 P2P2 P3P3 P4P4 H1H1 P1P1 P2P2 P3P3 P4P4 H2H2 P1P1 P2P2 P3P3 P4P4 H4H4 P1P1 P2P2 P3P3 P4P4 H3H3 v1v1 v2v2 v3v3 v4v4 …v √h
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Completing the Proof For each super-cluster S i : Either build a large grid minor inside S i Or show that S i is a good cluster
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Finding the Path-of-Sets System C1C1 C2C2 C3C3 …ChCh …
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Routing Problems
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Node-Disjoint Paths (NDP) Input: Graph G, source-sink pairs (s 1,t 1 ),…,(s k,t k ). Goal: Route as many pairs as possible via node- disjoint paths
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Node-Disjoint Paths (NDP) Input: Graph G, source-sink pairs (s 1,t 1 ),…,(s k,t k ). Goal: Route as many pairs as possible via node- disjoint paths
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Node-Disjoint Paths (NDP) Input: Graph G, source-sink pairs (s 1,t 1 ),…,(s k,t k ). Goal: Route as many pairs as possible via node- disjoint paths Solution value: 2
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Edge-Disjoint Paths (EDP) Input: Graph G, source-sink pairs (s 1,t 1 ),…,(s k,t k ). Goal: Route as many pairs as possible via edge- disjoint paths
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Edge-Disjoint Paths (EDP) Input: Graph G, source-sink pairs (s 1,t 1 ),…,(s k,t k ). Goal: Route as many pairs as possible via edge- disjoint paths Solution value: 3
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Edge-Disjoint Paths (EDP) Input: Graph G, source-sink pairs (s 1,t 1 ),…,(s k,t k ). Goal: Route as many pairs as possible via edge- disjoint paths NDP is more general than EDP
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Edge-Disjoint Paths (EDP) Input: Graph G, source-sink pairs (s 1,t 1 ),…,(s k,t k ). Goal: Route as many pairs as possible via edge- disjoint paths n – number of graph vertices k – number of demand pairs terminals – vertices participating in the demand pairs n – number of graph vertices k – number of demand pairs terminals – vertices participating in the demand pairs
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EDP and NDP Efficient algorithm when k is constant [Robertson, Seymour ‘90]. – running time: f(k) n 2 [Kawarabayashi,Kobayashi, Reed] General k: both problems are NP-hard [Karp ’72] An α-approximation algorithm: efficient algorithm always produces solutions of value at least OPT/α. An α-approximation algorithm: efficient algorithm always produces solutions of value at least OPT/α.
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Approximation Algorithm [Kolliopoulos, Stein ‘98] While there is a path P connecting any demand pair that has not been routed yet: Add such a path of smallest length to the solution (Delete from OPT all paths sharing vertices with P or routing the same demand pair) Analysis If the length of P is less than – at most paths are deleted from OPT. If the length of P is more than – at most paths remain in OPT.
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Approximation Algorithm [Kolliopoulos, Stein ‘98] While there is a path P connecting any demand pair that has not been routed yet: Add such a path of smallest length to the solution (Delete from OPT all paths sharing vertices with P or routing the same demand pair) Analysis If the length of P is less than – at most paths are deleted from OPT. If the length of P is more than – at most paths remain in OPT. -approximation This algorithm gives an – approximation for EDP. An -approximation is known [Chekuri, Khanna, Shepherd ’06]. This algorithm gives an – approximation for EDP. An -approximation is known [Chekuri, Khanna, Shepherd ’06].
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Approximation Status of NDP -approximation algorithm – Even on planar graphs – Even on grid graphs -hardness of approximation for any[Andrews, Zhang ‘05], [Andrews, C, Guruswami, Khanna, Talwar, Zhang ’10]
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Open Problem: NDP on a Grid Graph
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s1s1 t1t1 s2s2 t2t2 s3s3 t3t3
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-approximation algorithm [Chekuri, Khanna, Shepherd ’06]. The problem is NP-hard Ongoing work: O(n 1/4 )-approximation [builds on Aggarwal, Kleinberg, Williamson ‘96] Hard to approximate up to some constant c.
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Open Problem: EDP on Wall Graphs
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EDP with Congestion (EDPwC) A factor- approximation algorithm with congestion c routes. demand pairs with congestion at most c. optimum number of pairs with no congestion allowed
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EDPwC Congestion O(log n/log log n): constant approximation [Raghavan, Thompson ’87] - approximation with congestion c [Azar, Regev ’01], [Baveja, Srinivasan ’00], [Kolliopoulos, Stein ‘04] polylog(n)-approximation with congestion poly(log log n) [Andrews ‘10] polylog(k)-approximation with congestion 14 [C, ‘11] polylog(k)-approximation with congestion 2 [C, Li, ‘12] polylog(k)-approximation with constant congestion for NDP [Chekuri, Ene ’13]
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Edge-Disjoint Paths Input: Graph G, source-sink pairs (s 1,t 1 ),…,(s k,t k ). Goal: Connect as many pairs as possible by edge-disjoint paths. An instance is well-linked iff the set of all terminals is well-linked in G. Theorem [Chekuri, Khanna Shepherd ‘04]: an α - approximation algorithm on well-linked instances gives an O(α log 2 k)-approximation on any instance. terminals
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If an instance is well-linked, its treewidth is Ω(k) If the treewidth of G is k, can find a well-linked set of size Algorithms for Edge-Disjoint Paths well-linked instance large crossbar find the routing graph of treewidth k “similar” to path-of-sets system
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Crossbar Number of clusters poly(log k), not poly(k) The paths are not disjoint from each other and from the clusters, but cause a constant edge congestion Want: Path-of-sets system Can get: Tree-of-sets system … degree-3 tree ✔ ✔
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Tree-of-Sets System A degree-3 tree with h vertices Every vertex a connected cluster of G Every edge – a collection of h paths in G – the blue paths are node-disjoint from each other and internally disjoint from the clusters For each cluster, its interface is well-linked. h=k ε If the tree has height h 1/3 – done Otherwise it has h 1/3 leaves Will build a path-of-sets system on a subset of h 1/3 leaves Assumption: the tree has h 1/3 leaves
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High-Level Idea
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Stage 1: connect every leaf to the root by many disjoint paths Stage 2: exploit these paths to build a path- of-sets system
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Stage 1 h 1/3 leaves h parallel blue edges each leaf gets h 3/4 green paths h 1/3 leaves h parallel blue edges each leaf gets h 3/4 green paths
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Stage 1 h 1/3 leaves h parallel blue edges each leaf gets h 3/4 green paths h 1/3 leaves h parallel blue edges each leaf gets h 3/4 green paths
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Stage 1 h 1/3 leaves h parallel blue edges each leaf gets h 3/4 green paths h 1/3 leaves h parallel blue edges each leaf gets h 3/4 green paths
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Stage 1 h 1/3 leaves h parallel blue edges each leaf gets h 2/3 green paths h 1/3 leaves h parallel blue edges each leaf gets h 2/3 green paths
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Stage 2 Every leaf receives h 2/3 flow units from the root Will exploit these flows to build a path-of-sets system Process the tree from top to bottom
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Stage 2
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AB
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AB
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A B C D X R
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A B A’ B’ C D C’ D’ X R
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Stage 2 A B A’ B’ C D C’ D’ h 1/3 blue paths intersect at most h 1/3 green paths from each set h 1/3 blue paths intersect at most h 1/3 green paths from each set R X
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Stage 2 A B A’ B’ C D C’ D’ B’ C C’ h 1/3 leaves each leaf had h 2/3 green paths want h 1/3 parallel paths in path-of-sets system tree height ≤ h 1/3 h 1/3 leaves each leaf had h 2/3 green paths want h 1/3 parallel paths in path-of-sets system tree height ≤ h 1/3 R X
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Proof Summary 1.Path-of-sets system gives a large grid minor [Leaf, Seymour ‘12] 2.If G has large treewidth, can build a large tree-of-sets system: extension of [C ‘11], [C, Li ‘12], [Chekuri, Ene ‘12] 3.Can build a path-of-sets system from a tree- of-sets system polylog(k)-approximation for Node-Disjoint Paths with congestion 2
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Bypassing the Grid-Minor Theorem?
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Large-Treewidth Graph Decomposition [C, Chekuri ‘12] treewidth ≥ r G Treewidth k G Treewidth k h
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Example of Use: Feedback Vertex Set Feedback Vertex Set: given a graph G, select a min-cardinality subset U of vertices, such that G\U has no cycles. k: size of feedback vertex set Want: a fixed-parameter tractable algorithm, with running time f(k) poly(n).
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The Algorithm If treewidth of G at most g(k) – dynamic programming on the tree decomposition – running time: otherwise: G contains a grid minor of size, so feedback vertex set value more than k. Can choose, running time. What is g(k)? Typical use of grid-minor theorem in fixed parameter tractability algorithms. Bi-dimentionality theory Typical use of grid-minor theorem in fixed parameter tractability algorithms. Bi-dimentionality theory
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If treewidth of G at most g(k) – dynamic programming on the tree decomposition – running time: otherwise: G contains a grid minor of size, so feedback vertex set value more than k. Can choose, running time. The Algorithm What is g(k)?
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Large-Treewidth Graph Decomposition treewidth ≥ 2 G G k+1 feedback vertex set value at least k+1
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If treewidth of G at most g(k) – dynamic programming on the tree decomposition – running time: otherwise: G contains a grid minor of size, so feedback vertex set value more than k. Can choose, running time. The Algorithm What is g(k)?
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Conclusion First polynomial bound on grid minor size,, Best current negative result: Better upper/lower bounds? Better/simpler constructions of path-of-sets or tree-of-sets systems?
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More Open Questions Approximability of NDP/EDP: – general graphs – planar graphs – grid/wall graphs Congestion minimization – -approximation [Raghavan, Thompson ‘87] – -hard to approximate [Andrews, Zhang ‘07] Many more… Thank you!
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