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ETH-Tight Exact Algorithms for Hard Geometric Problems Using Geometric Separators

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ETH-Tight Exact Algorithms for Hard Geometric Problems Using Geometric Separators Exact Algorithms for Hard Geometric Problems Using Geometric Separators subexponential subexponential algorithms for algorithm for planar graphs subexponential Euclidean TSP algorithms for disk graphs ETH-based lower bound for Euclidean TSP Mark de Berg (TU Eindhoven) TU e Old and New (Geometric) Separator Theorems subexponential subexponential algorithms for algorithm for planar graphs subexponential Euclidean TSP algorithms for disk graphs ETH-based lower bound for Euclidean TSP Mark de Berg (TU Eindhoven) TU e Hardness: A Traditional Algorithmic View pspace np p Assumption: p 6= np • P = tractable • np-hard = untractable TU e Hardness: A Traditional Algorithmic View Algorithmic approach to problem solving pspace • Determine np-hardness by reduction from known np-hard problem np • give approximation algorithm p or • try to give fastest possible Assumption: p 6= np polynomial-time algorithm • sometimes lower bounds • P = tractable • np-hard = untractable TU e A More Refined View Algorithmic approach to problem solving pspace • Determine np-hardness by reduction from known np-hard problem np • give approximation algorithm • study parameterized complexity • study subexponential algorithms p or • try to give fastest possible Assumption: p 6= np polynomial-time algorithm • sometimes lower bounds • P = tractable • np-hard = untractable and relate problems to each other through conditional lower bounds TU e Talk Overview subexponential subexponential algorithms for algorithm for planar graphs subexponential Euclidean TSP algorithms for disk graphs ETH-based lower bound for Euclidean TSP TU e TALK OVERVIEW subexponential algorithms for planar graphs ETH-based lower bound for subexponential Euclidean TSP algorithms for subexponential disk graphs algorithm for Euclidean TSP subexponential subexponential algorithms for algorithm for planar graphs subexponential Euclidean TSP algorithms for disk graphs ETH-based lower bound for Euclidean TSP TU e Subexponential algorithms Subexponential algorithms: running time 2o(n) p p • Typical examples: 2O( n) or nO( n) or 2O(n= log n) TU e Subexponential algorithms Subexponential algorithms: running time 2o(n) p p • Typical examples: 2O( n) or nO( n) or 2O(n= log n) Do np-hard problems have subexponential algorithms? • 3-SAT: Is given 3-CNF formula witn n variables satisfiable? (x1 _ x2 _ x5) ^ (x3 _ x4 _ x6) ^ (x2 _ x3 _ x5) TU e Subexponential algorithms Subexponential algorithms: running time 2o(n) p p • Typical examples: 2O( n) or nO( n) or 2O(n= log n) Do np-hard problems have subexponential algorithms? • 3-SAT: Is given 3-CNF formula witn n variables satisfiable? (x1 _ x2 _ x5) ^ (x3 _ x4 _ x6) ^ (x2 _ x3 _ x5) Exponential-Time Hypothesis (ETH) (Impagliazzo, Paturi and Zane 2001) There is no sub-exponential algorithm for 3-SAT, that is, 3-SAT cannot be solved in 2o(n) time. What about other np-hard problems? TU e Three classic np-hard graph problems Input: graph G = (V; E) with n vertices Independent Set • Find largest set I ⊆ V such that no two vertices in I are connected by an edge. Vertex Cover • Find smallest set C ⊆ V that contains at least one endpoint of every edge in E. Dominating Set • Find smallest D ⊆ V such that for each v 2 V either v or one of its neighbors is in D. TU e Three classic np-hard graph problems Input: graph G = (V; E) with n vertices Independent Set • Find largest set I ⊆ V such that no two vertices in I are connected by an edge. Vertex Cover • Find smallest set C ⊆ V that contains at least one endpoint of every edge in E. Dominating Set • Find smallest D ⊆ V such that for each v 2 V either v or one of its neighbors is in D. TU e Three classic np-hard graph problems Input: graph G = (V; E) with n vertices Independent Set • Find largest set I ⊆ V such that no two vertices in I are connected by an edge. Vertex Cover • Find smallest set C ⊆ V that contains at least one endpoint of every edge in E. Dominating Set • Find smallest D ⊆ V such that for each v 2 V either v or one of its neighbors is in D. TU e Three classic np-hard graph problems Input: graph G = (V; E) with n vertices Independent Set • Find largest set I ⊆ V such that no two vertices in I are connected by an edge. complement Vertex Cover • Find smallest set C ⊆ V that contains at least one endpoint of every edge in E. Dominating Set • Find smallest D ⊆ V such that for each v 2 V either v or one of its neighbors is in D. TU e Three classic np-hard graph problems Input: graph G = (V; E) with n vertices Independent Set • Find largest set I ⊆ V such that no two vertices in I are connected by an edge. complement Vertex Cover • Find smallest set C ⊆ V that contains at least one endpoint of every edge in E. Dominating Set • Find smallest D ⊆ V such that for each v 2 V either v or one of its neighbors is in D. TU e Three classic np-hard graph problems Are there subexponential algorithms for these classic graph problems? TU e Three classic np-hard graph problems Are there subexponential algorithms for these classic graph problems? Theorem. Independent Set, Vertex Cover, and Dominating Set cannot be solved in 2o(n) time, under ETH. TU e Three classic np-hard graph problems Are there subexponential algorithms for these classic graph problems? Theorem. Independent Set, Vertex Cover, and Dominating Set cannot be solved in 2o(n) time, under ETH. but we can get subexponential algorithms on planar graphs TU e Planar graphs Planar graphs: graphs that can be drawn without crossing edges TU e Planar graphs Planar graphs: graphs that can be drawn without crossing edges Planar Separator Theorem (Lipton,Tarjan 1979) TU e Planar graphs Planar graphs: graphs that can be drawn without crossing edges Planar Separator Theorem (Lipton,Tarjan 1979) For anyp planar graph G = (V; E) there is a separator S ⊂ V of size O( n) such that V n S can be partitioned into subsets A and 2 B, each of size at most 3 n and with no edges between them. TU e Planar graphs Planar graphs: graphs that can be drawn without crossing edges S A B Planar Separator Theorem (Lipton,Tarjan 1979) For anyp planar graph G = (V; E) there is a separator S ⊂ V of size O( n) such that V n S can be partitioned into subsets A and 2 B, each of size at most 3 n and with no edges between them. TU e Planar graphs Planar graphs: graphs that can be drawn without crossing edges S A B Planar Separator Theorem (Lipton,Tarjan 1979) For anyp planar graph G = (V; E) there is a separator S ⊂ V of size O( n) such that V n S can be partitioned into subsets A and 2 B, each of size at most 3 n and with no edges between them. Such a (2=3)-balanced separator can be computed in O(n) time. TU e Subexponential algorithms on planar graphs p Theorem. Independent Set can be solved in 2O( n) time in planar graphs. TU e Subexponential algorithms on planar graphs p Theorem. Independent Set can be solved in 2O( n) time in planar graphs. p 1. Compute (2=3)-balanced separator S of size O( n). 2. For each independent set IS ⊆ S (including empty set) do (a) Recursively find max independent set IA for A n fneighbors of ISg (b) Recursively find max independent set IB for B n fneighbors of ISg (c) I(S) := IS [ IA [ IB 3. Return the largest of the sets I(S) found in Step 2. TU e Subexponential algorithms on planar graphs p Theorem. Independent Set can be solved in 2O( n) time in planar graphs. p 1. Compute (2=3)-balanced separator S of size O( n). 2. For each independent set IS ⊆ S (including empty set) do (a) Recursively find max independent set IA for A n fneighbors of ISg (b) Recursively find max independent set IB for B n fneighbors of ISg (c) I(S) := IS [ IA [ IB 3. Return the largest of the sets I(S) found in Step 2. TU e Subexponential algorithms on planar graphs p Theorem. Independent Set can be solved in 2O( n) time in planar graphs. p 1. Compute (2=3)-balanced separator S of size O( n). 2. For each independent set IS ⊆ S (including empty set) do (a) Recursively find max independent set IA for A n fneighbors of ISg (b) Recursively find max independent set IB for B n fneighbors of ISg (c) I(S) := IS [ IA [ IB 3. Return the largest of the sets I(S) found in Step 2. TU e Subexponential algorithms on planar graphs p Theorem. Independent Set can be solved in 2O( n) time in planar graphs. p 1. Compute (2=3)-balanced separator S of size O( n). 2. For each independent set IS ⊆ S (including empty set) do (a) Recursively find max independent set IA for A n fneighbors of ISg (b) Recursively find max independent set IB for B n fneighbors of ISg (c) I(S) := IS [ IA [ IB 3. Return the largest of the sets I(S) found in Step 2. TU e Subexponential algorithms on planar graphs p Theorem. Independent Set can be solved in 2O( n) time in planar graphs. p 1. Compute (2=3)-balanced separator S of size O( n). 2. For each independent set IS ⊆ S (including empty set) do (a) Recursively find max independent set IA for A n fneighbors of ISg (b) Recursively find max independent set IB for B n fneighbors of ISg (c) I(S) := IS [ IA [ IB 3.
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