Parameterized Approximation Schemes for Steiner Trees with Small - PowerPoint PPT Presentation

Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices ak 1 , Andreas Emil Feldmann 1 , Duˇ san Knop 1 , 2 ,Tom´ Pavel Dvoˇ r´ aˇ s ık 1 , Tom´ s Toufar 1 , Pavel Vesel´ y 1 Masaˇ r´ aˇ 1 Charles University,Prague, Czech Republic 2 University of Bergen, Bergen, Norway HALG 2018 Amsterdam, Netherlands The research leading to these results has received funding from the European Research Coun- cil under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 616787. EPAS for Steiner Trees HALG 2018 1 / 5

Steiner Tree Input: ◮ Graph G = ( V , E ). ◮ Edge weights: w : E → R + 0 . ◮ Terminals R ⊆ V (vertices in V \ R are called Steiner vertices). Goal: find a Steiner tree T ⊆ G . ◮ R ⊆ V ( T ). ◮ Minimize weight. EPAS for Steiner Trees HALG 2018 2 / 5

Steiner Tree Input: ◮ Graph G = ( V , E ). ◮ Edge weights: w : E → R + 0 . ◮ Terminals R ⊆ V (vertices in V \ R are called Steiner vertices). Goal: find a Steiner tree T ⊆ G . ◮ R ⊆ V ( T ). ◮ Minimize weight. EPAS for Steiner Trees HALG 2018 2 / 5

Steiner Tree Input: ◮ Graph G = ( V , E ). ◮ Edge weights: w : E → R + 0 . ◮ Terminals R ⊆ V (vertices in V \ R are called Steiner vertices). Goal: find a Steiner tree T ⊆ G . ◮ R ⊆ V ( T ). ◮ Minimize weight. 1 2 2 1 1 2 EPAS for Steiner Trees HALG 2018 2 / 5

Steiner Tree Input: ◮ Graph G = ( V , E ). ◮ Edge weights: w : E → R + 0 . ◮ Terminals R ⊆ V (vertices in V \ R are called Steiner vertices). Goal: find a Steiner tree T ⊆ G . ◮ R ⊆ V ( T ). ◮ Minimize weight. 1 2 2 1 1 2 EPAS for Steiner Trees HALG 2018 2 / 5

Steiner Tree Input: ◮ Graph G = ( V , E ). ◮ Edge weights: w : E → R + 0 . ◮ Terminals R ⊆ V (vertices in V \ R are called Steiner vertices). Goal: find a Steiner tree T ⊆ G . ◮ R ⊆ V ( T ). ◮ Minimize weight. 1 2 2 1 1 2 EPAS for Steiner Trees HALG 2018 2 / 5

Steiner Tree Input: ◮ Graph G = ( V , E ). ◮ Edge weights: w : E → R + 0 . ◮ Terminals R ⊆ V (vertices in V \ R are called Steiner vertices). Goal: find a Steiner tree T ⊆ G . ◮ R ⊆ V ( T ). ◮ Minimize weight. 1 2 2 1 1 2 EPAS for Steiner Trees HALG 2018 2 / 5

Steiner Tree Input: ◮ Graph G = ( V , E ). ◮ Edge weights: w : E → R + 0 . ◮ Terminals R ⊆ V (vertices in V \ R are called Steiner vertices). Goal: find a Steiner tree T ⊆ G . ◮ R ⊆ V ( T ). ◮ Minimize weight. 1 2 2 1 1 2 EPAS for Steiner Trees HALG 2018 2 / 5

Steiner Tree Input: ◮ Graph G = ( V , E ). ◮ Edge weights: w : E → R + 0 . ◮ Terminals R ⊆ V (vertices in V \ R are called Steiner vertices). Goal: find a Steiner tree T ⊆ G . ◮ R ⊆ V ( T ). ◮ Minimize weight. 1 2 2 1 1 2 EPAS for Steiner Trees HALG 2018 2 / 5

Known Results Parameterized Complexity Number of terminals | R | . ◮ FPT-algorithm [Dreyfus and Wagner ’71, M¨ olle et al. ’06]. ◮ (1 + ε )-approximate polynomial size kernel [Lokshtanov et al. ’16]. Number of Steiner vertices in the optimum | V ( T ) \ R | = p . ◮ W[2]-hard [folklore]. EPAS for Steiner Trees HALG 2018 3 / 5

Known Results Parameterized Complexity Number of terminals | R | . ◮ FPT-algorithm [Dreyfus and Wagner ’71, M¨ olle et al. ’06]. ◮ (1 + ε )-approximate polynomial size kernel [Lokshtanov et al. ’16]. Number of Steiner vertices in the optimum | V ( T ) \ R | = p . ◮ W[2]-hard [folklore]. EPAS for Steiner Trees HALG 2018 3 / 5

Known Results Parameterized Complexity Number of terminals | R | . ◮ FPT-algorithm [Dreyfus and Wagner ’71, M¨ olle et al. ’06]. ◮ (1 + ε )-approximate polynomial size kernel [Lokshtanov et al. ’16]. Number of Steiner vertices in the optimum | V ( T ) \ R | = p . ◮ W[2]-hard [folklore]. EPAS for Steiner Trees HALG 2018 3 / 5

Known Results Parameterized Complexity Number of terminals | R | . ◮ FPT-algorithm [Dreyfus and Wagner ’71, M¨ olle et al. ’06]. ◮ (1 + ε )-approximate polynomial size kernel [Lokshtanov et al. ’16]. Number of Steiner vertices in the optimum | V ( T ) \ R | = p . ◮ W[2]-hard [folklore]. EPAS for Steiner Trees HALG 2018 3 / 5

Known Results Parameterized Complexity Number of terminals | R | . ◮ FPT-algorithm [Dreyfus and Wagner ’71, M¨ olle et al. ’06]. ◮ (1 + ε )-approximate polynomial size kernel [Lokshtanov et al. ’16]. Number of Steiner vertices in the optimum | V ( T ) \ R | = p . ◮ W[2]-hard [folklore]. EPAS for Steiner Trees HALG 2018 3 / 5

Known Results Parameterized Complexity Number of terminals | R | . ◮ FPT-algorithm [Dreyfus and Wagner ’71, M¨ olle et al. ’06]. ◮ (1 + ε )-approximate polynomial size kernel [Lokshtanov et al. ’16]. Number of Steiner vertices in the optimum | V ( T ) \ R | = p . ◮ W[2]-hard [folklore]. EPAS for Steiner Trees HALG 2018 3 / 5

Known Results Parameterized Complexity Number of terminals | R | . ◮ FPT-algorithm [Dreyfus and Wagner ’71, M¨ olle et al. ’06]. ◮ (1 + ε )-approximate polynomial size kernel [Lokshtanov et al. ’16]. Number of Steiner vertices in the optimum | V ( T ) \ R | = p . ◮ W[2]-hard [folklore]. Approximation 96 / 95-approximation is NP-hard [Chleb´ ık and Chleb´ ıkov´ a ’02]. 1.39-approximation algorithm [Byrka et al. ’13]. EPAS for Steiner Trees HALG 2018 3 / 5

Known Results Parameterized Complexity Number of terminals | R | . ◮ FPT-algorithm [Dreyfus and Wagner ’71, M¨ olle et al. ’06]. ◮ (1 + ε )-approximate polynomial size kernel [Lokshtanov et al. ’16]. Number of Steiner vertices in the optimum | V ( T ) \ R | = p . ◮ W[2]-hard [folklore]. Approximation 96 / 95-approximation is NP-hard [Chleb´ ık and Chleb´ ıkov´ a ’02]. 1.39-approximation algorithm [Byrka et al. ’13]. EPAS for Steiner Trees HALG 2018 3 / 5

Known Results Parameterized Complexity Number of terminals | R | . ◮ FPT-algorithm [Dreyfus and Wagner ’71, M¨ olle et al. ’06]. ◮ (1 + ε )-approximate polynomial size kernel [Lokshtanov et al. ’16]. Number of Steiner vertices in the optimum | V ( T ) \ R | = p . ◮ W[2]-hard [folklore]. Approximation 96 / 95-approximation is NP-hard [Chleb´ ık and Chleb´ ıkov´ a ’02]. 1.39-approximation algorithm [Byrka et al. ’13]. EPAS for Steiner Trees HALG 2018 3 / 5

Known Results Parameterized Complexity Number of terminals | R | . ◮ FPT-algorithm [Dreyfus and Wagner ’71, M¨ olle et al. ’06]. ◮ (1 + ε )-approximate polynomial size kernel [Lokshtanov et al. ’16]. Number of Steiner vertices in the optimum | V ( T ) \ R | = p . ◮ W[2]-hard [folklore]. Approximation 96 / 95-approximation is NP-hard [Chleb´ ık and Chleb´ ıkov´ a ’02]. 1.39-approximation algorithm [Byrka et al. ’13]. We study the parameter p = | V ( T ) \ R | – good problem for parameterized approximation . EPAS for Steiner Trees HALG 2018 3 / 5

Our Results Existence of 1 Efficient parameterized approximation scheme – it returns (1 + ε )-approximation in time f ( p , ε ) · poly ( n ). EPAS for Steiner Trees HALG 2018 4 / 5

Our Results Existence of 1 Efficient parameterized approximation scheme – it returns (1 + ε )-approximation in time f ( p , ε ) · poly ( n ). 2 Polynomial size approximate kernelization scheme. EPAS for Steiner Trees HALG 2018 4 / 5

Our Results Existence of 1 Efficient parameterized approximation scheme – it returns (1 + ε )-approximation in time f ( p , ε ) · poly ( n ). 2 Polynomial size approximate kernelization scheme. Unweighted Weighted � � � � Undirected × ∗ × ∗∗ × ∗∗ � Directed ∗ Unless NP ⊆ coNP / poly. ∗∗ Unless FPT= W[2]. EPAS for Steiner Trees HALG 2018 4 / 5

Main Algorithmic Idea 1 Reduce the number of terminals under some bound f ( p , ε ). EPAS for Steiner Trees HALG 2018 5 / 5

Main Algorithmic Idea 1 Reduce the number of terminals under some bound f ( p , ε ). 2 Use some existing algorithm or kernel for the parameter | R | . EPAS for Steiner Trees HALG 2018 5 / 5

Main Algorithmic Idea 1 Reduce the number of terminals under some bound f ( p , ε ). 2 Use some existing algorithm or kernel for the parameter | R | . Undirected Unweighted Case d ≥ 1 /ε 1 2 Q EPAS for Steiner Trees HALG 2018 5 / 5

Main Algorithmic Idea 1 Reduce the number of terminals under some bound f ( p , ε ). 2 Use some existing algorithm or kernel for the parameter | R | . Undirected Unweighted Case d ≥ 1 /ε 1 2 Q Reduction Rule 1: We can assume any such edge is in the optimal solution. EPAS for Steiner Trees HALG 2018 5 / 5

Main Algorithmic Idea 1 Reduce the number of terminals under some bound f ( p , ε ). 2 Use some existing algorithm or kernel for the parameter | R | . Undirected Unweighted Case d ≥ 1 /ε 1 2 Q Reduction Rule 2: The optimal solution uses at least d edges to connect terminals in Q . Our solution uses at most d + 1 edges. EPAS for Steiner Trees HALG 2018 5 / 5

Main Algorithmic Idea 1 Reduce the number of terminals under some bound f ( p , ε ). 2 Use some existing algorithm or kernel for the parameter | R | . Undirected Unweighted Case d ≥ 1 /ε 1 2 Q Reduction Rule 2: The optimal solution uses at least d edges to connect terminals in Q . Our solution uses at most d + 1 edges. ALG OPT EPAS for Steiner Trees HALG 2018 5 / 5