Parameterized Maximum Path Coloring Michael Lampis September 9, - PowerPoint PPT Presentation

Parameterized Maximum Path Coloring Michael Lampis September 9, 2011 1 / 17

Path Coloring Definition Path Coloring ❖ Path Coloring ❖ Example ❖ Known results Input : A graph G and a multi-set of paths on that graph ❖ Edge slicing Constraint : Assign colors from { 1 , . . . , W } to the paths ❖ Max PC ❖ Max PC hardness so that paths that share an edge receive different colors. results Objective : min W ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ ( p ∆ , pW, pT ) - MaxPC ❖ ( pT ) -MaxPC binary trees ❖ ( pT ) -MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 2 / 17

Path Coloring Definition Path Coloring ❖ Path Coloring ❖ Example ❖ Known results Input : A graph G and a multi-set of paths on that graph ❖ Edge slicing Constraint : Assign colors from { 1 , . . . , W } to the paths ❖ Max PC ❖ Max PC hardness so that paths that share an edge receive different colors. results Objective : min W ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ● Graph could be undirected or bi-directed ❖ ( p ∆ , pW, pT ) - MaxPC ● Instead of paths we could be given endpoints (Routing ❖ ( pT ) -MaxPC binary trees and Path Coloring) ❖ ( pT ) -MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 2 / 17

Path Coloring Definition Path Coloring ❖ Path Coloring ❖ Example ❖ Known results Input : A graph G and a multi-set of paths on that graph ❖ Edge slicing Constraint : Assign colors from { 1 , . . . , W } to the paths ❖ Max PC ❖ Max PC hardness so that paths that share an edge receive different colors. results Objective : min W ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ● Graph could be undirected or bi-directed ❖ ( p ∆ , pW, pT ) - MaxPC ● Instead of paths we could be given endpoints (Routing ❖ ( pT ) -MaxPC binary trees and Path Coloring) ❖ ( pT ) -MaxPC binary trees ❖ Algoritm cont’d ● We’ll mostly talk about trees ( → unique routing) ❖ Open problems 2 / 17

Example ❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ ( p ∆ , pW, pT ) - MaxPC ❖ ( pT ) -MaxPC binary trees ❖ ( pT ) -MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 3 / 17

Example ❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ ( p ∆ , pW, pT ) - MaxPC ❖ ( pT ) -MaxPC binary trees ❖ ( pT ) -MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 3 / 17

Example ❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ ( p ∆ , pW, pT ) - MaxPC ❖ ( pT ) -MaxPC binary trees ❖ ( pT ) -MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 3 / 17

Example ❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ ( p ∆ , pW, pT ) - MaxPC ❖ ( pT ) -MaxPC binary trees ❖ ( pT ) -MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 3 / 17

Example ❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ ( p ∆ , pW, pT ) - MaxPC ❖ ( pT ) -MaxPC binary trees ❖ ( pT ) -MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 3 / 17

Known results ❖ Path Coloring ● PC is very hard! ❖ Example ❖ Known results ✦ NP-hard on stars [Erlebach, Jansen 2001] ❖ Edge slicing ❖ Max PC ✦ NP-hard on rings [Garey, Johnson, Miller, ❖ Max PC hardness results Papadimitriou 1980] ❖ DNP ❖ Reduction ✦ NP-hard on bi-directed binary trees [Kumar, ❖ Reduction Panigrahy, Russel, Sundaram 1997] ❖ Complexity jump ❖ ( p ∆ , pW, pT ) - MaxPC ❖ ( pT ) -MaxPC ● Good news: Thanks to a simple trick undirected trees binary trees are no harder than stars. ❖ ( pT ) -MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 4 / 17

Edge slicing ❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ ( p ∆ , pW, pT ) - MaxPC ❖ ( pT ) -MaxPC binary trees ❖ ( pT ) -MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 5 / 17

Edge slicing ❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ ( p ∆ , pW, pT ) - MaxPC ❖ ( pT ) -MaxPC binary trees ❖ ( pT ) -MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 5 / 17

Edge slicing ❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ ( p ∆ , pW, pT ) - MaxPC ❖ ( pT ) -MaxPC binary trees ❖ ( pT ) -MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 5 / 17

Edge slicing ❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ ( p ∆ , pW, pT ) - MaxPC ❖ ( pT ) -MaxPC binary trees ❖ ( pT ) -MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 5 / 17

Edge slicing ❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ ( p ∆ , pW, pT ) - MaxPC ❖ ( pT ) -MaxPC binary trees ❖ ( pT ) -MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 5 / 17

Edge slicing ❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ ( p ∆ , pW, pT ) - MaxPC ❖ ( pT ) -MaxPC binary trees ❖ ( pT ) -MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 5 / 17

Edge slicing ❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ ( p ∆ , pW, pT ) - MaxPC ❖ ( pT ) -MaxPC binary trees ❖ ( pT ) -MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 5 / 17

Edge slicing ❖ Path Coloring ● Repeated edge slicing can break ❖ Example down any undirected tree to a star ❖ Known results ❖ Edge slicing ❖ Max PC ● If we could solve PC on stars → poly- ❖ Max PC hardness results time algorithm (we can’t!) ❖ DNP ❖ Reduction ✦ But FPT algorithm when parame- ❖ Reduction terized by ∆ . [Erlebach, Jansen ❖ Complexity jump ❖ ( p ∆ , pW, pT ) - 2001] MaxPC ❖ ( pT ) -MaxPC binary trees ● Ironically, this doesn’t work for bi- ❖ ( pT ) -MaxPC binary trees directed trees, where stars are easy. ❖ Algoritm cont’d ❖ Open problems ✦ But FPT algorithm when param- eterized by ∆ + W . [Erlebach, Jansen 2001] 5 / 17

Max PC Max Path Coloring ❖ Path Coloring ❖ Example ❖ Known results Input : A graph G and a multi-set of paths on that graph, ❖ Edge slicing ❖ Max PC color buget W ❖ Max PC hardness Constraint : Assign colors from { 1 , . . . , W } to B of the results ❖ DNP paths so that paths that share an edge receive different ❖ Reduction colors. ❖ Reduction Objective : max B ❖ Complexity jump ❖ ( p ∆ , pW, pT ) - MaxPC ❖ ( pT ) -MaxPC ● Strict generalization of PC as a decision problem binary trees ❖ ( pT ) -MaxPC binary trees ✦ → At least as hard to solve exactly ❖ Algoritm cont’d ❖ Open problems 6 / 17

Max PC Max Path Coloring ❖ Path Coloring ❖ Example ❖ Known results Input : A graph G and a multi-set of paths on that graph, ❖ Edge slicing ❖ Max PC color buget W ❖ Max PC hardness Constraint : Assign colors from { 1 , . . . , W } to B of the results ❖ DNP paths so that paths that share an edge receive different ❖ Reduction colors. ❖ Reduction Objective : max B ❖ Complexity jump ❖ ( p ∆ , pW, pT ) - MaxPC ❖ ( pT ) -MaxPC ● Strict generalization of PC as a decision problem binary trees ❖ ( pT ) -MaxPC binary trees ✦ → At least as hard to solve exactly ❖ Algoritm cont’d ❖ Open problems ● Max PC is solvable in n ∆ W on trees. [Erlebach, Jansen 1998] ● Can we do this in FPT time for either parameter? 6 / 17

Max PC hardness results ❖ Path Coloring ● An n ∆ W algorithm is known to solve Max PC exactly on ❖ Example ❖ Known results trees. Can we do better? ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ ( p ∆ , pW, pT ) - MaxPC ❖ ( pT ) -MaxPC binary trees ❖ ( pT ) -MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 7 / 17