Everything you always wanted to know about the parameterized - PowerPoint PPT Presentation

Everything you always wanted to know about the parameterized complexity of Subgraph Isomorphism (but were afraid to ask) D´ aniel Marx, Micha� l Pilipczuk STACS’14, Lyon, March 6 th , 2014 Marx, Pilipczuk Subgraph Isomorphism 1/22

The problem Subgraph Isomorphism Given H and G , decide if H is a subgraph of G . Marx, Pilipczuk Subgraph Isomorphism 2/22

The problem Subgraph Isomorphism Given H and G , decide if H is a subgraph of G . NP-hard, as it generalizes Hamiltonian Path . Marx, Pilipczuk Subgraph Isomorphism 2/22

The problem Subgraph Isomorphism Given H and G , decide if H is a subgraph of G . NP-hard, as it generalizes Hamiltonian Path . Natural question : What if H is ‘small’, or ‘simple’. Marx, Pilipczuk Subgraph Isomorphism 2/22

The problem Subgraph Isomorphism Given H and G , decide if H is a subgraph of G . NP-hard, as it generalizes Hamiltonian Path . Natural question : What if H is ‘small’, or ‘simple’. Parameterized complexity Marx, Pilipczuk Subgraph Isomorphism 2/22

The problem Subgraph Isomorphism Given H and G , decide if H is a subgraph of G . NP-hard, as it generalizes Hamiltonian Path . Natural question : What if H is ‘small’, or ‘simple’. Parameterized complexity Let k := | V ( H ) | . Marx, Pilipczuk Subgraph Isomorphism 2/22

The problem Subgraph Isomorphism Given H and G , decide if H is a subgraph of G . NP-hard, as it generalizes Hamiltonian Path . Natural question : What if H is ‘small’, or ‘simple’. Parameterized complexity Let k := | V ( H ) | . Trivial n O ( k ) algorithm. Marx, Pilipczuk Subgraph Isomorphism 2/22

The problem Subgraph Isomorphism Given H and G , decide if H is a subgraph of G . NP-hard, as it generalizes Hamiltonian Path . Natural question : What if H is ‘small’, or ‘simple’. Parameterized complexity Let k := | V ( H ) | . Trivial n O ( k ) algorithm. Generalization of Clique ⇒ no f ( k ) · n O (1) algorithm (FPT), unless FPT=W[1]. Marx, Pilipczuk Subgraph Isomorphism 2/22

Tree-like H But if H is a path on k vertices, then there is a 1 . 66 k · n O (1) algorithm [Bj¨ orklund]. Marx, Pilipczuk Subgraph Isomorphism 3/22

Tree-like H But if H is a path on k vertices, then there is a 1 . 66 k · n O (1) algorithm [Bj¨ orklund]. Idea : measure the treewidth of H (denoted tw( H )). Marx, Pilipczuk Subgraph Isomorphism 3/22

Tree-like H But if H is a path on k vertices, then there is a 1 . 66 k · n O (1) algorithm [Bj¨ orklund]. Idea : measure the treewidth of H (denoted tw( H )). Theorem [Alon, Yuster, Zwick] Subgraph Isomorphism is solvable in time 2 O ( | V ( H ) | ) · n O (tw( H )) . Marx, Pilipczuk Subgraph Isomorphism 3/22

Tree-like H But if H is a path on k vertices, then there is a 1 . 66 k · n O (1) algorithm [Bj¨ orklund]. Idea : measure the treewidth of H (denoted tw( H )). Theorem [Alon, Yuster, Zwick] Subgraph Isomorphism is solvable in time 2 O ( | V ( H ) | ) · n O (tw( H )) . We need small | V ( H ) | and even smaller tw( H ), but G can be arbitrarily complicated. Marx, Pilipczuk Subgraph Isomorphism 3/22

Tree-like H But if H is a path on k vertices, then there is a 1 . 66 k · n O (1) algorithm [Bj¨ orklund]. Idea : measure the treewidth of H (denoted tw( H )). Theorem [Alon, Yuster, Zwick] Subgraph Isomorphism is solvable in time 2 O ( | V ( H ) | ) · n O (tw( H )) . We need small | V ( H ) | and even smaller tw( H ), but G can be arbitrarily complicated. Question : Could we get something better if we assume that tw( G ) is small, instead of tw( H )? Marx, Pilipczuk Subgraph Isomorphism 3/22

Tree-like G Theorem [FO model checking] Subgraph Isomorphism can be solved in time f ( | V ( H ) | , tw( G )) · n for some computable function f . Marx, Pilipczuk Subgraph Isomorphism 4/22

Tree-like G Theorem [FO model checking] Subgraph Isomorphism can be solved in time f ( | V ( H ) | , tw( G )) · n for some computable function f . We need small tw( G ), but | V ( H ) | must be also small. Marx, Pilipczuk Subgraph Isomorphism 4/22

Tree-like G Theorem [FO model checking] Subgraph Isomorphism can be solved in time f ( | V ( H ) | , tw( G )) · n for some computable function f . We need small tw( G ), but | V ( H ) | must be also small. Theorem [Matouˇ sek, Thomas] Subgraph Isomorphism for connected H can be solved in time f (∆( H )) · n O (tw( G )) for some computable function f . Marx, Pilipczuk Subgraph Isomorphism 4/22

Tree-like G Theorem [FO model checking] Subgraph Isomorphism can be solved in time f ( | V ( H ) | , tw( G )) · n for some computable function f . We need small tw( G ), but | V ( H ) | must be also small. Theorem [Matouˇ sek, Thomas] Subgraph Isomorphism for connected H can be solved in time f (∆( H )) · n O (tw( G )) for some computable function f . We can have unbounded size of V ( H ), but at the cost of: Marx, Pilipczuk Subgraph Isomorphism 4/22

Tree-like G Theorem [FO model checking] Subgraph Isomorphism can be solved in time f ( | V ( H ) | , tw( G )) · n for some computable function f . We need small tw( G ), but | V ( H ) | must be also small. Theorem [Matouˇ sek, Thomas] Subgraph Isomorphism for connected H can be solved in time f (∆( H )) · n O (tw( G )) for some computable function f . We can have unbounded size of V ( H ), but at the cost of: bounding ∆( H ); Marx, Pilipczuk Subgraph Isomorphism 4/22

Tree-like G Theorem [FO model checking] Subgraph Isomorphism can be solved in time f ( | V ( H ) | , tw( G )) · n for some computable function f . We need small tw( G ), but | V ( H ) | must be also small. Theorem [Matouˇ sek, Thomas] Subgraph Isomorphism for connected H can be solved in time f (∆( H )) · n O (tw( G )) for some computable function f . We can have unbounded size of V ( H ), but at the cost of: bounding ∆( H ); having tw( G ) in the exponent; Marx, Pilipczuk Subgraph Isomorphism 4/22

Tree-like G Theorem [FO model checking] Subgraph Isomorphism can be solved in time f ( | V ( H ) | , tw( G )) · n for some computable function f . We need small tw( G ), but | V ( H ) | must be also small. Theorem [Matouˇ sek, Thomas] Subgraph Isomorphism for connected H can be solved in time f (∆( H )) · n O (tw( G )) for some computable function f . We can have unbounded size of V ( H ), but at the cost of: bounding ∆( H ); having tw( G ) in the exponent; assuming that H is connected. Marx, Pilipczuk Subgraph Isomorphism 4/22

Motivation Question : How does treewidth influence the parameterized complexity of Subgraph Isomorphism ? Marx, Pilipczuk Subgraph Isomorphism 5/22

Motivation Question : How does treewidth influence the parameterized complexity of Subgraph Isomorphism ? Answer : It depence... Marx, Pilipczuk Subgraph Isomorphism 5/22

Motivation Question : How does treewidth influence the parameterized complexity of Subgraph Isomorphism ? Answer : It depence... whether it’s treewidth of H or of G ; Marx, Pilipczuk Subgraph Isomorphism 5/22

Motivation Question : How does treewidth influence the parameterized complexity of Subgraph Isomorphism ? Answer : It depence... whether it’s treewidth of H or of G ; what other assumptions about other parameters we make. Marx, Pilipczuk Subgraph Isomorphism 5/22

Motivation Question : How does treewidth influence the parameterized complexity of Subgraph Isomorphism ? Answer : It depence... whether it’s treewidth of H or of G ; what other assumptions about other parameters we make. What about other constraints? Like planarity, cliquewidth, excluded minors... Marx, Pilipczuk Subgraph Isomorphism 5/22

Motivation Question : How does treewidth influence the parameterized complexity of Subgraph Isomorphism ? Answer : It depence... whether it’s treewidth of H or of G ; what other assumptions about other parameters we make. What about other constraints? Like planarity, cliquewidth, excluded minors... Our motivation : Marx, Pilipczuk Subgraph Isomorphism 5/22

Motivation Question : How does treewidth influence the parameterized complexity of Subgraph Isomorphism ? Answer : It depence... whether it’s treewidth of H or of G ; what other assumptions about other parameters we make. What about other constraints? Like planarity, cliquewidth, excluded minors... Our motivation : It’s a mess. Marx, Pilipczuk Subgraph Isomorphism 5/22

Motivation Question : How does treewidth influence the parameterized complexity of Subgraph Isomorphism ? Answer : It depence... whether it’s treewidth of H or of G ; what other assumptions about other parameters we make. What about other constraints? Like planarity, cliquewidth, excluded minors... Our motivation : It’s a mess. Let’s try to clean it. Marx, Pilipczuk Subgraph Isomorphism 5/22

Parameters We consider the following 10 parameters for H and G : Number of vertices | V ( · ) | (only H ). 1 Number of connected components cc( · ). 2 Maximum degree ∆( · ). 3 Treewidth tw( · ). 4 Pathwidth pw( · ). 5 Feedback vertex set number fvs( · ). 6 Cliquewidth cw( · ). 7 Genus genus( · ). 8 Hadwiger number (largest clique minor) hadw( · ). 9 10 Topological Hadwiger number (largest topological clique minor) hadw T ( · ). Marx, Pilipczuk Subgraph Isomorphism 6/22

Constraints Known results indicate that there is a complexity shift change between tw( G ) = 1 and tw( G ) = 2. Marx, Pilipczuk Subgraph Isomorphism 7/22