Counting and Finding Homomorphisms is Universal for Parameterized - PowerPoint PPT Presentation

Counting and Finding Homomorphisms is Universal for Parameterized Complexity Theory Marc Roth (Merton College, Oxford University, United Kingdom) , Philip Wellnitz (MPII, SIC, Saarbrücken, Germany) SODA 2020

Basic Definitions and General Overview Main Result Open Problems Finding Graph Homomorphisms Graph Homomorphism Mapping from graph H to G that preserves edges; Write Hom( H → G ) for the set of all graph hom’s from H to G . Φ = bipartite H G | V ( H ) | = 4 Marc Roth and Philip Wellnitz Counting Homomorphisms is Universal for Parameterized Complexity Theory

Basic Definitions and General Overview Main Result Open Problems Finding Graph Homomorphisms Graph Homomorphism Mapping from graph H to G that preserves edges; Write Hom( H → G ) for the set of all graph hom’s from H to G . Φ = bipartite H G | V ( H ) | = 4 Marc Roth and Philip Wellnitz Counting Homomorphisms is Universal for Parameterized Complexity Theory

Basic Definitions and General Overview Main Result Open Problems Finding Graph Homomorphisms Graph Homomorphism Mapping from graph H to G that preserves edges; Write Hom( H → G ) for the set of all graph hom’s from H to G . Φ = bipartite H G | V ( H ) | = 4 Marc Roth and Philip Wellnitz Counting Homomorphisms is Universal for Parameterized Complexity Theory

Basic Definitions and General Overview Main Result Open Problems Finding Graph Homomorphisms Graph Homomorphism Mapping from graph H to G that preserves edges; Write Hom( H → G ) for the set of all graph hom’s from H to G . Φ = bipartite H G | V ( H ) | = 4 #Hom( H → G ) = 14 Marc Roth and Philip Wellnitz Counting Homomorphisms is Universal for Parameterized Complexity Theory

Basic Definitions and General Overview Main Result Open Problems Finding Graph Homomorphisms Graph Homomorphism Mapping from graph H to G that preserves edges; Write Hom( H → G ) for the set of all graph hom’s from H to G . Φ = bipartite H G | V ( H ) | = 4 Marc Roth and Philip Wellnitz Counting Homomorphisms is Universal for Parameterized Complexity Theory

Basic Definitions and General Overview Main Result Open Problems Finding Graph Homomorphisms Graph Homomorphism Mapping from graph H to G that preserves edges; Write Hom( H → G ) for the set of all graph hom’s from H to G . Φ = bipartite H G | V ( H ) | = 4 No homomorphisms from H to G . Marc Roth and Philip Wellnitz Counting Homomorphisms is Universal for Parameterized Complexity Theory

Basic Definitions and General Overview Main Result Open Problems Finding Graph Homomorphisms H OM ( H → G ) Given graphs H ∈ H and G ∈ G , check if there is a graph hom from H to G . Marc Roth and Philip Wellnitz Counting Homomorphisms is Universal for Parameterized Complexity Theory

Basic Definitions and General Overview Main Result Open Problems Finding Graph Homomorphisms H OM ( H → G ) Given graphs H ∈ H and G ∈ G , check if there is a graph hom from H to G . H G , Graph classes Marc Roth and Philip Wellnitz Counting Homomorphisms is Universal for Parameterized Complexity Theory

Basic Definitions and General Overview Main Result Open Problems Finding Graph Homomorphisms H OM ( H → G ) Given graphs H ∈ H and G ∈ G , check if there is a graph hom from H to G . H G , Graph classes · · · · · · · · · · · · · · · · · · All Graphs ( ⊤ ) All Bipartite Graphs All Cliques Marc Roth and Philip Wellnitz Counting Homomorphisms is Universal for Parameterized Complexity Theory

Basic Definitions and General Overview Main Result Open Problems Finding Graph Homomorphisms H OM ( H → G ) Given graphs H ∈ H and G ∈ G , check if there is a graph hom from H to G . H G , Graph classes set of graphs · · · · · · · · · · · · · · · · · · All Graphs ( ⊤ ) All Bipartite Graphs All Cliques Marc Roth and Philip Wellnitz Counting Homomorphisms is Universal for Parameterized Complexity Theory

Basic Definitions and General Overview Main Result Open Problems Known Results H OM ( H → G ) Given graphs H ∈ H and G ∈ G , check if there is a graph hom from H to G . Marc Roth and Philip Wellnitz Counting Homomorphisms is Universal for Parameterized Complexity Theory

Basic Definitions and General Overview Main Result Open Problems Known Results H OM ( H → G ) Given graphs H ∈ H and G ∈ G , check if there is a graph hom from H to G . NP-complete H OM ( ⊤ → { K 3 } ) H OM ( ⊤ → ⊤ ) 3- COLORABLE 3- COLORABLE Marc Roth and Philip Wellnitz Counting Homomorphisms is Universal for Parameterized Complexity Theory

Basic Definitions and General Overview Main Result Open Problems Known Results H OM ( H → G ) Given graphs H ∈ H and G ∈ G , check if there is a graph hom from H to G . NP-complete H OM ( ⊤ → { K 3 } ) H OM ( ⊤ → ⊤ ) 3- COLORABLE 3- COLORABLE Marc Roth and Philip Wellnitz Counting Homomorphisms is Universal for Parameterized Complexity Theory

Basic Definitions and General Overview Main Result Open Problems Known Results H OM ( H → G ) Given graphs H ∈ H and G ∈ G , check if there is a graph hom from H to G . NP-complete H OM ( ⊤ → { K 3 } ) H OM ( ⊤ → { } ) 3- COLORABLE Marc Roth and Philip Wellnitz Counting Homomorphisms is Universal for Parameterized Complexity Theory

Basic Definitions and General Overview Main Result Open Problems Known Results H OM ( H → G ) Given graphs H ∈ H and G ∈ G , check if there is a graph hom from H to G . Are there fast algorithms for special cases of H OM ( ⊤ → ⊤ )? Marc Roth and Philip Wellnitz Counting Homomorphisms is Universal for Parameterized Complexity Theory

Basic Definitions and General Overview Main Result Open Problems Known Results H OM ( H → G ) Given graphs H ∈ H and G ∈ G , check if there is a graph hom from H to G . What makes H OM ( ⊤ → ⊤ ) hard? Marc Roth and Philip Wellnitz Counting Homomorphisms is Universal for Parameterized Complexity Theory

Basic Definitions and General Overview Main Result Open Problems Known Results H OM ( H → G ) Given graphs H ∈ H and G ∈ G , check if there is a graph hom from H to G . poly-time solvable NP-complete H OM ( ⊤ → G ) G contains only G contains a bipartite graphs non-bipartite graph [Hell, Nešetˇ ril ’90] [Hell, Nešetˇ ril ’90] Marc Roth and Philip Wellnitz Counting Homomorphisms is Universal for Parameterized Complexity Theory

Basic Definitions and General Overview Main Result Open Problems Known Results #H OM ( H → G ) Given graphs H ∈ H and G ∈ G , count all graph homomorphisms from H to G . poly-time solvable #P-complete #H OM ( ⊤ → G ) (explicit criterion exists) (explicit criterion exists) [Dyer, Greenhill ’00] [Dyer, Greenhill ’00] Marc Roth and Philip Wellnitz Counting Homomorphisms is Universal for Parameterized Complexity Theory

Basic Definitions and General Overview Main Result Open Problems Known Results H OM ( H → G ) Given graphs H ∈ H and G ∈ G , check if there is a graph hom from H to G . What about the other side , H OM ( H → ⊤ )? Marc Roth and Philip Wellnitz Counting Homomorphisms is Universal for Parameterized Complexity Theory

Basic Definitions and General Overview Main Result Open Problems Known Results Parameter: | V ( H ) | H OM ( H → G ) Given graphs H ∈ H and G ∈ G , check if there is a graph hom from H to G . What about the other side , H OM ( H → ⊤ )? Marc Roth and Philip Wellnitz Counting Homomorphisms is Universal for Parameterized Complexity Theory

Basic Definitions and General Overview Main Result Open Problems Known Results Parameter: | V ( H ) | H OM ( H → G ) Given graphs H ∈ H and G ∈ G , check if there is a graph hom from H to G . FPT W[1]-hard ( f ( | V ( H ) | ) · poly ( | V ( G ) | ) time) (not faster than K -C LIQUE ) H OM ( H → ⊤ ) “ H contains only graphs “ H contains graphs with with small treewidth” arbitrary large tw” [Grohe ’03] [Grohe ’03] Marc Roth and Philip Wellnitz Counting Homomorphisms is Universal for Parameterized Complexity Theory

Basic Definitions and General Overview Main Result Open Problems Known Results Parameter: | V ( H ) | #H OM ( H → G ) Given graphs H ∈ H and G ∈ G , count all graph homomorphisms from H to G . FPT #W[1]-hard ( f ( | V ( H ) | ) · poly ( | V ( G ) | ) time) (not faster than # K -C LIQUE ) #H OM ( H → ⊤ ) “ H contains only graphs “ H contains a graph with small treewidth” with large treewidth” [Dalmau, Jonsson ’04] [Dalmau, Jonsson ’04] Marc Roth and Philip Wellnitz Counting Homomorphisms is Universal for Parameterized Complexity Theory

Basic Definitions and General Overview Main Result Open Problems Known Results Parameter: | V ( H ) | #H OM ( H → G ) Given graphs H ∈ H and G ∈ G , count all graph homomorphisms from H to G . Complexity dichotomies when restricting either G or H . Marc Roth and Philip Wellnitz Counting Homomorphisms is Universal for Parameterized Complexity Theory

Basic Definitions and General Overview Main Result Open Problems Known Results Parameter: | V ( H ) | #H OM ( H → G ) Given graphs H ∈ H and G ∈ G , count all graph homomorphisms from H to G . Complexity dichotomies when restricting either G or H . What if we restrict both sides ? Marc Roth and Philip Wellnitz Counting Homomorphisms is Universal for Parameterized Complexity Theory