Structural and Logical Approaches to the Graph Isomorphism Problem - PowerPoint PPT Presentation

Structural and Logical Approaches to the Graph Isomorphism Problem Martin Grohe RWTH Aachen

Graph Isomorphism (GI) Given two graphs, decide if they are isomorphic. 2

Graph Isomorphism (GI) Given two graphs, decide if they are isomorphic. 2

Graph Isomorphism (GI) Given two graphs, decide if they are isomorphic. 1 8 2 2 7 4 7 3 1 5 6 4 5 8 3 6 8 1 6 5 3 1 2 3 4 5 6 7 8 4 2 7 2

Status of the Problem GI is in NP, but not known to be in PTIME or NP-complete. 3

Status of the Problem GI is in NP, but not known to be in PTIME or NP-complete. ◮ One of the few natural problems with this property 3

Status of the Problem GI is in NP, but not known to be in PTIME or NP-complete. ◮ One of the few natural problems with this property ◮ GI was studied in the chemistry literature in the 1950s 3

Status of the Problem GI is in NP, but not known to be in PTIME or NP-complete. ◮ One of the few natural problems with this property ◮ GI was studied in the chemistry literature in the 1950s ◮ Open problem in [Karp, 1972] and [Garey and Johnson, 1979] 3

Status of the Problem GI is in NP, but not known to be in PTIME or NP-complete. ◮ One of the few natural problems with this property ◮ GI was studied in the chemistry literature in the 1950s ◮ Open problem in [Karp, 1972] and [Garey and Johnson, 1979] ◮ Can be solved fairly well in practice. 3

This Talk 1. A Brief Survey 2. Colour Refinement and Weisfeiler-Lehman 3. A Linear Programming Approach to Graph Isomorphism 4. Concluding Remarks 4

A Brief Survey 5

Complexity 6

Complexity GI is unlikely to be NP-complete: Theorem (Boppana, Hastad, Zachos 1987; Schöning 1988) If GI is NP-complete then the polynomial hierarchy collapses to its second level. 6

Complexity GI is unlikely to be NP-complete: Theorem (Boppana, Hastad, Zachos 1987; Schöning 1988) If GI is NP-complete then the polynomial hierarchy collapses to its second level. Theorem (Toran 2004) GI is hard for the class DET (and hence for NL). 6

Upper Bounds Worst Case (Zemlyachenko; Babai 1981; Babai, Luks 1983) GI can be solved in time 2 O ( √ n · log n ) . 7

Upper Bounds Worst Case (Zemlyachenko; Babai 1981; Babai, Luks 1983) GI can be solved in time 2 O ( √ n · log n ) . Average Case (Babai, Erdös, Selkow 1980) GI can be solved in expected linear time (in the G ( n , 1 / 2 ) model). 7

Tractable Classes GI can be solved in polynomial time when restricted to classes of: 8

Tractable Classes GI can be solved in polynomial time when restricted to classes of: ◮ planar graphs Hopcroft, Tarjan 1972 in linear time Hopcroft, Wong 1974 in logspace Datta et al. 2009 8

Tractable Classes GI can be solved in polynomial time when restricted to classes of: ◮ planar graphs Hopcroft, Tarjan 1972 in linear time Hopcroft, Wong 1974 in logspace Datta et al. 2009 ◮ bounded genus Filotti, Mayer 1980; Miller 1980 8

Tractable Classes GI can be solved in polynomial time when restricted to classes of: ◮ planar graphs Hopcroft, Tarjan 1972 in linear time Hopcroft, Wong 1974 in logspace Datta et al. 2009 ◮ bounded genus Filotti, Mayer 1980; Miller 1980 ◮ bounded eigenvalue multiplicities Babai et al. 1982 8

Tractable Classes GI can be solved in polynomial time when restricted to classes of: ◮ planar graphs Hopcroft, Tarjan 1972 in linear time Hopcroft, Wong 1974 in logspace Datta et al. 2009 ◮ bounded genus Filotti, Mayer 1980; Miller 1980 ◮ bounded eigenvalue multiplicities Babai et al. 1982 ◮ bounded degree Luks 1982 8

Tractable Classes GI can be solved in polynomial time when restricted to classes of: ◮ planar graphs Hopcroft, Tarjan 1972 in linear time Hopcroft, Wong 1974 in logspace Datta et al. 2009 ◮ bounded genus Filotti, Mayer 1980; Miller 1980 ◮ bounded eigenvalue multiplicities Babai et al. 1982 ◮ bounded degree Luks 1982 ◮ graphs with excluded minors Ponomarenko 1988 8

Tractable Classes GI can be solved in polynomial time when restricted to classes of: ◮ planar graphs Hopcroft, Tarjan 1972 in linear time Hopcroft, Wong 1974 in logspace Datta et al. 2009 ◮ bounded genus Filotti, Mayer 1980; Miller 1980 ◮ bounded eigenvalue multiplicities Babai et al. 1982 ◮ bounded degree Luks 1982 ◮ graphs with excluded minors Ponomarenko 1988 ◮ bounded tree width Bodlaender 1990 8

Tractable Classes GI can be solved in polynomial time when restricted to classes of: ◮ planar graphs Hopcroft, Tarjan 1972 in linear time Hopcroft, Wong 1974 in logspace Datta et al. 2009 ◮ bounded genus Filotti, Mayer 1980; Miller 1980 ◮ bounded eigenvalue multiplicities Babai et al. 1982 ◮ bounded degree Luks 1982 ◮ graphs with excluded minors Ponomarenko 1988 ◮ bounded tree width Bodlaender 1990 ◮ interval graphs in AC 2 Klein 1996 in logspace Köbler et al. 2010 8

Tractable Classes GI can be solved in polynomial time when restricted to classes of: ◮ planar graphs Hopcroft, Tarjan 1972 in linear time Hopcroft, Wong 1974 in logspace Datta et al. 2009 ◮ bounded genus Filotti, Mayer 1980; Miller 1980 ◮ bounded eigenvalue multiplicities Babai et al. 1982 ◮ bounded degree Luks 1982 ◮ graphs with excluded minors Ponomarenko 1988 ◮ bounded tree width Bodlaender 1990 ◮ interval graphs in AC 2 Klein 1996 in logspace Köbler et al. 2010 ◮ graphs with excluded topological subgraphs G., Marx 2012 8

excluded top. subgraph excluded minor bounded degree bounded genus interval graphs bounded tree width planar trees paths 9

Hard Classes GI restricted to the following classes is as hard as the general problem: ◮ bipartite graphs ◮ chordal graphs ◮ rectangle intersection graphs (Uehara 2008) ◮ graphs of bounded degeneracy ◮ graphs of bounded expansion 10

bounded degeneracy rectangle intersection graphs bounded expansion excluded top. subgraphs excluded minors bounded degree bounded genus interval graphs bounded tree width planar trees paths 11

Algorithms GI-algorithms can be divided into three groups: ◮ graph theoretic algorithms ◮ group theoretic algorithms ◮ combinatorial algorithms 12

Graph Theoretic Algorithms The algorithms for the following classes are typical graph algorithms exploiting the graph theoretic properties of the classes: 13

Graph Theoretic Algorithms The algorithms for the following classes are typical graph algorithms exploiting the graph theoretic properties of the classes: ◮ planar graphs (Hopcroft, Tarjan 1972; Hopcroft, Wong 1974; Datta et al. 2009) ◮ bounded genus (Filotti, Mayer 1980; Miller 1980) ◮ bounded tree width (Bodlaender 1990) ◮ interval graphs (Klein 1996; Köbler et al. 2010) 13

Group Theoretic Algorithms The following algorithms are based on computing generators for the automorphism groups; they exploit properties of the automorphism groups: 14

Group Theoretic Algorithms The following algorithms are based on computing generators for the automorphism groups; they exploit properties of the automorphism groups: ◮ coloured graphs with bounded colour-class-size (Babai 1979 randomized; Furst, Hopcroft Luks 1980 deterministic) ◮ graphs with bounded eigenvalue multiplicities (Babai, Grigoriev, Mount 1982) ◮ graphs of bounded degree (Luks 1982) ◮ k -contractible graphs (Miller 1983) ◮ graphs with excluded minors (Ponomarenko 1988) ◮ general graphs in time 2 O ( √ n · log n ) (Zemlyachenko; Babai 1981; Babai, Luks 1983) 14

Group Theoretic Algorithms The following algorithms are based on computing generators for the automorphism groups; they exploit properties of the automorphism groups: ◮ coloured graphs with bounded colour-class-size (Babai 1979 randomized; Furst, Hopcroft Luks 1980 deterministic) ◮ graphs with bounded eigenvalue multiplicities (Babai, Grigoriev, Mount 1982) ◮ graphs of bounded degree (Luks 1982) ◮ k -contractible graphs (Miller 1983) ◮ graphs with excluded minors (Ponomarenko 1988) ◮ general graphs in time 2 O ( √ n · log n ) (Zemlyachenko; Babai 1981; Babai, Luks 1983) The group theoretic approach dominated research on the graph isomorphism problem since the early 1980s. 14

Combinatorial Algorithms Colour Refinement Weisfeiler Individualisation Lehman Refinement 15

Combinatorial Algorithms Colour Refinement Weisfeiler Individualisation Lehman Refinement Simple and generic algorithms that do not use the properties of specific graph classes. 15

Combinatorial Algorithms Colour Refinement Weisfeiler Individualisation Lehman Refinement Simple and generic algorithms that do not use the properties of specific graph classes. Most practical GI-tools, for example Nauty (McKay 1981), are based on Individualisation Refinement. 15

Colour Refinement and Weisfeiler-Lehman 16

Colour Refinement Iteratively compute colouring of vertices of graph G 17

Colour Refinement Iteratively compute colouring of vertices of graph G Initialisation All vertices get the same colour. 17

Colour Refinement Iteratively compute colouring of vertices of graph G Initialisation All vertices get the same colour. Refinement Step Two nodes v , w get different colours if there is some colour c such that v and w have different numbers of neighbours of colour c . 17