List H -Colouring Problems Given a fixed digraph H Each vertex x of the input digraph G has a list L ( x ) ⊆ V ( H ) Is there a homomorphism f : G → H for which all f ( x ) ∈ L ( x ) ? This is a CSP H is a structure with one binary relation E ( H ) and all 2 | V ( H ) |− 1 possible unary relations R i ( H ) (non-empty subsets of V ( H ) ) Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
List H -Colouring Problems Given a fixed digraph H Each vertex x of the input digraph G has a list L ( x ) ⊆ V ( H ) Is there a homomorphism f : G → H for which all f ( x ) ∈ L ( x ) ? This is a CSP H is a structure with one binary relation E ( H ) and all 2 | V ( H ) |− 1 possible unary relations R i ( H ) (non-empty subsets of V ( H ) ) Instead of having a list L ( x ) , put x in the corresponding unary relation R i ( G ) Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
List H -Colouring Problem for Reflexive Graphs For a reflexive graph H If H is an interval graph, then the list H -colouring problem is polynomial; Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
List H -Colouring Problem for Reflexive Graphs For a reflexive graph H If H is an interval graph, then the list H -colouring problem is polynomial; otherwise it is NP-complete Feder+H 1998 Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
List H -Colouring Problem for Reflexive Graphs For a reflexive graph H If H is an interval graph, then the list H -colouring problem is polynomial; otherwise it is NP-complete Feder+H 1998 Lekkerkerker-Boland 1962 H is an interval graph if and only if it does not have a hole or an AT ... ... Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
List H -Colouring Problem for Reflexive Graphs For a reflexive graph H If H is an interval graph, then the list H -colouring problem is polynomial; otherwise it is NP-complete Feder+H 1998 Lekkerkerker-Boland 1962 H is an interval graph if and only if it does not have a hole or an AT ... ... Structural characterization = ⇒ dichotomy Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
List H -Colouring Problem for Irreflexive Graphs For an irreflexive graph H Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
List H -Colouring Problem for Irreflexive Graphs For an irreflexive graph H If H is a bipartite and H circular arc, then the list H -colouring problem is polynomial; otherwise it is NP-complete Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
List H -Colouring Problem for Irreflexive Graphs For an irreflexive graph H If H is a bipartite and H circular arc, then the list H -colouring problem is polynomial; otherwise it is NP-complete For a bipartite graph H H is a circular arc ⇐ ⇒ H does not have an even hole > 4 or an edge-asteroid Feder+H+Huang 1999 ... ... Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
List H -Colouring Problem for Irreflexive Graphs For an irreflexive graph H If H is a bipartite and H circular arc, then the list H -colouring problem is polynomial; otherwise it is NP-complete For a bipartite graph H H is a circular arc ⇐ ⇒ H does not have an even hole > 4 or an edge-asteroid Feder+H+Huang 1999 ... ... Similarly for general graphs Feder+H+Huang 2007 Structural characterization = ⇒ dichotomy Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
Complements of Circular Arc Graphs For a bipartite graph H H is a circular arc graph ⇔ H is an intersection graph of a family of two-directional rays Shresta+Tayu+Ueno 2010 Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
Complements of Circular Arc Graphs For a bipartite graph H H is a circular arc graph ⇔ H is an intersection graph of a family of two-directional rays Shresta+Tayu+Ueno 2010 a b c a w b y c x x y z w z Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
Complements of Circular Arc Graphs For a bipartite graph H H is a circular arc graph ⇔ H is an intersection graph of a family of two-directional rays Shresta+Tayu+Ueno 2010 a b c a w b y c x x y z w z OPEN: O ( m + n ) recognition Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
General Digraphs Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
General Digraphs A k -ary polymorphism on a structure H A homomorphism φ : H k → H Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
General Digraphs A k -ary polymorphism on a structure H A homomorphism φ : H k → H A conservative polymorphism f A polymorphism φ : H k → H with φ ( u 1 , . . . , u k ) ∈ { u 1 , . . . , u k } Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
Min Ordering Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
Min Ordering A min ordering < of the vertices of a digraph H uv , u ′ v ′ ∈ E ( H ) = ⇒ min ( u , u ′ ) min ( v , v ′ ) ∈ E ( H ) Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
Min Ordering A min ordering < of the vertices of a digraph H uv , u ′ v ′ ∈ E ( H ) = ⇒ min ( u , u ′ ) min ( v , v ′ ) ∈ E ( H ) u’ v v’ u Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
Min Ordering A min ordering < of the vertices of a digraph H uv , u ′ v ′ ∈ E ( H ) = ⇒ min ( u , u ′ ) min ( v , v ′ ) ∈ E ( H ) u’ v v’ u Each interval graph has a min ordering By the left endpoints Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
Min Ordering A min ordering < of the vertices of a digraph H uv , u ′ v ′ ∈ E ( H ) = ⇒ min ( u , u ′ ) min ( v , v ′ ) ∈ E ( H ) u’ v v’ u Each interval graph has a min ordering By the left endpoints A’ B A B’ B A B’ Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
Min Ordering A min ordering < of V ( H ) uv , u ′ v ′ ∈ E ( H ) = ⇒ min ( u , u ′ ) min ( v , v ′ ) ∈ E ( H ) u’ v u v’ Theorem Each interval graph has a min ordering Theorem If H admits a min ordering, then the list H -colouring problem is polynomial Gutjahr, Welzl, Woeginger 1992 Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
Conservative Ternary Polymorphisms A conservative majority on H A conservative ternary polymorphism g : H 3 → H such that g ( u , u , v ) = g ( u , v , u ) = g ( v , u , u ) = u Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
Conservative Ternary Polymorphisms A conservative majority on H A conservative ternary polymorphism g : H 3 → H such that g ( u , u , v ) = g ( u , v , u ) = g ( v , u , u ) = u Theorem If H admits a conservative majority, then the list H -colouring problem is polynomial Feder+Vardi 1993; Jeavons 1998 Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
Conservative Ternary Polymorphisms A conservative Maltsev on H A conservative ternary polymorphism h : H 3 → H such that h ( u , u , v ) = h ( v , u , u ) = v Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
Conservative Ternary Polymorphisms A conservative Maltsev on H A conservative ternary polymorphism h : H 3 → H such that h ( u , u , v ) = h ( v , u , u ) = v Theorem If H admits a conservative Maltsev, then the list H -colouring problem is polynomial Jeavons, Cohen, Gyssens 1997 Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
An Algebraic Classification For a general relational structure H If each pair u , v ∈ V ( H ) admits a conservative polymorphism f of H such that f |{ u , v } is min-ordering, majority, or Maltsev, then the list H -colouring problem is polynomial. Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
An Algebraic Classification For a general relational structure H If each pair u , v ∈ V ( H ) admits a conservative polymorphism f of H such that f |{ u , v } is min-ordering, majority, or Maltsev, then the list H -colouring problem is polynomial. Otherwise it is NP-complete Bulatov 2011, Barto 2012 Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
An Algebraic Classification For a general relational structure H If each pair u , v ∈ V ( H ) admits a conservative polymorphism f of H such that f |{ u , v } is min-ordering, majority, or Maltsev, then the list H -colouring problem is polynomial. Otherwise it is NP-complete Bulatov 2011, Barto 2012 Is NP-completeness again caused by obstructions? (How to certify H is "bad"?) Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
Existence of Polymorphisms of Digraphs Invertible pair u , v � � � � � u v � � � � � v u �� �� �� �� �� �� �� �� �� �� v u �� �� �� �� �� �� �� �� �� �� u v Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
Existence of Polymorphisms of Digraphs Invertible pair u , v � � � � � u v � � � � � v u �� �� �� �� �� �� �� �� �� �� v u �� �� �� �� �� �� �� �� �� �� u v Min ordering A digraph H admits a min ordering if and only if it has no invertible pair Feder+H+Huang+Rafiey 2009 Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
Existence of Polymorphisms of Digraphs Invertible pair u , v � � � � � u v � � � � � v u �� �� �� �� �� �� �� �� �� �� v u �� �� �� �� �� �� �� �� �� �� u v Necessity v’ u u’ u u’ v v v’ Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
Existence of Polymorphisms of Digraphs Majority A digraph H admits a conservative majority if and only if it has no permutable triple H+Rafiey 2010 Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
Existence of Polymorphisms of Digraphs Majority A digraph H admits a conservative majority if and only if it has no permutable triple H+Rafiey 2010 Recall g ( u , u , v ) = g ( u , v , u ) = g ( v , u , u ) = u Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
Existence of Polymorphisms of Digraphs Majority A digraph H admits a conservative majority if and only if it has no permutable triple H+Rafiey 2010 Recall g ( u , u , v ) = g ( u , v , u ) = g ( v , u , u ) = u g ( u , v , w ) � = u w u v s(u) b(u) Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
Existence of Polymorphisms of Digraphs Majority A digraph H admits a conservative majority if and only if it has no permutable triple H+Rafiey 2010 Recall g ( u , u , v ) = g ( u , v , u ) = g ( v , u , u ) = u g ( u , v , w ) � = u u v w s(u) b(u) Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
Existence of Polymorphisms of Digraphs Majority A digraph H admits a conservative majority if and only if it has no permutable triple H+Rafiey 2010 Permutable triple u v w s(u) b(u) s(v)b(v) s(w) b(w) Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
Existence of Polymorphisms of Digraphs Maltsev A digraph H admits a conservative Maltsev if and only if it has no end triple End triple u v w b(w) s(w) s(u) b(u) Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
Digraph Asteroidal Triples Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
Digraph Asteroidal Triples An obstruction for both min ordering and conservative majority: Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
Digraph Asteroidal Triples An obstruction for both min ordering and conservative majority: DAT A permutable triple u , v , w with each pair ( s ( u ) , b ( u )) , ( s ( v ) , b ( v )) , ( s ( w ) , b ( w )) being invertible Recall permutable triple u v w s(v)b(v) s(w) s(u) b(u) b(w) Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
Digraph Asteroidal Triples A DAT a b a’ a a’ b a a’ c b’ c a a’ c b’ b a’ a a’ c b’ b’ c a’ a’ b’ a a’ a c c a’ a’ b b b Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
Polynomial Dichotomy Classification for Digraphs For a digraph H If H is DAT-free, the list H -colouring problem is polynomial Otherwise, the problem is NP-complete Testing for the existence of a DAT is polynomial H+Rafiey 2010 Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
What happened to Maltsev? Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
What happened to Maltsev? Maltsev not needed! If H is DAT-free then each pair u , v ∈ V ( H ) admits a conservative polymorphism f of H such that f |{ u , v } is min-ordering or majority. Otherwise the problem is NP-complete H+Rafiey 2010 Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
What happened to Maltsev? Maltsev not needed! If H is DAT-free then each pair u , v ∈ V ( H ) admits a conservative polymorphism f of H such that f |{ u , v } is min-ordering or majority. Otherwise the problem is NP-complete H+Rafiey 2010 Conservative Maltsev weaker than conservative majority If a digraph H has a global conservative Maltsev polymorphism, it also has a global conservative majority polymorphism Kazda 2011 Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
What happened to Maltsev? Maltsev not needed! If H is DAT-free then each pair u , v ∈ V ( H ) admits a conservative polymorphism f of H such that f |{ u , v } is min-ordering or majority. Otherwise the problem is NP-complete H+Rafiey 2010 Conservative Maltsev weaker than conservative majority If a digraph H has a global conservative Maltsev polymorphism, it also has a global conservative majority polymorphism Kazda 2011 Digraphs H with conservative Maltsev are simple Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
What happened to Maltsev? Maltsev not needed! If H is DAT-free then each pair u , v ∈ V ( H ) admits a conservative polymorphism f of H such that f |{ u , v } is min-ordering or majority. Otherwise the problem is NP-complete H+Rafiey 2010 Conservative Maltsev weaker than conservative majority If a digraph H has a global conservative Maltsev polymorphism, it also has a global conservative majority polymorphism Kazda 2011 Digraphs H with conservative Maltsev are simple Admit a logspace algorithm for the list H -colouring problem Carvalho, Egri, Jackson, Niven 2015 Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
How about space? Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
How about space? Trichotomy for list H -colouring when H is reflexive Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
How about space? Trichotomy for list H -colouring when H is reflexive If H is not an interval graph, then the problem is NP-complete Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
How about space? Trichotomy for list H -colouring when H is reflexive If H is not an interval graph, then the problem is NP-complete If H is an interval graph but not a cograph, then the problem is polynomial time solvable, but NL-complete Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
How about space? Trichotomy for list H -colouring when H is reflexive If H is not an interval graph, then the problem is NP-complete If H is an interval graph but not a cograph, then the problem is polynomial time solvable, but NL-complete If H is an interval graph and a cograph, then the problem is solvable in logspace Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
How about space? Trichotomy for list H -colouring when H is reflexive If H is not an interval graph, then the problem is NP-complete If H is an interval graph but not a cograph, then the problem is polynomial time solvable, but NL-complete If H is an interval graph and a cograph, then the problem is solvable in logspace Cograph = no induced P 4 = recursively built from unions and joins Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
How about space? Trichotomy for list H -colouring when H is reflexive If H is not an interval graph, then the problem is NP-complete If H is an interval graph but not a cograph, then the problem is polynomial time solvable, but NL-complete If H is an interval graph and a cograph, then the problem is solvable in logspace Cograph = no induced P 4 = recursively built from unions and joins Interval + Cograph = "Trivially Perfect" = no induced P 4 , C 4 = recursively built from unions, and joins with K ∗ 1 Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
How about space? Trichotomy for list H -colouring when H is reflexive If H is not an interval graph, then the problem is NP-complete If H is an interval graph but not a cograph, then the problem is polynomial time solvable, but NL-complete If H is an interval graph and a cograph, then the problem is solvable in logspace Cograph = no induced P 4 = recursively built from unions and joins Interval + Cograph = "Trivially Perfect" = no induced P 4 , C 4 = recursively built from unions, and joins with K ∗ 1 Similarly for irreflexive graphs, and general graphs. Reformulation of Egri+Krokhin+Larose+Tesson 2012 Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
A General Classification for Digraphs Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
A General Classification for Digraphs Trichotomy for list H -colouring problems when H is any digraph If H contains a DAT, then the problem is NP-complete Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
A General Classification for Digraphs Trichotomy for list H -colouring problems when H is any digraph If H contains a DAT, then the problem is NP-complete If H is DAT-free but contains a circular N , then the problem is polynomial time solvable, but NL-complete Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
A General Classification for Digraphs Trichotomy for list H -colouring problems when H is any digraph If H contains a DAT, then the problem is NP-complete If H is DAT-free but contains a circular N , then the problem is polynomial time solvable, but NL-complete If H contains no circular N , then the problem is solvable in logspace Egri+H+Larose+Rafiey 2013 Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
A General Classification for Digraphs A circular N a a ... ... b b Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
A General Classification for Digraphs A circular N a a ... ... b b Testing for the existence of a circular N is polynomial Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
Algebraic Classification for H -colouring A Maltsev on H A ternary polymorphism h : H 3 → H such that h ( u , u , v ) = h ( v , u , u ) = v Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
Algebraic Classification for H -colouring A Maltsev on H A ternary polymorphism h : H 3 → H such that h ( u , u , v ) = h ( v , u , u ) = v A Hagemann-Mitschke chain on H Ternary polymorphisms h i : H 3 → H such that h 1 ( u , v , v ) = u h i ( u , u , v ) = h i + 1 ( u , v , v ) h k ( u , u , v ) = v Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
Algebraic Classification for H -colouring A Maltsev on H A ternary polymorphism h : H 3 → H such that h ( u , u , v ) = h ( v , u , u ) = v A Hagemann-Mitschke chain on H Ternary polymorphisms h i : H 3 → H such that h 1 ( u , v , v ) = u h i ( u , u , v ) = h i + 1 ( u , v , v ) h k ( u , u , v ) = v Algebraic classification conjecture for logspace If H admits a Hagemann-Mitschke chain, then H -colouring is in logspace. Otherwise it is NL-complete. Larose+Tesson 2009 Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
Syntactic Restrictions for H -colouring Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
Syntactic Restrictions for H -colouring Examples the non-existence of a K 1 -colouring is expressible in first order logic the non-existence of a K 2 -colouring is certifiable by obstructions of treewidth two the non-existence of a K 2 -colouring is expressible in datalog Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
Syntactic Restrictions for H -colouring Examples the non-existence of a K 1 -colouring is expressible in first order logic the non-existence of a K 2 -colouring is certifiable by obstructions of treewidth two the non-existence of a K 2 -colouring is expressible in datalog Datalog example oddpath(u,v) <- (u ∼ v) oddpath(u,v) <- oddpath(u,w), u ∼ z, z ∼ v test <- oddpath(u,u) Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
Syntactic Restrictions for H -colouring Examples the non-existence of a K 1 -colouring is expressible in first order logic the non-existence of a K 2 -colouring is certifiable by obstructions of treewidth two the non-existence of a K 2 -colouring is expressible in datalog Symmetric datalog example oddpath(u,v) <- (u ∼ v) oddpath(u,v) <- oddpath(u,w), w ∼ z, z ∼ v oddpath(u,w) <- oddpath(u,v), w ∼ z, z ∼ v test <- oddpath(u,u) Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space
Recommend
More recommend
