(BJARNE-FEST) Pavol Hell, SFU and Charles U List Homomorphisms, - - PowerPoint PPT Presentation

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(BJARNE-FEST) Pavol Hell, SFU and Charles U List Homomorphisms, - - PowerPoint PPT Presentation

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List Homomorphisms, Time and Space Pavol Hell, SFU and Charles U GRAPH THEORY 2015 MEETING IN DENMARK (BJARNE-FEST) Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space Homomorphisms Given digraphs G and H A homomorphism f : G

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List Homomorphisms, Time and Space

Pavol Hell, SFU and Charles U GRAPH THEORY 2015 MEETING IN DENMARK

(BJARNE-FEST)

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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Homomorphisms

Given digraphs G and H A homomorphism f : G → H is a mapping f : V(G) → V(H) such that xy ∈ E(G) = ⇒ f(x)f(y) ∈ E(H)

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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Homomorphisms

Given digraphs G and H A homomorphism f : G → H is a mapping f : V(G) → V(H) such that xy ∈ E(G) = ⇒ f(x)f(y) ∈ E(H) Undirected graphs are symmetric digraphs

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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Homomorphisms

Given digraphs G and H A homomorphism f : G → H is a mapping f : V(G) → V(H) such that xy ∈ E(G) = ⇒ f(x)f(y) ∈ E(H) Undirected graphs are symmetric digraphs General relational structures G, H with vertex-sets V(G), V(H) For each i, corresponding relations Ri(G), Ri(H) of arity ri;

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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Homomorphisms

Given digraphs G and H A homomorphism f : G → H is a mapping f : V(G) → V(H) such that xy ∈ E(G) = ⇒ f(x)f(y) ∈ E(H) Undirected graphs are symmetric digraphs General relational structures G, H with vertex-sets V(G), V(H) For each i, corresponding relations Ri(G), Ri(H) of arity ri; a homomorphism f : V(G) → V(H) satisfies, for each i, (x1, x2, . . . , xri) ∈ Ri(G) = ⇒ (f(x1), f(x2), . . . , f(xri)) ∈ Ri(H)

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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Homomorphisms

Given digraphs G and H A homomorphism f : G → H is a mapping f : V(G) → V(H) such that xy ∈ E(G) = ⇒ f(x)f(y) ∈ E(H) Undirected graphs are symmetric digraphs General relational structures G, H with vertex-sets V(G), V(H) For each i, corresponding relations Ri(G), Ri(H) of arity ri; a homomorphism f : V(G) → V(H) satisfies, for each i, (x1, x2, . . . , xri) ∈ Ri(G) = ⇒ (f(x1), f(x2), . . . , f(xri)) ∈ Ri(H) Finitely many relations of finite arities

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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Unary Relations

What if arity ri = 1?

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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Unary Relations

What if arity ri = 1? Then Ri(H) ⊆ V(H) and Ri(G) ⊆ V(G) Any homomorphism f of G to H has v ∈ Ri(G) = ⇒ f(v) ∈ Ri(H)

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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H-Colouring Problems

The H-colouring problem for a fixed digraph H Does an input digraph G admit a homomorphism to H?

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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H-Colouring Problems

The H-colouring problem for a fixed digraph H Does an input digraph G admit a homomorphism to H? Example: H = Kt Does an input graph G admit a t-colouring?

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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H-Colouring Problems

The H-colouring problem for a fixed digraph H Does an input digraph G admit a homomorphism to H? Example: H = Kt Does an input graph G admit a t-colouring? Dichotomy for t-colourings Polynomial for t ≤ 2, NP-complete for t > 2

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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H-Colouring Problems

The H-colouring problem for a fixed digraph H Does an input digraph G admit a homomorphism to H? Example: H = Kt Does an input graph G admit a t-colouring? Dichotomy for t-colourings Polynomial for t ≤ 2, NP-complete for t > 2 The H-colouring problem for a fixed relational structure H

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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H-Colouring Problems

The H-colouring problem for a fixed digraph H Does an input digraph G admit a homomorphism to H? Example: H = Kt Does an input graph G admit a t-colouring? Dichotomy for t-colourings Polynomial for t ≤ 2, NP-complete for t > 2 The H-colouring problem for a fixed relational structure H The "constraint satisfaction problem" with template H

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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H-Colouring Problems

The H-colouring problem for a fixed digraph H Does an input digraph G admit a homomorphism to H? Example: H = Kt Does an input graph G admit a t-colouring? Dichotomy for t-colourings Polynomial for t ≤ 2, NP-complete for t > 2 The H-colouring problem for a fixed relational structure H The "constraint satisfaction problem" with template H

Is there dichotomy?

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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Dichotomy Conjecture

The Feder - Vardi CSP dichotomy conjecture, 1993 For each relational structure H, the H-colouring problem is polynomial or is NP-complete

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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Dichotomy Conjecture

The Feder - Vardi CSP dichotomy conjecture, 1993 For each relational structure H, the H-colouring problem is polynomial or is NP-complete True if |V(H)| = 2

Schaeffer 1978 Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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Dichotomy Conjecture

The Feder - Vardi CSP dichotomy conjecture, 1993 For each relational structure H, the H-colouring problem is polynomial or is NP-complete True if |V(H)| = 2

Schaeffer 1978

True for undirected graphs H

H+Nešetˇ ril 1990: polynomial if H is bipartite or has a loop, NP-complete otherwise Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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Dichotomy Conjecture

The Feder - Vardi CSP dichotomy conjecture, 1993 For each relational structure H, the H-colouring problem is polynomial or is NP-complete True if |V(H)| = 2

Schaeffer 1978

True for undirected graphs H

H+Nešetˇ ril 1990: polynomial if H is bipartite or has a loop, NP-complete otherwise

Sufficient to show for all digraphs H

Feder-Vardi 1993 Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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Algebraic Classification

A k-ary polymorphism on H A homomorphism φ : Hk → H

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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Algebraic Classification

A k-ary polymorphism on H A homomorphism φ : Hk → H Dichotomy Classification Conjecture H-colouring is polynomial if H has a polymorphism f such that f(v, u, . . . , u) = f(u, v, . . . , u) = · · · = f(u, u, . . . , v) f(u, u, . . . , u) = u,

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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Algebraic Classification

A k-ary polymorphism on H A homomorphism φ : Hk → H Dichotomy Classification Conjecture H-colouring is polynomial if H has a polymorphism f such that f(v, u, . . . , u) = f(u, v, . . . , u) = · · · = f(u, u, . . . , v) f(u, u, . . . , u) = u, and is NP-complete otherwise

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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Algebraic Classification

A k-ary polymorphism on H A homomorphism φ : Hk → H Dichotomy Classification Conjecture H-colouring is polynomial if H has a polymorphism f such that f(v, u, . . . , u) = f(u, v, . . . , u) = · · · = f(u, u, . . . , v) f(u, u, . . . , u) = u, and is NP-complete otherwise A weak near unanimity polymorphism

Jeavons, Bulatov, Krokhin 2000 Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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Algebraic Classification

A k-ary polymorphism on H A homomorphism φ : Hk → H Dichotomy Classification Conjecture H-colouring is polynomial if H has a polymorphism f such that f(v, u, . . . , u) = f(u, v, . . . , u) = · · · = f(u, u, . . . , v) f(u, u, . . . , u) = u, and is NP-complete otherwise A weak near unanimity polymorphism

Jeavons, Bulatov, Krokhin 2000

Conjecture "WNU = P"

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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List H-Colouring Problems

Given a fixed digraph H Each vertex x of the input digraph G has a list L(x) ⊆ V(H)

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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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)?

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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List Homomorphism Problems

Fixed graph H Processors and connections

1 2 3 4 H

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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List Homomorphism Problems

3 23 03 012 12

G

. .

134

Input graph G Tasks and communications

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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List Homomorphism Problems

3 23 03 012 12

G

. .

134

1 2 3 4 H

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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List Homomorphism Problems

134 3 23 03 012 12

G

. .

1 2 3 4 H

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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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)?

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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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

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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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)|−1possible unary relations Ri(H) (non-empty subsets of V(H))

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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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)|−1possible unary relations Ri(H) (non-empty subsets of V(H)) Instead of having a list L(x), put x in the corresponding unary relation Ri(G)

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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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

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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

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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

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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

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List H-Colouring Problem for Irreflexive Graphs

For an irreflexive graph H

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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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

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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

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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

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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

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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 w b a c x y z w a b c x y z

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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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 w b a c x y z w a b c x y z OPEN: O(m + n) recognition

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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General Digraphs

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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General Digraphs

A k-ary polymorphism on a structure H A homomorphism φ : Hk → H

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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General Digraphs

A k-ary polymorphism on a structure H A homomorphism φ : Hk → H A conservative polymorphism f A polymorphism φ : Hk → H with φ(u1, . . . , uk) ∈ {u1, . . . , uk}

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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Min Ordering

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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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

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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)

v’ u’ v u Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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SLIDE 51

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)

v’ u’ v u

Each interval graph has a min ordering By the left endpoints

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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SLIDE 52

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)

v’ u’ v u

Each interval graph has a min ordering By the left endpoints

B A’ A A B’ B B’

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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Min Ordering

A min ordering < of V(H) uv, u′v′ ∈ E(H) = ⇒ min(u, u′) min(v, v′) ∈ E(H)

v’ u’ v u

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

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Conservative Ternary Polymorphisms

A conservative majority on H A conservative ternary polymorphism g : H3 → 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

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Conservative Ternary Polymorphisms

A conservative majority on H A conservative ternary polymorphism g : H3 → 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

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Conservative Ternary Polymorphisms

A conservative Maltsev on H A conservative ternary polymorphism h : H3 → H such that h(u, u, v) = h(v, u, u) = v

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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Conservative Ternary Polymorphisms

A conservative Maltsev on H A conservative ternary polymorphism h : H3 → 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

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An Algebraic Classification

For a general relational structure H If each pair u, v ∈ V(H) admits a conservative polymorphism f

  • f 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

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SLIDE 59

An Algebraic Classification

For a general relational structure H If each pair u, v ∈ V(H) admits a conservative polymorphism f

  • f 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

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SLIDE 60

An Algebraic Classification

For a general relational structure H If each pair u, v ∈ V(H) admits a conservative polymorphism f

  • f 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

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Existence of Polymorphisms of Digraphs

Invertible pair u, v v

  • u

v v u u u v

  • Pavol Hell, SFU and Charles U

List Homomorphisms, Time and Space

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SLIDE 62

Existence of Polymorphisms of Digraphs

Invertible pair u, v v

  • u

v v u 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

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SLIDE 63

Existence of Polymorphisms of Digraphs

Invertible pair u, v v

  • u

v v u u u v

  • Necessity

u v’ u’ u v’ v u’ v

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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SLIDE 64

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

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SLIDE 65

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

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SLIDE 66

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 s(u) b(u) u v

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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SLIDE 67

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 s(u) b(u) u v

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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SLIDE 68

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 b(w) u s(u) b(u) v s(v)b(v) w s(w)

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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SLIDE 69

Existence of Polymorphisms of Digraphs

Maltsev A digraph H admits a conservative Maltsev if and only if it has no end triple End triple b(u) u v w b(w) s(w) s(u)

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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Digraph Asteroidal Triples

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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Digraph Asteroidal Triples

An obstruction for both min ordering and conservative majority:

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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SLIDE 72

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 b(w) u s(u) b(u) v s(v)b(v) w s(w)

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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Digraph Asteroidal Triples

A DAT

a a’ a c b b’ a a’ b a b a’ c b’ a c a’ a’ a a c b b b a’ a’ a’ c b’ a’ a’ a’ c b’ b’ c

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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SLIDE 74

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

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SLIDE 75

What happened to Maltsev?

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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SLIDE 76

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

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SLIDE 77

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

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SLIDE 78

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

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SLIDE 79

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

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SLIDE 80

How about space?

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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SLIDE 81

How about space?

Trichotomy for list H-colouring when H is reflexive

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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SLIDE 82

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

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SLIDE 83

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

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SLIDE 84

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

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SLIDE 85

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 P4 = recursively built from unions and joins

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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SLIDE 86

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 P4 = recursively built from unions and joins Interval + Cograph = "Trivially Perfect" = no induced P4, C4 = recursively built from unions, and joins with K ∗

1

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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SLIDE 87

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 P4 = recursively built from unions and joins Interval + Cograph = "Trivially Perfect" = no induced P4, C4 = 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

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SLIDE 88

A General Classification for Digraphs

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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SLIDE 89

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

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SLIDE 90

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

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SLIDE 91

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

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SLIDE 92

A General Classification for Digraphs

A circular N

b ... ... a a b

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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SLIDE 93

A General Classification for Digraphs

A circular N

b ... ... a a b

Testing for the existence of a circular N is polynomial

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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SLIDE 94

Algebraic Classification for H-colouring

A Maltsev on H A ternary polymorphism h : H3 → H such that h(u, u, v) = h(v, u, u) = v

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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SLIDE 95

Algebraic Classification for H-colouring

A Maltsev on H A ternary polymorphism h : H3 → H such that h(u, u, v) = h(v, u, u) = v A Hagemann-Mitschke chain on H Ternary polymorphisms hi : H3 → H such that h1(u, v, v) = u hi(u, u, v) = hi+1(u, v, v) hk(u, u, v) = v

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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SLIDE 96

Algebraic Classification for H-colouring

A Maltsev on H A ternary polymorphism h : H3 → H such that h(u, u, v) = h(v, u, u) = v A Hagemann-Mitschke chain on H Ternary polymorphisms hi : H3 → H such that h1(u, v, v) = u hi(u, u, v) = hi+1(u, v, v) hk(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

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SLIDE 97

Syntactic Restrictions for H-colouring

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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SLIDE 98

Syntactic Restrictions for H-colouring

Examples the non-existence of a K1-colouring is expressible in first

  • rder logic

the non-existence of a K2-colouring is certifiable by

  • bstructions of treewidth two

the non-existence of a K2-colouring is expressible in datalog

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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SLIDE 99

Syntactic Restrictions for H-colouring

Examples the non-existence of a K1-colouring is expressible in first

  • rder logic

the non-existence of a K2-colouring is certifiable by

  • bstructions of treewidth two

the non-existence of a K2-colouring is expressible in datalog Datalog example

  • ddpath(u,v) <- (u∼v)
  • ddpath(u,v) <- oddpath(u,w), u∼z, z∼v

test <- oddpath(u,u)

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

slide-100
SLIDE 100

Syntactic Restrictions for H-colouring

Examples the non-existence of a K1-colouring is expressible in first

  • rder logic

the non-existence of a K2-colouring is certifiable by

  • bstructions of treewidth two

the non-existence of a K2-colouring is expressible in datalog Symmetric datalog example

  • ddpath(u,v) <- (u∼v)
  • ddpath(u,v) <- oddpath(u,w), w∼z, z∼v
  • ddpath(u,w) <- oddpath(u,v), w∼z, z∼v

test <- oddpath(u,u)

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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SLIDE 101

Syntactic Restrictions for H-colouring

Symmetric datalog If the non-existence of H-colouring is expressible in symmetric datalog, then H admits a Hagemann-Mitschke chain, and the H-colouring problem can be solved in logspace

Larose-Tesson 2009 Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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SLIDE 102

Syntactic Restrictions for H-colouring

Symmetric datalog If the non-existence of H-colouring is expressible in symmetric datalog, then H admits a Hagemann-Mitschke chain, and the H-colouring problem can be solved in logspace

Larose-Tesson 2009

LT Conjecture If H admits a Hagemann-Mitschke chain, then the non-existence of an H-colouring is expressible in symmetric datalog

Larose+Tesson 2009 Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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SLIDE 103

List H-Colouring when H is a Digraph∗

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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SLIDE 104

List H-Colouring when H is a Digraph∗

The following statements are equivalent H does not contain a circular N H admits a conservative Hagemann-Mitschke chain the non-existence of a list H-colouring is expressible in symmetric datalog

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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SLIDE 105

List H-Colouring when H is a Digraph∗

The following statements are equivalent H does not contain a circular N H admits a conservative Hagemann-Mitschke chain the non-existence of a list H-colouring is expressible in symmetric datalog In that case the list H-colouring problem is in logspace.

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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SLIDE 106

List H-Colouring when H is a Digraph∗

The following statements are equivalent H does not contain a circular N H admits a conservative Hagemann-Mitschke chain the non-existence of a list H-colouring is expressible in symmetric datalog In that case the list H-colouring problem is in logspace. Otherwise it is NL-complete

Larose+Tesson 2009 Egri+H+Larose+Rafiey 2013 Dalmau+Egri+H+Larose+Rafiey 2015 Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space

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SLIDE 107

List H-Colouring when H is a Digraph∗

The following statements are equivalent H does not contain a circular N H admits a conservative Hagemann-Mitschke chain the non-existence of a list H-colouring is expressible in symmetric datalog In that case the list H-colouring problem is in logspace. Otherwise it is NL-complete

Larose+Tesson 2009 Egri+H+Larose+Rafiey 2013 Dalmau+Egri+H+Larose+Rafiey 2015

"Symmetric Datalog = HM Chains = Logspace" The LT conjecture holds for digraph∗ list H-colouring

Pavol Hell, SFU and Charles U List Homomorphisms, Time and Space