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Constraint Satisfaction
and
Graph Theory
Constraint Satisfaction
and
Graph Theory
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PlanPlan
• How much is lost by focusing on graphs
• How graphs help intuition
• How graph theory gets used
• How much is lost by focusing on graphs
• How graphs help intuition
• How graph theory gets used
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Graph ColouringGraph Colouring
• One of the most common illustrations of constrain satisfaction problems
• All variables have the same domain, consisting of k colours
• One binary constraint requires adjacent pairs of variables to obtain different colours
• One of the most common illustrations of constrain satisfaction problems
• All variables have the same domain, consisting of k colours
• One binary constraint requires adjacent pairs of variables to obtain different colours
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Exam Scheduling via Graph ColouringExam Scheduling via Graph Colouring
G = courses and their conflicts
Colours = examination times
A colouring of G = a schedule of exam periods in which conflicting courses are scheduled at different times
G = courses and their conflicts
Colours = examination times
A colouring of G = a schedule of exam periods in which conflicting courses are scheduled at different times
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CS100Math101
Phys150 Chem110
Monday 9am
Monday 1pm
Tuesday 9am
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More realistic modelsMore realistic models
are possible with the full power of
constraint satisfaction…
are possible with the full power of
constraint satisfaction…
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More realistic modelsMore realistic models
are possible with the full power of
constraint satisfaction…
BUT
graph theory is not just colouring
are possible with the full power of
constraint satisfaction…
BUT
graph theory is not just colouring
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Graph HomomorphismsGraph Homomorphisms
Suppose G and H are graphs.
A homomorphism of G to H is a mapping f : V(G) V(H) such that uv G implies f(u)f(v) H
Adjacent vertices have adjacent images
Suppose G and H are graphs.
A homomorphism of G to H is a mapping f : V(G) V(H) such that uv G implies f(u)f(v) H
Adjacent vertices have adjacent images
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Exam SchedulingExam Scheduling
G = courses and their conflicts H = periods and their compatibility
A homomorphism of G to H = a schedule of exam periods so that conflicting courses are scheduled at compatible times
G = courses and their conflicts H = periods and their compatibility
A homomorphism of G to H = a schedule of exam periods so that conflicting courses are scheduled at compatible times
Via Graph HomomorphismsVia Graph Homomorphisms
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CS100
Phys150 Chem110
Monday 9am
Monday 1pm
Tuesday 9am
Biol101
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General CSP’sGeneral CSP’s
• Given a vocabulary (finite number of relation symbols, each of finite arity)
• and two structures G and H over the vocabulary, with ground sets V(G) and V(H), and interpretations of all the relation symbols, as relations over the set of the stated arities
• Given a vocabulary (finite number of relation symbols, each of finite arity)
• and two structures G and H over the vocabulary, with ground sets V(G) and V(H), and interpretations of all the relation symbols, as relations over the set of the stated arities
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Homomorphism of G to HHomomorphism of G to H
A mapping f : V(G) V(H) such that
(v1,…,vr) R(G) implies (f(v1),…,f(vr)) R(H),
for all relation symbols R of the vocabulary
A mapping f : V(G) V(H) such that
(v1,…,vr) R(G) implies (f(v1),…,f(vr)) R(H),
for all relation symbols R of the vocabulary
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Graph Theory in CSPGraph Theory in CSP
Take the vocabulary to consist of
one binary relation symbol E
(possibly with restricted interpretations,
to reflexive, symmetric, etc, relations)
Take the vocabulary to consist of
one binary relation symbol E
(possibly with restricted interpretations,
to reflexive, symmetric, etc, relations)
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•Conflicting courses at compatible times
(R(G) and R(H) binary relations)
At most three chemistry exams in one day
(R(G) and R(H) quaternary relations)
All physics exams on the first ten days (R(G) and R(H) unary relations)
•Conflicting courses at compatible times
(R(G) and R(H) binary relations)
At most three chemistry exams in one day
(R(G) and R(H) quaternary relations)
All physics exams on the first ten days (R(G) and R(H) unary relations)
V(G) = courses V(H) = times V(G) = courses V(H) = times
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G changes every semester H tends to stay constant for a while
The Constraint Satisfaction Problem
for a fixed a structure H:
CSP(H)
Given a structure G (same vocabulary as H)
Is there a homomorphism of G to H ?
G changes every semester H tends to stay constant for a while
The Constraint Satisfaction Problem
for a fixed a structure H:
CSP(H)
Given a structure G (same vocabulary as H)
Is there a homomorphism of G to H ?
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Dichotomy Conjecture [FV]
Dichotomy Conjecture [FV]
For each H, the problem CSP(H) is in P
or is NP-complete
For each H, the problem CSP(H) is in P
or is NP-complete
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When H is a graph (one binary relation symbol E)
CSP(H) is called the H-colouring problem
graphs … E(G), E(H) symmetric digraphs … unrestricted interpretations
Also denoted COL(H) and HOM(H)
When H is a graph (one binary relation symbol E)
CSP(H) is called the H-colouring problem
graphs … E(G), E(H) symmetric digraphs … unrestricted interpretations
Also denoted COL(H) and HOM(H)
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CSP(Kn)CSP(Kn)
(one binary relation, `distinct’)
CSP(Kn) is the n-colouring problem
(one binary relation, `distinct’)
CSP(Kn) is the n-colouring problem
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DICHOTOMYDICHOTOMY
The n-colouring problem is in P for n = 1, 2 NP-complete for all other n
The graph H-colouring problem is
in P for H bipartite or with a loop
NP-complete for all other H
The n-colouring problem is in P for n = 1, 2 NP-complete for all other n
The graph H-colouring problem is
in P for H bipartite or with a loop
NP-complete for all other H
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DICHOTOMYDICHOTOMY
The H-colouring problem is [HN] in P for H bipartite or
with a loop NP-complete for all other H
The H-colouring problem is [HN] in P for H bipartite or
with a loop NP-complete for all other H
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H-colouringH-colouring
• May assume G is connected
• May assume H is connected
• May assume H is a core (there is no homomorphism to a proper subgraph)
• May assume G is connected
• May assume H is connected
• May assume H is a core (there is no homomorphism to a proper subgraph)
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Polynomial casesPolynomial cases
• H has a loop (some vv E(H))All G admit a homomorphism to
H
H is bipartite (V(H) = two independent sets)Thus we may assume H = K2
(core)
• H has a loop (some vv E(H))All G admit a homomorphism to
H
H is bipartite (V(H) = two independent sets)Thus we may assume H = K2
(core)
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Deciding if a 2-colouring exists Deciding if a 2-colouring exists
Obvious algorithm
1
1 1 1
2 22
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Deciding if a 2-colouring exists Deciding if a 2-colouring exists
Algorithm succeeds
2-colouring existsNo odd cycles
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Deciding if a 2-colouring exists Deciding if a 2-colouring exists
Obvious algorithm
1
1 1 1
2 22
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Deciding if a 2-colouring exists Deciding if a 2-colouring exists
Algorithm succeeds
2-colouring existsNo odd cycles
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Deciding if a 2-colouring exists Deciding if a 2-colouring exists
Algorithm succeeds
2-colouring existsNo odd cycles
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G has a 2-colouring(is bipartite)
G has a 2-colouring(is bipartite)
if and only if it contains no induced
if and only if it contains no induced
7 . . .35
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NP-complete casesNP-complete cases
• Cliques Kn with n 3
• Pentagon C5
• Cliques Kn with n 3
• Pentagon C5
1
2
34
5
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C5 - colouring is NP-complete
C5 - colouring is NP-complete
Reduce from 5-colourability:Reduce from 5-colourability:
G
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C5 - colouring is NP-complete
C5 - colouring is NP-complete
Reduce from 5-colourability:Reduce from 5-colourability:
G*
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C5 - colouring is NP-complete
C5 - colouring is NP-complete
Reduce from 5-colourability:
G is 5-colourableif and only if G* is C5-colourable
Reduce from 5-colourability:
G is 5-colourableif and only if G* is C5-colourable
G*
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C5 - colouring is NP-complete
C5 - colouring is NP-complete
Reduce from K5-colourability:
G is K5-colourable
if and only if G* is C5-
colourable
From K5 to C5
Reduce from K5-colourability:
G is K5-colourable
if and only if G* is C5-
colourable
From K5 to C5
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C5 - colouring is NP-complete
C5 - colouring is NP-complete
Reduce from K5-colourability:
G is K5-colourable
if and only if G* is C5-colourable
From K5 to C5
From denser to sparser…
Reduce from K5-colourability:
G is K5-colourable
if and only if G* is C5-colourable
From K5 to C5
From denser to sparser…
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W5 - colouring is NP-complete
W5 - colouring is NP-complete
From to
C5 W5
From smaller to larger…
From to
C5 W5
From smaller to larger…
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maps to maps toiff
…
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maps to maps toiff
…
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Enough to prove forEnough to prove for
…
…
[HN] 1988 [B] 2005 (Monotonicity)
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Digraph H-colouringDigraph H-colouring
• Dichotomy not known
• No classification in terms of digraph properties proposed
• Each CSP(H) is polynomially equivalent to some digraph H’-colouring problem [FV]
Monotonicity fails
• Dichotomy not known
• No classification in terms of digraph properties proposed
• Each CSP(H) is polynomially equivalent to some digraph H’-colouring problem [FV]
Monotonicity fails
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Monotonicity failsMonotonicity fails
NP-complete In P
[GWW]
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CSP(H) as H’-COLCSP(H) as H’-COL
• We may assume that H is a core
• We may replace CSP(H) by RET(H)
(H is a substructure of G and we seek a homomorphism of G to H that keeps vertices of H fixed)
• We may assume that H is a core
• We may replace CSP(H) by RET(H)
(H is a substructure of G and we seek a homomorphism of G to H that keeps vertices of H fixed)
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RET(H)RET(H)
H
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RET(H)RET(H)
H
G
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RET(H)RET(H)
Retraction impossible
G
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RET(H)RET(H)
Ha
b
Retraction impossible, but homomorphism exists
a
a
a
a
b
bb
G
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RET(H)RET(H)
If H is a core then CSP(H) and RET(H) are polynomially equivalentIf H is a core then CSP(H) and RET(H) are polynomially equivalent
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Each CSP(H) is polynomially equivalent to
some digraph H’-colouring problem
• Each RET(H) is polynomially equivalent to some bipartite graph retraction problem
• Each bipartite graph retraction problem is polynomially equivalent to some digraph H’-colouring problem
Each CSP(H) is polynomially equivalent to
some digraph H’-colouring problem
• Each RET(H) is polynomially equivalent to some bipartite graph retraction problem
• Each bipartite graph retraction problem is polynomially equivalent to some digraph H’-colouring problem
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RET(H) as RET(H’)RET(H) as RET(H’)
(H is an arbitrary structure, H’ is a bip graph)
(H is an arbitrary structure, H’ is a bip graph)
H’
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RET(H) as RET(H’)RET(H) as RET(H’)
(H is an arbitrary structure, H’ is a bip graph)(H is an arbitrary structure, H’ is a bip graph)
H’
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From bip RET(H) to digraph H’-colouringFrom bip RET(H) to
digraph H’-colouring
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CSP(H) is polynomially equivalent
CSP(H) is polynomially equivalent
• to CSP(H’) for some digraph H’
• to RET(H’) for some digraph H’
• to RET(H’) for some bipartite graph H’
• to RET(H’) for some reflexive graph H’
• to RET(H’) for some partial order H’
• to CSP(H’) for some digraph H’
• to RET(H’) for some digraph H’
• to RET(H’) for some bipartite graph H’
• to RET(H’) for some reflexive graph H’
• to RET(H’) for some partial order H’
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Monotonicity failsMonotonicity fails
NP-complete In P
[GWW]
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NP-completenessNP-completeness
Reduce ONE-IN-THREE-SATReduce ONE-IN-THREE-SAT
T
F F
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Polynomial algorithmPolynomial algorithm
is in P iffis
(and it is)
is in P iffis
(and it is)
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Deciding if a C3-colouring exists Deciding if a C3-colouring exists
Obvious algorithm
0
2 0 1
1 210
12
1
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Deciding if a C3-colouring exists Deciding if a C3-colouring exists
Algorithm succeeds
C3-colouring existsNo bad cycles
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Deciding if a C3-colouring exists Deciding if a C3-colouring exists
Obvious algorithm
0
2 0 1
1 210
12
1
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Deciding if a C3-colouring exists Deciding if a C3-colouring exists
Obvious algorithm
0
2 0 1
1 210
12
1
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Deciding if a C3-colouring exists Deciding if a C3-colouring exists
Algorithm succeeds
C3-colouring existsNo bad cycles
A cycle is good if it has net length 3k
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Algorithm succeeds
C3-colouring existsNo bad cycles
Deciding if a C3-colouring exists Deciding if a C3-colouring exists
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Algorithm succeeds
C3-colouring existsNo bad cycles
Deciding if a C3-colouring exists Deciding if a C3-colouring exists
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G has a Ck-colouring G has a Ck-colouring
if and only if
G contains no cycle of net length ≠0(mod k)
if and only if
G contains no cycle of net length ≠0(mod k)
01
2
…
kCk
[BB]
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Monotonicity failsMonotonicity fails
NP-complete In P
Not hereditarily hard (Polynomial extension)(irreflexive)
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Another perspective on the undirected
dichotomy
Another perspective on the undirected
dichotomy
• Codd is hereditarily hard
enough to show
• C3 is hereditarily hard
• Codd is hereditarily hard
enough to show
• C3 is hereditarily hard
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Two kinds of difficultyTwo kinds of difficulty
• `Few’ directed cycles– monotonicity fails (oscilation)– unclear distinctions
• `Few’ directed cycles– monotonicity fails (oscilation)– unclear distinctions
In P NP-complete
[HNZ] [GWW]
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NP-complete caseNP-complete case
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Two kinds of difficultyTwo kinds of difficulty
• `Few’ directed cycles– monotonicity fails (oscilation)– unclear distinctions– BUT
• `Few’ directed cycles– monotonicity fails (oscilation)– unclear distinctions– BUT
NP-c P - extension
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Two kinds of difficultyTwo kinds of difficulty
• `Few’ directed cycles• `Few’ directed cycles
NP-complete[BM] [GWW]
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Two kinds of difficultyTwo kinds of difficulty
• `Few’ directed cycles
– BUT
• `Few’ directed cycles
– BUT
NP-complete P - extension
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Two kinds of difficultyTwo kinds of difficulty
• `Many’ directed cycles• `Many’ directed cycles
H0
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Two kinds of difficultyTwo kinds of difficulty
• `Many’ directed cycles• `Many’ directed cycles
Hereditarily hard
(if H contains H0 and H has no loops, then CSP(H) is NP-complete)
H0
[BHM]
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Classification Conjecture [BHM]
Classification Conjecture [BHM]
• If H* (H with sources and sinks recursively removed) admits a homomorphism to some Ck (k>1) then H has a polynomial extension
(`few cycles’)
• Otherwise, H is hereditarily hard (`many cycles’)
• If H* (H with sources and sinks recursively removed) admits a homomorphism to some Ck (k>1) then H has a polynomial extension
(`few cycles’)
• Otherwise, H is hereditarily hard (`many cycles’)
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Status of the conjecture
Status of the conjecture
• True for several graph families
Open for
• True for several graph families
Open for
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Status of the conjecture
Status of the conjecture
• True for several graph families
Open for
• True for several graph families
Open for
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Status of the conjecture
Status of the conjecture
• True for several graph families
Open for
True for
• True for several graph families
Open for
True for
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True for partitionable graphs
True for partitionable graphs
If the bi-directed edges of H form a nonbipartite graph, then CSP(H) is NPcIf the bi-directed edges of H form a nonbipartite graph, then CSP(H) is NPc
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• Each v incident with a bi-directed edge
• Bi-directed edges between parts• Uni-directed edges within parts
• Each v incident with a bi-directed edge
• Bi-directed edges between parts• Uni-directed edges within parts
True for partitionable graphs
True for partitionable graphs
[BHM]
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True for partitionable graphs
True for partitionable graphs
• Each v incident with a bi-directed edge
• Bi-directed edges between parts• Uni-directed edges within parts
• Each v incident with a bi-directed edge
• Bi-directed edges between parts• Uni-directed edges within parts
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True for partitionable graphs
True for partitionable graphs
• Each v incident with a bi-directed edge
• Bi-directed edges between parts• Uni-directed edges within parts
• Each v incident with a bi-directed edge
• Bi-directed edges between parts• Uni-directed edges within parts
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True for partitionable graphs
True for partitionable graphs
• Each v incident with a bi-directed edge• Bi-directed edges between parts• Uni-directed edges within parts ?
• Each v incident with a bi-directed edge• Bi-directed edges between parts• Uni-directed edges within parts ?
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What algorithms?What algorithms?
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G has a Ck-colouring G has a Ck-colouring
if and only if
G contains no cycle of net length ≠0(mod k)
if and only if
G contains no cycle of net length ≠0(mod k)
01
2
…
kCk
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G has a Ck-colouring G has a Ck-colouring
if and only if
there is no homomorphism to G from a
cycle of net length ≠0 (mod k)
if and only if
there is no homomorphism to G from a
cycle of net length ≠0 (mod k)
…(obstacles are oriented cycles)
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G has a Pk-colouring G has a Pk-colouring
if and only if
there is no homomorphism to G from a
path of net length > k[BB]
if and only if
there is no homomorphism to G from a
path of net length > k[BB]
(obstacles are oriented paths)
… Pk
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G has a P-colouring G has a P-colouring
if and only if
there is no homomorphism to G from a
path P’ which is bad(does not admit a homomorphism to P)
if and only if
there is no homomorphism to G from a
path P’ which is bad(does not admit a homomorphism to P)
(obstacles are oriented paths)
[HZ] (cf also [GWW])
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G has a Tk-colouring G has a Tk-colouring
if and only if
there is no homomorphism to G from Pk
if and only if
there is no homomorphism to G from Pk
(obstacles are oriented paths)
… Tk
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What algorithms?What algorithms?
• H has tree duality
a family H of oriented trees
such that G admits an H-colouring iff there is no homomorphism from a T H to G
(obstacles are oriented trees)
• H has tree duality
a family H of oriented trees
such that G admits an H-colouring iff there is no homomorphism from a T H to G
(obstacles are oriented trees)
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What algorithms?What algorithms?
• H has treewidth k duality
a family H of digraphs of treewidth k
such that G admits an H-colouring iff there is no homomorphism from a T H to G
(obstacles are digraphs of small treewidth)
• H has treewidth k duality
a family H of digraphs of treewidth k
such that G admits an H-colouring iff there is no homomorphism from a T H to G
(obstacles are digraphs of small treewidth)
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Bounded treewidth duality
Bounded treewidth duality
k such that obstacles are graphs of treewidth k
If H has bounded treewidth duality,
then CSP(H) is in P[HNZ] [FV] width, datalog
k such that obstacles are graphs of treewidth k
If H has bounded treewidth duality,
then CSP(H) is in P[HNZ] [FV] width, datalog
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What algorithms?What algorithms?
• Tree duality (obstacles are oriented trees)
• Treewidth two duality (obstacles are graphs of treewidth 2)
• Bounded treewidth duality
• There exist H without bounded treewidth duality but with CSP(H) in P [A]
• Tree duality (obstacles are oriented trees)
• Treewidth two duality (obstacles are graphs of treewidth 2)
• Bounded treewidth duality
• There exist H without bounded treewidth duality but with CSP(H) in P [A]
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G has a Tk-colouring G has a Tk-colouring
if and only if
there is no homomorphism to G from Pk
if and only if
there is no homomorphism to G from Pk
(obstacles are oriented paths)
… Tk
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Finitary dualitiesFinitary dualities
• If H has finitary duality, then H has tree duality [NT]
• This happens if and only if H-colourability is first-order definable [A]Also follows from [R]
• If H has finitary duality, then H has tree duality [NT]
• This happens if and only if H-colourability is first-order definable [A]Also follows from [R]
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Add unary relationsAdd unary relations (still graph theory)
Fix a graph H with k vertices.
Vocabulary has one binary relation name E and a set of unary relation names U1, U2,…, U2k-1
Interpret H with Ui(H) the subsets of V(H)
(conservative structure H)
(still graph theory)
Fix a graph H with k vertices.
Vocabulary has one binary relation name E and a set of unary relation names U1, U2,…, U2k-1
Interpret H with Ui(H) the subsets of V(H)
(conservative structure H)
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List homomorphism problem
List homomorphism problemLHOM(H) CSP(H*)
Fixed graph H
Given an input graph G, with lists L(v) V(H), v V(G)
Is there a homomorphism f of G to H
with all f(v) L(v) ?
LHOM(H) CSP(H*)
Fixed graph H
Given an input graph G, with lists L(v) V(H), v V(G)
Is there a homomorphism f of G to H
with all f(v) L(v) ?
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RET(H)RET(H)
LHOM(H) restricted to inputs G containing
H with L(v)={v} for all vertices v of H
LHOM(H) restricted to inputs G containing
H with L(v)={v} for all vertices v of H
u u
v v
w w x yx y
u,v,w,x,y u,v,w,x,yu,v,w,x,y
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List homomorphism problem
List homomorphism problem
Assume E(H) is reflexive
If H is an interval graph, thenLHOM(H) is in P
Otherwise, LHOM(H) is NP-complete
[FH]
Assume E(H) is reflexive
If H is an interval graph, thenLHOM(H) is in P
Otherwise, LHOM(H) is NP-complete
[FH]
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Interval graphsInterval graphs
a
ec d
b
dc e
b a
(reflexive)
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Interval graphsInterval graphs
a
ec d
b
dc e
b a
have conservative majority polymorphisms
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Interval graphsInterval graphs
a
ec d
b
dc e
b a
conservative majority polymorphisms
m(a,b,c) = a, b, or c
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Structural characterization
Structural characterization
• H is an interval graph if and only if it does not have an induced cycle of length >3, or an asteroidal triple of vertices [LB]
• H is an interval graph if and only if it does not have an induced cycle of length >3, or an asteroidal triple of vertices [LB]
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List homomorphism problem
List homomorphism problem
Assume E(H) is reflexive
If H is an interval graph, thenLHOM(H) is in P
Otherwise, LHOM(H) is NP-complete
[FH]
Assume E(H) is reflexive
If H is an interval graph, thenLHOM(H) is in P
Otherwise, LHOM(H) is NP-complete
[FH]
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Conservative majority polymorphisms
Conservative majority polymorphisms
• Reflexive interval graphs have them
• They imply polynomiality of LHOM(H)
• LHOM(H) is NP-complete for other reflexive graphs
• Reflexive interval graphs have them
• They imply polynomiality of LHOM(H)
• LHOM(H) is NP-complete for other reflexive graphs
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Predicted theoremPredicted theorem
A reflexive graph H has a conservative
majority function iff it is an interval graph
[BFHHM]
A reflexive graph H has a conservative
majority function iff it is an interval graph
[BFHHM]
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Predicted theoremPredicted theorem
A reflexive graph H has a conservative majority function iff it is an interval graph
[BFHHM]
A reflexive graph H has a majority function
if and only if it is a retract of a product of
interval graphs [JMP] [HR]
A reflexive graph H has a conservative majority function iff it is an interval graph
[BFHHM]
A reflexive graph H has a majority function
if and only if it is a retract of a product of
interval graphs [JMP] [HR]
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Absolute retractsAbsolute retracts
• A reflexive graph H is a retract of every G which contains H as an isometric subgraph iff H has a majority polymorphism [JMP] [HR]
• A reflexive graph H is a retract of every G which contains H as an isometric subgraph iff H has a majority polymorphism [JMP] [HR]
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Absolute retractsAbsolute retracts
• A reflexive graph H is a retract of every G which contains H as an isometric subgraph iff H has a majority polymorphism [JMP] [HR]
• But RET(H) is polynomial in other cases
• A reflexive graph H is a retract of every G which contains H as an isometric subgraph iff H has a majority polymorphism [JMP] [HR]
• But RET(H) is polynomial in other cases
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List homomorphism problem
List homomorphism problem
In general
If H is a bi-arc graph, then LHOM(H) is in P
Otherwise, LHOM(H) is NP-complete[FHH]
G has cons. majority iff it is a bi-arc graph
[BFHHM]
In general
If H is a bi-arc graph, then LHOM(H) is in P
Otherwise, LHOM(H) is NP-complete[FHH]
G has cons. majority iff it is a bi-arc graph
[BFHHM]
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List homomorphism problem
List homomorphism problem
In general
If H is a bi-arc graph, then LHOM(H) is in P
Otherwise, LHOM(H) is NP-complete[FHH]
G has cons. near un. iff it is a bi-arc graph
[BFHHM]
In general
If H is a bi-arc graph, then LHOM(H) is in P
Otherwise, LHOM(H) is NP-complete[FHH]
G has cons. near un. iff it is a bi-arc graph
[BFHHM]
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Dichotomy for conservative HDichotomy for conservative H
(not graph theory)
CSP(H) is in P or is NP-complete [B]
(not graph theory)
CSP(H) is in P or is NP-complete [B]
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Digraphs HDigraphs H• Conjecture 1 [FH]
– For reflexive digraphs, LHOM(H) Pwhen H has the X-underbar property and is NP-complete otherwise
• Conjecture 2 [FH]
– For irreflexive digraphs, LHOM(H) Pwhen H has a majority function, and is NP-complete otherwise
• Conjecture 1 [FH]
– For reflexive digraphs, LHOM(H) Pwhen H has the X-underbar property and is NP-complete otherwise
• Conjecture 2 [FH]
– For irreflexive digraphs, LHOM(H) Pwhen H has a majority function, and is NP-complete otherwise
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Restricted listsRestricted lists
Suppose each list induces a connected
subgraph of H
–If H is chordal CLHOM(H) P
–Otherwise CLHOM(H) is NP-complete
[FH]
Suppose each list induces a connected
subgraph of H
–If H is chordal CLHOM(H) P
–Otherwise CLHOM(H) is NP-complete
[FH]
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Chordal graphChordal graph
• Intersection graph of subtrees in a tree
• No induced cycle of length > 3
• Perfect elimination ordering of vertices
• Have near unanimity polymorphisms [BFHHM]
• Intersection graph of subtrees in a tree
• No induced cycle of length > 3
• Perfect elimination ordering of vertices
• Have near unanimity polymorphisms [BFHHM]
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Minimum Cost HomomorphismsMinimum Cost Homomorphisms
• Let cv(w) = the cost of mapping
v V(G) to w
V(H)
Compute minf ∑v cv(f(v))
MCHOM(H) (H is fixed, input is G and c)
• Let cv(w) = the cost of mapping
v V(G) to w
V(H)
Compute minf ∑v cv(f(v))
MCHOM(H) (H is fixed, input is G and c)
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DichotomyDichotomy
• If each component of H is a reflexive interval graph or an irreflexive interval bigraph, then MCHOM(H) P
[GHRY,CCJK]
• Otherwise, MCHOM(H) is NP-complete[GHRY]
• If each component of H is a reflexive interval graph or an irreflexive interval bigraph, then MCHOM(H) P
[GHRY,CCJK]
• Otherwise, MCHOM(H) is NP-complete[GHRY]
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Relation to Soft Constraints
Relation to Soft Constraints
• Unary constraints (lists) are soft
• Binary constraint (edges) are crisp
• Unary constraints (lists) are soft
• Binary constraint (edges) are crisp
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Proper interval graphsProper interval graphs
• Representable by an inclusion-free family of intervals
• A reflexive graph G is a proper interval graph iff it has no induced
• Representable by an inclusion-free family of intervals
• A reflexive graph G is a proper interval graph iff it has no induced
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Proper interval graphsProper interval graphs
• Representable by an inclusion-free family of intervals
• A reflexive graph G is a proper interval graph iff it has a Min-Max polymorphism
• Representable by an inclusion-free family of intervals
• A reflexive graph G is a proper interval graph iff it has a Min-Max polymorphism
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Reflexive graphs HReflexive graphs H
• HOM(H) (=CSP(H)) always easy
• RET(H) dichotomy open
• LHOM(H) easy just for interval graphs
• MCHOM(H) easy just for proper interval graphs
• HOM(H) (=CSP(H)) always easy
• RET(H) dichotomy open
• LHOM(H) easy just for interval graphs
• MCHOM(H) easy just for proper interval graphs
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ReviewReview
• How much is lost by focusing on graphs
• How graphs help intuition
• How graph theory gets used
• How much is lost by focusing on graphs
• How graphs help intuition
• How graph theory gets used
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OverviewsOverviews
G. Hahn, G. MacGillivray, Graph homomorphisms: computational aspects and infinite graphs, 2006
• P. Hell, J. Nesetril, Graph Homomorphisms, OUP, 2004
• P.Hell, From graph colouring to constraint satisfaction: there and back again, 2006
• P. Hell, Algorithmic aspects of graph homomorphisms, Surveys in Combinatorics 2003, LMS Lecture Note Series 307
G. Hahn, G. MacGillivray, Graph homomorphisms: computational aspects and infinite graphs, 2006
• P. Hell, J. Nesetril, Graph Homomorphisms, OUP, 2004
• P.Hell, From graph colouring to constraint satisfaction: there and back again, 2006
• P. Hell, Algorithmic aspects of graph homomorphisms, Surveys in Combinatorics 2003, LMS Lecture Note Series 307
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SourcesSources• A. Atserias, On digraph coloring problems and treewidth
duality, LICS 2005• J. Bang-Jensen, P. Hell, On the effect of two cycles on the
complexity of colouring, DAM 26 (1990) J. Bang Jensen, G. MacGillivray, On the complexity of
colouring by digraphs with at most one directed cycle, Ars Combin. 35A (1993)
• J. Bang Jensen, P. Hell, G. MacGillivray, Hereditarily hard H-colouring problems, DM 138 (1995)
• J. Bang Jensen, P. Hell, G. MacGillivray, On the complexity of colouring by superdigraphs of bipartite graphs, DM 109 (1992)
• A. Bulatov, Tractable conservative constraint satisfaction problems, ACM TCL, to appear
• A. Bulatov, H-coloring dichotomy revisited, manuscript 2005• A. Bulatov, A. Krokhin, P. Jeavons, Constraint satisfaction
problems and finite algebras, ICALP 2000
• A. Atserias, On digraph coloring problems and treewidth duality, LICS 2005
• J. Bang-Jensen, P. Hell, On the effect of two cycles on the complexity of colouring, DAM 26 (1990)
J. Bang Jensen, G. MacGillivray, On the complexity of colouring by digraphs with at most one directed cycle, Ars Combin. 35A (1993)
• J. Bang Jensen, P. Hell, G. MacGillivray, Hereditarily hard H-colouring problems, DM 138 (1995)
• J. Bang Jensen, P. Hell, G. MacGillivray, On the complexity of colouring by superdigraphs of bipartite graphs, DM 109 (1992)
• A. Bulatov, Tractable conservative constraint satisfaction problems, ACM TCL, to appear
• A. Bulatov, H-coloring dichotomy revisited, manuscript 2005• A. Bulatov, A. Krokhin, P. Jeavons, Constraint satisfaction
problems and finite algebras, ICALP 2000
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• G. Bloom, S. Burr, On unavoidable digraphs in orientations of graphs, JGT 11 (1987)
• R. Brewster, T. Feder, P. Hell, J. Huang, G. MacGillivray, NUF’s, 2004
• D. Cohen, M. Cooper, P. Jeavons, A. Krokhin, A maximal tractable class of soft constraints, JAIR 22 (2004)G. Bloom, S. Burr, On unavoidable digraphs in orientations of graphs, JGT 11 (1987)
• R. Brewster, T. Feder, P. Hell, J. Huang, G. MacGillivray, NUF’s, 2004
• D. Cohen, M. Cooper, P. Jeavons, A. Krokhin, A maximal tractable class of soft constraints, JAIR 22 (2004)T. Feder, P. Hell, J. Huang, Bi-arc graphs and the complexity of list homomorphisms, JGT 42 (2003)
• T. Feder, P. Hell, List homomorphisms to reflexive graphs, JCT B 72 (1998)
• T. Feder, P. Hell, J. Huang, List homomorphisms to reflexive digraphs, manuscript 2003
• T. Feder, M. Vardi, The computational structure of monotone monadic SNP and constraint satisfaction, SICOMP 28 (1998)
• G. Gutin, A. Rafei, P. Hell, A. Yeo, Minimum cost homomorphisms, manuscript 2005
• G. Bloom, S. Burr, On unavoidable digraphs in orientations of graphs, JGT 11 (1987)
• R. Brewster, T. Feder, P. Hell, J. Huang, G. MacGillivray, NUF’s, 2004
• D. Cohen, M. Cooper, P. Jeavons, A. Krokhin, A maximal tractable class of soft constraints, JAIR 22 (2004)G. Bloom, S. Burr, On unavoidable digraphs in orientations of graphs, JGT 11 (1987)
• R. Brewster, T. Feder, P. Hell, J. Huang, G. MacGillivray, NUF’s, 2004
• D. Cohen, M. Cooper, P. Jeavons, A. Krokhin, A maximal tractable class of soft constraints, JAIR 22 (2004)T. Feder, P. Hell, J. Huang, Bi-arc graphs and the complexity of list homomorphisms, JGT 42 (2003)
• T. Feder, P. Hell, List homomorphisms to reflexive graphs, JCT B 72 (1998)
• T. Feder, P. Hell, J. Huang, List homomorphisms to reflexive digraphs, manuscript 2003
• T. Feder, M. Vardi, The computational structure of monotone monadic SNP and constraint satisfaction, SICOMP 28 (1998)
• G. Gutin, A. Rafei, P. Hell, A. Yeo, Minimum cost homomorphisms, manuscript 2005
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• W. Gutjahr, E. Welzl, G. Woeginger, Polynomial graph-colorings, DAM 35 (1992)
• P. Hell, J. Huang, Interval bigraphs and circular arc graphs, JGT 46 (2004)
• P. Hell, J. Nesetril, On the complexity of H-colouring, JCT B 48 (1990)
• P. Hell, J. Nesetril, X. Zhu, Duality and polynomial testing of tree homomorphisms, TAMS 348 (1996)
• P. Hell, J. Nesetril, X. Zhu, Complexity of tree homomorphisms, DAM 70 (1996)
• P. Hell, I. Rival, Retracts in graphs, Canad. J. Math. 39 (1987)
• P. Hell, X. Zhu, Homomorphisms to oriented paths, DM 132 (1994)
• E.M. Jawhari, D. Misane, M. Pouzet, Contemp. Math. 57 (1986)
• J. Nesetril, C. Tardif, Duality theorems for finite structures, JCT B 80 (2000)
• B. Rossman, Existential positive types and preservation under homomorphisms, LICS 2005
• W. Gutjahr, E. Welzl, G. Woeginger, Polynomial graph-colorings, DAM 35 (1992)
• P. Hell, J. Huang, Interval bigraphs and circular arc graphs, JGT 46 (2004)
• P. Hell, J. Nesetril, On the complexity of H-colouring, JCT B 48 (1990)
• P. Hell, J. Nesetril, X. Zhu, Duality and polynomial testing of tree homomorphisms, TAMS 348 (1996)
• P. Hell, J. Nesetril, X. Zhu, Complexity of tree homomorphisms, DAM 70 (1996)
• P. Hell, I. Rival, Retracts in graphs, Canad. J. Math. 39 (1987)
• P. Hell, X. Zhu, Homomorphisms to oriented paths, DM 132 (1994)
• E.M. Jawhari, D. Misane, M. Pouzet, Contemp. Math. 57 (1986)
• J. Nesetril, C. Tardif, Duality theorems for finite structures, JCT B 80 (2000)
• B. Rossman, Existential positive types and preservation under homomorphisms, LICS 2005