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Graph Partitions

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Sets may be required to be independent. This kind of circle. depicts an ... Deciding if a k-colouring exists is. in P for k =1, 2. NP-complete ... [Berge] ... – PowerPoint PPT presentation

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Title: Graph Partitions


1
Graph Partitions
2
  • Vertex partitions
  • Partition V(G) into k sets

(k3)
3
This kind of circle depicts an arbitrary set
This kind of line means there may be edges
between the two sets
4
Special properties of partitions
  • Sets may be required to be independent

5
This kind of circle depicts an independent set
6
This is just a k-colouring
(k3)
7
Deciding if a k-colouring exists is
  • in P for k 1, 2
  • ? NP-complete for all other k

(k2)
8
Deciding if a 2-colouring exists
1
Obvious algorithm
2
2
2
1
1
1
1
2
(k2)
9
Deciding if a 2-colouring exists
Algorithm succeeds
2-colouring exists
No odd cycles
10
Deciding if a 2-colouring exists
Algorithm succeeds
2-colouring exists
No odd cycles
11
Deciding if a 2-colouring exists
Algorithm succeeds
2-colouring exists
No odd cycles
12
G has a 2-colouring(is bipartite)
  • if and only if it contains no induced

7 . . .
3
5
13
Special properties of partitions
  • ? Sets may be required to have no edges joining
    them

14
This kind of dotted line means there are no
edges joining the two sets
15
  • This is (corresponds to) a homomorphism.
  • Here a homomorphism to C5 - also known
  • as a C5-colouring.

16
A homomorphism of G to H(or an H-colouring of G)
  • is a mapping f V(G) ??V(H) such that
  • uv ? E(G) implies f(u)f(v) ? E(H).
  • A homomorphism f of G to C5 corresponds to
  • a partition of V(G) into five independent sets
  • with the right connections.

17
f -1(1)
f -1(2)
f -1(3)
f -1(5)
f -1(4)
1
2
3
5
C5
4
18
1
2
3
5
4
19
Special properties of partitions
  • Sets may be required to be cliques

20
This kind of circle depicts a clique
21
  • This is just a colouring of the
  • complement of G

22
G is a split graph
  • if it is partitionable as

23
Deciding if G is a split graph
  • ? is in P

24
  • G is split graph
  • if and only if
  • it contains no induced

5
4
25
Deciding if G is split
Algorithm succeeds
A splitting exists
No forbidden subgraphs
H-Klein-Nogueira-Protti
26
This is a clique cutset
  • (assuming all parts are nonempty)

27
Deciding if G has a clique cutset
  • is in P
  • has applications in solving optimization problems
    on chordal graphs
  • Tarjan, Whitesides,

28
G is a chordal graph
  • if it contains no induced

6 . . .
4
5
29
G is a chordal graph
  • if it contains no induced

6 . . .
4
5
if and only if every induced subgraph is either a
clique or has a clique cutset Dirac
30
G is a cograph
  • if it contains no induced

31
G is a cograph
  • if it contains no induced

if and only if every induced subgraph is
partitionable as or
Seinsche
32
This kind of line means all possible edges are
present
33
A homogeneous set (module)
  • Another well-known kind of partition

34
A homogeneous set (module)
  • finding one is in P
  • has applications in decomposition and recognition
    of comparability graphs (and in solving
    optimization problems on comparability graphs)
  • Gallai

35
G is a perfect graph
? ?
  • holds for G and all its induced subgraphs.

36
G is a perfect graph
? ?
  • holds for G and all its induced subgraphs.
  • G is perfect if and only if G and its complement
  • contain no induced

7 . . .
3
5
Chudnovsky, Robertson, Seymour, Thomas
37
Perfect graphs
  • contain bipartite graphs, line graphs of
    bipartite graphs, split graphs, chordal graphs,
    cographs, comparability graphs
  • and their complements, and
  • model many max-min relations.
  • Berge

38
Perfect graphs
  • contain bipartite graphs, line graphs of
    bipartite graphs, split graphs, chordal graphs,
    cographs, comparability graphs
  • and their complements, and
  • model many max-min relations.

Basic graphs
39
G is perfect
  • if and only if it is basic or it admits a
  • partition
  • all others

Chudnovsky, Robertson, Seymour, Thomas
40
Special properties of partitions
  • Sets may be required to be
  • Independent sets
  • cliques
  • or unrestricted
  • Between the sets we may require
  • no edges
  • all edges
  • or no restriction

41
The matrix M of a partition
  • 0 if Vi is independent
  • M(i,i) 1 if Vi is a clique
  • if Vi is unrestricted
  • 0 if Vi and Vj are not joined
  • M(i,j) 1 if Vi and Vj are fully joined
  • if Vi to Vj is unrestricted

42
The problem PART(M)
  • Instance A graph G
  • Question Does G admit a partition according to
    the matrix M ?

43
The problem SPART(M)
  • Instance A graph G
  • Question Does G admit a
  • surjective partition according to M ?
  • (the parts are non-empty)

44
The problem LPART(M)
  • Instance A graph G, with lists
  • Question Does G admit a
  • list partition according to M ?
  • (each vertex is placed to a set on its list)

45
For PART(M) we assume
  • NO DIAGONAL ASTERISKS
  • M has a diagonal of k zeros and l ones
  • ( k l n )

46
Small matrices M
  • When M 4 PART(M) classified as being in P or
    NP-complete
  • Feder-H-Klein-Motwani
  • When M 4 SPART(M) classified as being in P
    or NP-complete
  • deFigueiredo-Klein-Gravier-Dantas
  • except for one matrix M

47
Small matrices M with lists
  • When M 4 LPART(M) classified as being in P
    or NP-complete, except for one matrix
  • Feder-H-Klein-Motwani
  • de Figueiredo-Klein-Kohayakawa-Reed
  • Cameron-Eschen-Hoang-Sritharan
  • When M 3 digraph partition problems
    classified as being in P or NP-complete
  • Feder-H-Nally

48
Classified PART(M)
  • M has no 1s (or no 0s)
  • H-Nesetril, Feder-H-Huang

49
Classified PART(M)
  • M has no 1s (or no 0s)
  • H-Nesetril, Feder-H-Huang
  • PART(M) is in P if M corresponds to a graph
  • which has a loop or is bipartite, and it is
  • NP-complete otherwise
  • LPART(M) is in P if M corresponds to a bi-arc
  • graph, and it is NP-complete otherwise

50
Bi-Arc Graphs
  • Defined as (complements of) certain intersection
    graphs
  • A common generalization of interval graphs (with
    loops) and (complements of) circular arc graphs
    of clique covering number two (no loops).

51
Classified PART(M)
  • M has no 1s (or no 0s)
  • H-Nesetril, Feder-H-Huang
  • M has no s

52
Classified PART(M)
  • M has no 1s (or no 0s)
  • H-Nesetril, Feder-H-Huang
  • M has no s
  • All PART(M) and LPART(M) in P Feder-H

53
CSP(H)
  • Given a structure T with vertices V(H) and
    relations R1(H), Rk(H) of arities r1, , rk
  • Decide whether or not an input structure G with
    vertices V(G) and relations R1(G), Rk(G), of
    the same arities r1, , rk admits a homomorphism
    f of G to H.
  • DICHOTOMY CONJECTURE Feder-Vardi
  • Each CSP(H) is in P or is NP-complete

54
Can all PART(M) be classified?
  • If for every matrix M the problem PART(M) is
  • in P or is NP-complete, then the Dichotomy
  • Conjecture is true.
  • Feder-H
  • Thus hoping to classify all problems PART(M)
  • appears to be overly ambitious

55
G is complete bipartite
  • if and only if it contains no induced

56
G is a split graph
  • if and only if
  • it contains no induced

5
4
57
G is a bipartite graph
  • if and only if
  • It contains no induced

7 . . .
3
5
58
Another classification of PART(M)?
  • For which matrices M can the problem
  • PART(M) be described by finitely many
  • forbidden induced subgraphs?

59
Infinitely many forbidden induced subgraphs occur
  • whenever M contains
  • or
  • Feder-H-Xie

60
Do all others have finite sets of forbidden
induced subgraphs?
k
l
61
Do all others have finite sets of forbidden
induced subgraphs?
  • NO

62
For small matrices M
  • If M 5, all other partition problems have
    only finitely many forbidden induced subgraphs
  • If M 6, there are other partition problems
    that have infinitely many forbidden induced
    subgraphs
  • Feder-H-Xie

63
means without
  • If M 5, all other partition problems have
    only finitely many forbidden induced subgraphs
  • If M 6, there are other partition problems
    that have infinitely many forbidden induced
    subgraphs
  • Feder-H-Xie

64
Restrictions to inputs G
  • Since these partitions relate closely to
  • perfect graphs, we may want to restrict
  • attention to (classes of) perfect graphs G

65
If M is normal
  • The problem PART(M) restricted to
  • perfect graphs G is in P
  • Feder-H
  • (fmfs)

66
BUT
  • classifying PART(M), for perfect G, as
  • being in P or being NP-complete, would
  • still solve the dichotomy conjecture

67
If M is crossed
  • The problem PART(M) restricted to
  • chordal graphs G is in P
  • Feder-H-Klein-Nogueira-Protti

68
BUT
  • there are problems PART(M), restricted to
  • chordal graphs G, which are NP-complete
  • Feder-H-Klein-Nogueira-Protti

69
For all M
  • A cograph G has a partition if and
  • only if G does not contain one
  • of a finite set of forbidden induced subgraphs
  • Feder-H-Hochstadter

70
Are these problems CSPs?
  • Yes - two adjacent vertices of G have certain
    allowed images in H and two nonadjacent vertices
    of G have certain allowed images in H. (Two
    binary relations)

71
Are these problems CSPs?
  • Yes - two adjacent vertices of G have certain
    allowed images in H and two nonadjacent vertices
    of G have certain allowed images in H. (Two
    binary relations)
  • No - this is not a CSP(T), as inputs are
    restricted to have each pair of distinct
    variables in a unique binary relation.

72
Full CSPs
  • Given a set L of positive integers, an L-full
    structure G has each k ? L elements in a unique
    k-ary relation
  • CSPL(H) is CSP(H) restricted to L-full structures
    G

73
Example with m binary relations
  • Given a complete graph with edges coloured by
  • 1, 2, , m.
  • Given such a G, colour the vertices 1, 2, ,
    m,
  • without a monochromatic edge

?
i
i
i
74
  • When m 2, the problem is in P

75
  • When m 2, the problem is in P
  • When m ? 4, it is NP-complete

76
  • When m 2, the problem is in P
  • When m ? 4, it is NP-complete
  • When m 3, we only have algorithms of complexity
  • n O ( log n / log log n ) FHKS

77
  • An algorithm of complexity nO(log n) solving
    the (more general) problem with lists
  • Given a complete graph G with
  • edges coloured by 1, 2, 3, and
  • vertices equipped with lists ? 1,2,3

78
If all lists have size ? 2
  • Introduce a boolean variable for each vertex (use
    the first/second member of its list)
  • Express each edge-constraint as a clause of two
    variables
  • Solve by 2-SAT

79
In general
  • Let X be the set of vertices with lists 1,2,3
  • Recursively reduce X as follows
  • Try to colour G without giving any vertex its
    majority colour
  • Give each vertex in turn its majority colour

X
? (X-1) / 3
X
X
80
Analysis of Recursive Algorithm
  • Time to solve problem with X x
  • T(x) (1 x T(2x/3)) . T(2-SAT)

81
Analysis of Recursive Algorithm
  • Time to solve problem with X x
  • T(x) (1 x T(2x/3)) . T(2-SAT)
  • ? T(x) x O(log x)

82
Analysis of Recursive Algorithm
  • Time to solve problem with X x
  • T(x) (1 x T(2x/3)) . T(2-SAT)
  • ? T(x) x O(log x)
  • ? T(n) n O(log n)

83
Can we say anything ?
  • A kind of (quasi) dichotomy
  • If 1 ? L then every CSPL(H) is
  • ? quasi-polynomial or
  • ? NP-complete
  • Feder-H
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