Finding reproducible cluster partitions for the k-means algorithm
Graph Partitions. Partition V(G) into k sets (k=3) Vertex partitions.
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Transcript of Graph Partitions. Partition V(G) into k sets (k=3) Vertex partitions.
Graph Partitions
Partition V(G) into k sets
(k=3)
Vertex partitions
This kind of circledepicts an arbitrary set This kind of line means
there may be edges between the two sets
Special properties of partitions
• Sets may be required to be independent
This kind of circledepicts an independent set
This is just a k-colouring
(k=3)
Deciding if a k-colouring exists is
in P for k =1, 2
NP-complete for all other k
(k=2)
Deciding if a 2-colouring exists
Obvious algorithm:
(k=2)1 2
1
1 1 1
2 22
Deciding if a 2-colouring exists
Algorithm succeeds
2-colouring existsNo odd cycles
Deciding if a 2-colouring exists
Algorithm succeeds
2-colouring existsNo odd cycles
Deciding if a 2-colouring exists
Algorithm succeeds
2-colouring existsNo odd cycles
G has a 2-colouring(is bipartite)
if and only if it contains no induced
7 . . .35
Special properties of partitions
Sets may be required to have no edges joining them
This kind of dotted line means there are no edges
joining the two sets
This is (corresponds to) a homomorphism.
Here a homomorphism to C5 - also known
as a C5-colouring.
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.
1 2
3
4
5C5
f -1(1) f -1(2)
f -1(3)
f -1(4)
f -1(5)
1 2
3
4
5
Special properties of partitions
• Sets may be required to be cliques
This kind of circledepicts a clique
This is just a colouring of the
complement of G
if it is partitionable as
G is a split graph
is in P
Deciding if G is a split graph
G is split graph
if and only if
it contains no induced
4 5
Deciding if G is split
Algorithm succeeds
A splitting existsNo forbidden subgraphs
[H-Klein-Nogueira-Protti]
(assuming all parts are nonempty)
This is a clique cutset
Deciding if G has a clique cutset
• is in P
• has applications in solving optimization problems on chordal graphs
[Tarjan, Whitesides,…]
G is a chordal graph
if it contains no induced
6 . . .4 5
G is a chordal graph
if it contains no induced
if and only if every induced subgraph is eithera clique or has a clique cutset
[Dirac]
4 56 . . .
G is a cograph
if it contains no induced
G is a cograph
if it contains no induced
if and only if every induced subgraph is partitionable as or
[Seinsche]
This kind of line means allpossible edges are present
Another well-known kind of partition
A homogeneous set (module)
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…]
G is a perfect graph
holds for G and all its induced subgraphs. =
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 . . .35
[Chudnovsky, Robertson, Seymour, Thomas]
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]
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
G is perfect
if and only if it is basic or it admits a
partition …
all others
[Chudnovsky, Robertson, Seymour, Thomas]
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
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
The problem PART(M)
• Instance: A graph G
• Question: Does G admit a partition according to the matrix M ?
The problem SPART(M)
• Instance: A graph G
• Question: Does G admit a
surjective partition according to M ?(the parts are non-empty)
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)
For PART(M) we assume
NO DIAGONAL ASTERISKS *
M has a diagonal of k zeros and l ones
( k + l = n )
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
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]
• M has no 1’s (or no 0’s)
[H-Nesetril, Feder-H-Huang]
Classified PART(M)
• M has no 1’s (or no 0’s)
[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
Classified PART(M)
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).
• M has no 1’s (or no 0’s)[H-Nesetril, Feder-H-Huang]
• M has no *’s
Classified PART(M)
• M has no 1’s (or no 0’s)[H-Nesetril, Feder-H-Huang]
• M has no *’s
All PART(M) and LPART(M) in P [Feder-H]
Classified PART(M)
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
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…
Can all PART(M) be classified?
G is complete bipartite
if and only if it contains no induced
G is a split graph
if and only if
it contains no induced
4 5
G is a bipartite graph
if and only if
It contains no induced
7 . . .35
Another classification of PART(M)?
For which matrices M can the problem
PART(M) be described by finitely many
forbidden induced subgraphs?
Infinitely many forbidden induced subgraphs occur
• whenever M contains
• or
[Feder-H-Xie]
Do all others have finite sets of forbidden induced subgraphs?
k
l
Do all others have finite sets of forbidden induced subgraphs?
NO…
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]
• 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]
means without
Restrictions to inputs G
Since these partitions relate closely to
perfect graphs, we may want to restrict
attention to (classes of) perfect graphs G
If M is normal
The problem PART(M) restricted to
perfect graphs G is in P
[Feder-H]
(fmfs)
BUT…
…classifying PART(M), for perfect G, as
being in P or being NP-complete, would
still solve the dichotomy conjecture
If M is crossed
The problem PART(M) restricted to
chordal graphs G is in P
[Feder-H-Klein-Nogueira-Protti]
BUT…
…there are problems PART(M), restricted to
chordal graphs G, which are NP-complete
[Feder-H-Klein-Nogueira-Protti]
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]
Are these problems CSP’s?
• 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)
Are these problems CSP’s?
• 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.
Full CSP’s
• 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
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
When m = 2, the problem is in P
When m = 2, the problem is in P When m 4, it is NP-complete
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]
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}
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
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|-1) / 3X
X
X
Analysis of Recursive Algorithm
Time to solve problem with |X|= x
T(x) = (1 + x T(2x/3)) . T(2-SAT)
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)
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)
Can we say anything ?
A kind of (quasi) dichotomy
If 1 L then every CSPL(H) is
quasi-polynomial or
NP-complete[Feder-H]