The stable set problem is FPT in bull-free graphs
Nicolas Trotignon — CNRS — LIP
ENS de Lyon
GROW
6th workshop on Graph Classes, Optimization, and Width Parameters
(wonderful place of) Santorini, October 2013
Co-authors
Joint work with:
Stéphan Thomassé
ENS de Lyon
Kristina Vušković
Leeds University and
Union University, Belgrade
The bull
The bull:
The bull is self-complementary
Bull-free: no bull as an induced subgraph
Examples: triangle-free graphs, their complements
Consequence: many problems are hard for bull-free graphs,
for instance the Maximum Weighted Stable Set Problem
(MWSS for short)
The bull
The bull:
The bull is self-complementary
Bull-free: no bull as an induced subgraph
Examples: triangle-free graphs, their complements
Consequence: many problems are hard for bull-free graphs,
for instance the Maximum Weighted Stable Set Problem
(MWSS for short)
The bull
The bull:
The bull is self-complementary
Bull-free: no bull as an induced subgraph
Examples: triangle-free graphs, their complements
Consequence: many problems are hard for bull-free graphs,
for instance the Maximum Weighted Stable Set Problem
(MWSS for short)
Chudnovsky’s theorem
Theorem (Chudnovsky, 2012)
Every bull-free graph is basic or has a decomposition.
Our results
With Thomassé and Vušković, we proved the following properties
of bull-free graphs:
Bound on the chromatic number χ (in function of ω and the
maximum χ of a triangle-free subgraph)
χ(G ) ≤ f (ω(G ), max{χ(H); H ⊆ G and H triangle-free})
Polytime algorithm for the MWSS problem when restricted to
odd-hole-free graphs
FPT for the same problem in general bull-free graphs
Parameter: weight of the solution
No polynomial kernel, but what we call a
polynomial Turing-kernel.
Our results
With Thomassé and Vušković, we proved the following properties
of bull-free graphs:
Bound on the chromatic number χ (in function of ω and the
maximum χ of a triangle-free subgraph)
χ(G ) ≤ f (ω(G ), max{χ(H); H ⊆ G and H triangle-free})
Polytime algorithm for the MWSS problem when restricted to
odd-hole-free graphs
FPT for the same problem in general bull-free graphs
Parameter: weight of the solution
No polynomial kernel, but what we call a
polynomial Turing-kernel.
Our results
With Thomassé and Vušković, we proved the following properties
of bull-free graphs:
Bound on the chromatic number χ (in function of ω and the
maximum χ of a triangle-free subgraph)
χ(G ) ≤ f (ω(G ), max{χ(H); H ⊆ G and H triangle-free})
Polytime algorithm for the MWSS problem when restricted to
odd-hole-free graphs
FPT for the same problem in general bull-free graphs
Parameter: weight of the solution
No polynomial kernel, but what we call a
polynomial Turing-kernel.
Our results
With Thomassé and Vušković, we proved the following properties
of bull-free graphs:
Bound on the chromatic number χ (in function of ω and the
maximum χ of a triangle-free subgraph)
χ(G ) ≤ f (ω(G ), max{χ(H); H ⊆ G and H triangle-free})
Polytime algorithm for the MWSS problem when restricted to
odd-hole-free graphs
FPT for the same problem in general bull-free graphs
Parameter: weight of the solution
No polynomial kernel, but what we call a
polynomial Turing-kernel.
Decomposition 1: homogeneous set
A homogeneous set of a graph T is a set X like that:
Y
X
|X | ≥ 2
|Y ∪ Z | ≥ 1
Z
Decomposition 2: homogeneous pair
A homogeneous pair of a graph T is a pair (A, B) like that:
F
C
E
A
D
B
|A ∪ B| ≥ 3
|C ∪ D ∪ E ∪ F | ≥ 3
Between A and B: at least one edge and one antiedge
Blocks of decomposition
Y
Z
Y
→
X
x
F
C
F
E
A
Z
D
B
In fact: we work with trigraphs
→
C
E
a
D
b
Blocks of decomposition
Y
Z
Y
→
X
x
F
C
F
E
A
Z
D
B
In fact: we work with trigraphs
→
C
E
a
D
b
Chudnovsky’s theorem (the real one)
Theorem (Chudnovsky, 2012)
Every bull-free trigraph is basic or has a decomposition.
Extreme decomposition
An extreme decomposition is a decomposition one side of which is
basic
Theorem
For every non-basic bull-free trigraph, one of the following holds:
T admits a homogeneous set X such that T [X ] is basic
T admits a homogeneous pair (A, B) such that T [A ∪ B] is
basic
This is polynomial: the sets can be found in time O(n8 )
Faster implementation?
Extreme decomposition
An extreme decomposition is a decomposition one side of which is
basic
Theorem
For every non-basic bull-free trigraph, one of the following holds:
T admits a homogeneous set X such that T [X ] is basic
T admits a homogeneous pair (A, B) such that T [A ∪ B] is
basic
This is polynomial: the sets can be found in time O(n8 )
Faster implementation?
Keeping track of α
Y
Z
Y
→
X
α(X)
F
F
C
E
A
Z
D
B
→
C
α(A)
E
α(A ∪ B)
D
α(B)
Basic trigraphs
We are done (in polytime) if we can handle weighted basic
trigraphs. What are they?
Small trigraphs: at most 8 vertices
Class TFKS: a Triangle-Free trigraph + a disjoint union of
strong KliqueS with no edges between them
Class TFKS: complements of the formers
Theorem (Farrugia, 2004)
Deciding whether a graph is TFKS is NP-complete.
Basic trigraphs
We are done (in polytime) if we can handle weighted basic
trigraphs. What are they?
Small trigraphs: at most 8 vertices
Class TFKS: a Triangle-Free trigraph + a disjoint union of
strong KliqueS with no edges between them
Class TFKS: complements of the formers
Theorem (Farrugia, 2004)
Deciding whether a graph is TFKS is NP-complete.
Basic trigraphs
We are done (in polytime) if we can handle weighted basic
trigraphs. What are they?
Small trigraphs: at most 8 vertices
Class TFKS: a Triangle-Free trigraph + a disjoint union of
strong KliqueS with no edges between them
Class TFKS: complements of the formers
Theorem (Farrugia, 2004)
Deciding whether a graph is TFKS is NP-complete.
Basic trigraphs
We are done (in polytime) if we can handle weighted basic
trigraphs. What are they?
Small trigraphs: at most 8 vertices
Class TFKS: a Triangle-Free trigraph + a disjoint union of
strong KliqueS with no edges between them
Class TFKS: complements of the formers
Theorem (Farrugia, 2004)
Deciding whether a graph is TFKS is NP-complete.
Basic trigraphs
We are done (in polytime) if we can handle weighted basic
trigraphs. What are they?
Small trigraphs: at most 8 vertices
Class TFKS: a Triangle-Free trigraph + a disjoint union of
strong KliqueS with no edges between them
Class TFKS: complements of the formers
Theorem (Farrugia, 2004)
Deciding whether a graph is TFKS is NP-complete.
Handling basic trigraphs: TFKS
K1
K2
no
Kt
At most n3 maximal stable sets
So we are done by the classical enumeration of maximal stable
sets with polytime delay (Tsukiyama, Ide, Ariyoshi, and
Shirakawa 1977, or Makino and Uno 2004)
Handling basic trigraphs: TFKS
K1
K2
no
Kt
At most n3 maximal stable sets
So we are done by the classical enumeration of maximal stable
sets with polytime delay (Tsukiyama, Ide, Ariyoshi, and
Shirakawa 1977, or Makino and Uno 2004)
Handling basic trigraphs: TFKS
K1
K2
no
Kt
MWSS is NP-hard (because NP-hard for triangle-free graphs,
Poljak 1974)
But an FPT approach works: a sufficiently big number of
vertices ensures a big stable set. Technically:
|V (T )| ≤ O(α(T )5 )
Handling basic trigraphs: TFKS
K1
K2
no
Kt
MWSS is NP-hard (because NP-hard for triangle-free graphs,
Poljak 1974)
But an FPT approach works: a sufficiently big number of
vertices ensures a big stable set. Technically:
|V (T )| ≤ O(α(T )5 )
Main FPT result
Sum up of everything:
Theorem
Given a weighted bull-free trigraph T and an integer k, one can
5
decide in time 2O(k ) O(n9 ) whether T contains a stable set of
weight at least k.
This generalizes Dabrowski, Lozin, Müller and Rautenbach (2012)
who have a similar result for {bull, house}-free graphs.
House:
Main FPT result
Sum up of everything:
Theorem
Given a weighted bull-free trigraph T and an integer k, one can
5
decide in time 2O(k ) O(n9 ) whether T contains a stable set of
weight at least k.
This generalizes Dabrowski, Lozin, Müller and Rautenbach (2012)
who have a similar result for {bull, house}-free graphs.
House:
No polynomial kernel
Our method tastes like a polynomial kernel, but is not a
polynomial kernel!
Moreover, our problem is OR-compositional, so by
Bodlaender, Downey, Fellows and Hermelin (2009), MWSS for
bull-free graphs should not have a polynomial kernel, unless
some complexity classes collapse.
Would a kernel-like notion make sense for our problem?
No polynomial kernel
Our method tastes like a polynomial kernel, but is not a
polynomial kernel!
Moreover, our problem is OR-compositional, so by
Bodlaender, Downey, Fellows and Hermelin (2009), MWSS for
bull-free graphs should not have a polynomial kernel, unless
some complexity classes collapse.
Would a kernel-like notion make sense for our problem?
No polynomial kernel
Our method tastes like a polynomial kernel, but is not a
polynomial kernel!
Moreover, our problem is OR-compositional, so by
Bodlaender, Downey, Fellows and Hermelin (2009), MWSS for
bull-free graphs should not have a polynomial kernel, unless
some complexity classes collapse.
Would a kernel-like notion make sense for our problem?
Polynomial Turing-kernels
A parameterized problem has a polynomial Turing-kernel if
there exist constants c, c 0 such that
computing the solution of any instance (X , k) can be done in
time O(nc ) provided that we have access to an oracle which
can decide any instance (X 0 , k 0 ) where (X 0 , k 0 ) has size at
0
most O(k c )
A polynomial kernel is more like a polynomial Karp-kernel
Are there problems with no polynomial Turing-kernel?
Polynomial Turing-kernels
A parameterized problem has a polynomial Turing-kernel if
there exist constants c, c 0 such that
computing the solution of any instance (X , k) can be done in
time O(nc ) provided that we have access to an oracle which
can decide any instance (X 0 , k 0 ) where (X 0 , k 0 ) has size at
0
most O(k c )
A polynomial kernel is more like a polynomial Karp-kernel
Are there problems with no polynomial Turing-kernel?
Polynomial Turing-kernels
A parameterized problem has a polynomial Turing-kernel if
there exist constants c, c 0 such that
computing the solution of any instance (X , k) can be done in
time O(nc ) provided that we have access to an oracle which
can decide any instance (X 0 , k 0 ) where (X 0 , k 0 ) has size at
0
most O(k c )
A polynomial kernel is more like a polynomial Karp-kernel
Are there problems with no polynomial Turing-kernel?
MWSS has a polynomial Turing-kernel in bull-free graphs
Theorem
MWSS has an O(k 5 ) Turing-kernel for bull-free weighted trigraphs.
This can this be done for the unweighted version and for graphs.
Odd-hole-free graph
An odd hole in a trigraph is a chordless odd cycle of length at
least 5
If a trigraph is {bull, odd-hole-free}, TFKS’s are perfect
MWSS is therefore polynomial
Bounding χ for bull-free graphs
Let χT (G ) be the maximum chromatic number of a triangle-free
subgraph of G .
Bounding χ by a function of ω is impossible in general:
Zykov 1949, Mycielski, Blanche Descartes, Erdős-Hajnal . . .
Bounding χ by a function of χT is impossible: cliques
But what about bounding by a function of both?
This question is open in general. We answer it by “yes” for
bull-free graphs.
Clique-colouring
A clique-colouring of a graph is a colouring (possibly not
proper) such that no maximal clique of size ≥ 2 is
monochromatic
Every proper colouring is a clique-colouring
For triangle-free graphs, proper- and clique-colouring are the
same notion
Lemma
Let C be a hereditary class of graphs and suppose there exists a
function f such that every graph of C admits a clique-colouring
with at most f (χT (G ), ω(G )) colours.
Then there exits a function g such that every graph of C admits a
proper colouring with at most g (χT (G ), ω(G )) colours.
Proof: induction on ω
A bound on χ
Theorem
Therefore, there exists a function f such that every graph of C
admits a proper colouring with at most f (χT (G ), ω(G )) colours
Are there questions?
Yes, there are.
Is there a complexity-tool showing that problems have no
polynomial Turing-kernels?
Does there exists a function f such that every graph G
satisfies
χ(G ) ≤ f (χT (G ), ω(G ))?
Suppose that C is a hereditary class of graphs such that there
exists a polynomial time for MWSS. Does there exists a
function f such that every graph G in C satisfies
χ(G ) ≤ f (ω(G ))?
Are there questions?
Yes, there are.
Is there a complexity-tool showing that problems have no
polynomial Turing-kernels?
Does there exists a function f such that every graph G
satisfies
χ(G ) ≤ f (χT (G ), ω(G ))?
Suppose that C is a hereditary class of graphs such that there
exists a polynomial time for MWSS. Does there exists a
function f such that every graph G in C satisfies
χ(G ) ≤ f (ω(G ))?
Are there questions?
Yes, there are.
Is there a complexity-tool showing that problems have no
polynomial Turing-kernels?
Does there exists a function f such that every graph G
satisfies
χ(G ) ≤ f (χT (G ), ω(G ))?
Suppose that C is a hereditary class of graphs such that there
exists a polynomial time for MWSS. Does there exists a
function f such that every graph G in C satisfies
χ(G ) ≤ f (ω(G ))?
Are there questions?
Yes, there are.
Is there a complexity-tool showing that problems have no
polynomial Turing-kernels?
Does there exists a function f such that every graph G
satisfies
χ(G ) ≤ f (χT (G ), ω(G ))?
Suppose that C is a hereditary class of graphs such that there
exists a polynomial time for MWSS. Does there exists a
function f such that every graph G in C satisfies
χ(G ) ≤ f (ω(G ))?
Are there questions?
Yes, there are.
Is there a complexity-tool showing that problems have no
polynomial Turing-kernels?
Does there exists a function f such that every graph G
satisfies
χ(G ) ≤ f (χT (G ), ω(G ))?
Suppose that C is a hereditary class of graphs such that there
exists a polynomial time for MWSS. Does there exists a
function f such that every graph G in C satisfies
χ(G ) ≤ f (ω(G ))?
Thanks
Thanks for your attention.