Is it hard to color a graph as a 4 years old child?
Édouard Bonnet1 , Florent Foucaud2 , Eunjung Kim3 , and
Florian Sikora3 .
1
Hungarian Academy of Sciences
2 LIMOS – France
3 LAMSADE, Université Paris Dauphine, CNRS – France
JGA 2015
Is it hard to color a graph as a 4 years old child?
Édouard Bonnet1 , Florent Foucaud2 , Eunjung Kim3 , and
Florian Sikora3 .
1
Hungarian Academy of Sciences
2 LIMOS – France
3 LAMSADE, Université Paris Dauphine, CNRS – France
JGA 2015
(Wrote most of these slides)
Is it hard to color a graph as a 4 years old child?
Édouard Bonnet1 , Florent Foucaud2 , Eunjung Kim3 , and
Florian Sikora3 .
1
Hungarian Academy of Sciences
2 LIMOS – France
3 LAMSADE, Université Paris Dauphine, CNRS – France
JGA 2015
Is it hard to color a graph as a 4 years old child?
Édouard Bonnet1 , Florent Foucaud2 , Eunjung Kim3 , and
Florian Sikora3 .
1
Hungarian Academy of Sciences
2 LIMOS – France
3 LAMSADE, Université Paris Dauphine, CNRS – France
JGA 2015
Warm Up
Exact algorithms
Weak Grundy Coloring
Outline
Warm Up
Exact algorithms
Weak Grundy Coloring
2/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Outline
Warm Up
Exact algorithms
Weak Grundy Coloring
3/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Grundy Colorings – Definition(s)
The worst way of reasonably coloring a graph.
4/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Grundy Colorings – Definition(s)
The worst way of reasonably coloring a graph.
I
Order the vertices v1 , v2 . . . vn to maximize the number of
colors used by the greedy coloring: the Grundy Number (GN).
I
That is, vi is colored with c(vi ) the first color that is not in its
neighborhood (first-fit).
4/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Grundy Colorings – Definition(s)
The worst way of reasonably coloring a graph.
I
Order the vertices v1 , v2 . . . vn to maximize the number of
colors used by the greedy coloring: the Grundy Number (GN).
I
That is, vi is colored with c(vi ) the first color that is not in its
neighborhood (first-fit).
4/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Grundy Colorings – Definition(s)
The worst way of reasonably coloring a graph.
I
Order the vertices v1 , v2 . . . vn to maximize the number of
colors used by the greedy coloring: the Grundy Number (GN).
I
That is, vi is colored with c(vi ) the first color that is not in its
neighborhood (first-fit).
I
Connected version: ∀i, G[v1 ∪ . . . ∪ vi ] is connected.
4/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Grundy Colorings – Definition(s)
The worst way of reasonably coloring a graph.
I
Order the vertices v1 , v2 . . . vn to maximize the number of
colors used by the greedy coloring: the Grundy Number (GN).
I
That is, vi is colored with c(vi ) the first color that is not in its
neighborhood (first-fit).
I
Connected version: ∀i, G[v1 ∪ . . . ∪ vi ] is connected.
I
Weak version: vi can be colored with any color in
{1, . . . , c(vi )}.
4/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Algorithmic motivations
I
GN(G) upper bounds the number of colors used by any
greedy heuristic for Min Coloring.
I
GN(G) 6 C · χ(G) on some classes of graphs gives a
C -approximation for Min Coloring.
I
See Sampaio’s PhD thesis for further motivations.
5/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Can you achieve color 6?
6/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
1
Can you achieve color 6?
6/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
1
2
Can you achieve color 6?
6/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
1
2
1
Can you achieve color 6?
6/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
1
2
1
1
Can you achieve color 6?
6/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
1
2
1
2
1
Can you achieve color 6?
6/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
1
2
1
3
2
1
Can you achieve color 6?
6/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
1
2
1
4
3
2
1
Can you achieve color 6?
6/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
1
2
4
1
4
3
2
1
Can you achieve color 6?
6/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
1
2
4
1
4
5
3
2
1
Can you achieve color 6?
6/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
1
1
2
4
1
4
5
3
2
1
Can you achieve color 6?
6/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
1
1
2
4
1
4
5
3
2
2
1
Can you achieve color 6?
6/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
1
1
3
2
4
1
4
5
3
2
2
1
Can you achieve color 6?
6/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
1
Weak Grundy Coloring
6
1
3
2
4
1
4
5
3
2
2
1
Can you achieve color 6?
6/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Cycles
Grundy number =?
7/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Cycles
1
Grundy number =?
7/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Cycles
2
1
Grundy number =?
7/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Cycles
1
2
1
Grundy number =?
7/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Cycles
3
1
2
1
Grundy number = 3
7/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
(even) Cycles
1
Connected Grundy number =?
8/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
(even) Cycles
2
1
Connected Grundy number =?
8/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
(even) Cycles
2
1
2
Connected Grundy number =?
8/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
(even) Cycles
2
1
2
1
Connected Grundy number =?
8/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
(even) Cycles
2
1
2
2
1
Connected Grundy number =?
8/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
(even) Cycles
2
1
1
2
2
1
Connected Grundy number =?
8/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
(even) Cycles
2
2
1
1
2
2
1
Connected Grundy number =?
8/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
(even) Cycles
1
2
2
1
1
2
2
1
Connected Grundy number = 2
8/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Kn,m
Grundy number =?
9/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Kn,m
1
2
1
2
1
2
1
2
1
Grundy number = 2
9/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Kp,p minus a perfect matching
Grundy number =?
10/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Kp,p minus a perfect matching
1
Grundy number =?
10/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Kp,p minus a perfect matching
1
1
Grundy number =?
10/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Kp,p minus a perfect matching
1
1
2
Grundy number =?
10/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Kp,p minus a perfect matching
1
1
2
2
Grundy number =?
10/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Kp,p minus a perfect matching
1
1
2
2
3
Grundy number =?
10/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Kp,p minus a perfect matching
1
1
2
2
3
4
Grundy number = p
10/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Kp,p minus a perfect matching
Connected Grundy number =?
11/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Kp,p minus a perfect matching
1
Connected Grundy number =?
11/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Kp,p minus a perfect matching
1
2
Connected Grundy number =?
11/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Kp,p minus a perfect matching
1
2
1
Connected Grundy number =?
11/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Kp,p minus a perfect matching
1
2
1
2
1
2
1
2
Connected Grundy number = 2
11/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
(minimal) Witnesses
How many vertices (at least) did we need to achieve color k?
(easy)
12/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
(minimal) Witnesses
How many vertices (at least) did we need to achieve color k?
(easy)
k (Clique of size k).
12/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
(minimal) Witnesses
How many vertices (at most) did we need to achieve color k?
13/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
(minimal) Witnesses
How many vertices (at most) did we need to achieve color k?
4
13/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
(minimal) Witnesses
How many vertices (at most) did we need to achieve color k?
4
1
2
3
13/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
(minimal) Witnesses
How many vertices (at most) did we need to achieve color k?
4
1
2
1
3
1
2
13/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
(minimal) Witnesses
How many vertices (at most) did we need to achieve color k?
4
1
2
1
3
1
2
1
Binomial tree T4 .
13/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
(minimal) Witnesses
How many vertices (at most) did we need to achieve color k?
4
1
2
1
3
1
2
I
|Tk | = Σ16i6k−1 |Ti |, |T1 | = 1.
I
So |Tk | = 2k−1
1
Binomial tree T4 .
13/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
(minimal) Witnesses – Consequences
I
Algorithm:
I
For every subset of 2k−1 vertices, check if there is a witness.
Theorem (Zaker ’05)
k−1
The Grundy number can be computed in O(f (k)n2
).
XP algorithm: O(f (k)ng(k) ) : polynomial for fixed values of k.
14/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
(minimal) Witnesses – Consequences
I
Algorithm:
I
For every subset of 2k−1 vertices, check if there is a witness.
Theorem (Zaker ’05)
k−1
The Grundy number can be computed in O(f (k)n2
).
XP algorithm: O(f (k)ng(k) ) : polynomial for fixed values of k.
Can we do the same for the connected Grundy number?
14/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Witness for the connected version
15/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Witness for the connected version
1
15/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Witness for the connected version
2
1
15/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Witness for the connected version
2
1
2
15/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Witness for the connected version
2
1
1
2
15/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Witness for the connected version
2
2
1
1
2
15/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Witness for the connected version
1
2
2
1
1
2
15/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Witness for the connected version
3
1
2
2
1
1
2
Connected Grundy number = 3 but unbounded witness.
15/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Witness for the connected version
3
1
2
2
1
1
2
Connected Grundy number = 3 but unbounded witness.
Can’t do the previous trick!
15/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Witness for the connected version
3
1
2
2
1
1
2
Connected Grundy number = 3 but unbounded witness.
Can’t do the previous trick!
Theorem
Connected Grundy Coloring is NP-complete
15/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Witness for the connected version
3
1
2
2
1
1
2
Connected Grundy number = 3 but unbounded witness.
Can’t do the previous trick!
Theorem
Connected Grundy Coloring is NP-complete
even for k = 7.
15/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Outline
Warm Up
Exact algorithms
Weak Grundy Coloring
16/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Solving Grundy Coloring
Try all possible ordering of the vertices and check if at least k
colors are used by the greedy coloring: Θ(n!)-time.
17/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Solving Grundy Coloring
Try all possible ordering of the vertices and check if at least k
colors are used by the greedy coloring: Θ(n!)-time.
We can have a O(c n ) algorithm.
17/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Solving Grundy Coloring
In a witness:
I
C1
C2
1
2
3
1
2
3
1
2
3
1
2
C3
Any color class is an independent dominating set in the graph
induced by the next classes.
18/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Solving Grundy Coloring
In a witness:
C1
C2
1
2
3
1
2
3
1
2
3
1
2
C3
I
Any color class is an independent dominating set in the graph
induced by the next classes.
I
GN(S) = max{GN(S \ X ), X ind. dom. set of G[S]} + 1.
18/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Solving Grundy Coloring
I
I
A minimal independent dominating set is a maximal
independent set.
One can enumerate all maximal independent sets in O(1.45n )
time.
I
Filling a cell in the table takes O(1.45i ) time for a subset of
size i.
19/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Solving Grundy Coloring
I
I
A minimal independent dominating set is a maximal
independent set.
One can enumerate all maximal independent sets in O(1.45n )
time.
I
Filling a cell in the table takes O(1.45i ) time for a subset of
size i.
Theorem
One can solve grundy coloring in
n
i
i=0 i 1.45
Pn
= (1 + 1.45)n .
19/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Solving Grundy Coloring
I
I
A minimal independent dominating set is a maximal
independent set.
One can enumerate all maximal independent sets in O(1.45n )
time.
I
Filling a cell in the table takes O(1.45i ) time for a subset of
size i.
Theorem
One can solve grundy coloring in
n
i
i=0 i 1.45
Pn
= (1 + 1.45)n .
Cannot replace the n by the treewidth (even feedback vertex set).
Theorem
Under the ETH, Grundy Coloring cannot be solved in O ∗ (c tw ) for
any constant c (even O ∗ (2o(tw log tw ) )).
19/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Outline
Warm Up
Exact algorithms
Weak Grundy Coloring
20/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Are there graphs where weak Grundy exceeds Grundy number?
21/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Are there graphs where weak Grundy exceeds Grundy number?
Weak Grundy number = 3, (connected) Grundy number = 2.
21/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Are there graphs where weak Grundy exceeds Grundy number?
1
Weak Grundy number = 3, (connected) Grundy number = 2.
21/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Are there graphs where weak Grundy exceeds Grundy number?
2
1
Weak Grundy number = 3, (connected) Grundy number = 2.
21/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Are there graphs where weak Grundy exceeds Grundy number?
2
1
1
Weak Grundy number = 3, (connected) Grundy number = 2.
21/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Are there graphs where weak Grundy exceeds Grundy number?
2
3
1
1
Weak Grundy number = 3, (connected) Grundy number = 2.
21/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Color Coding
I
Add colors between 1 and k uniformly at random to the
instance.
22/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Color Coding
I
Add colors between 1 and k uniformly at random to the
instance.
I
The probability that a good solution was well colored is a
function of k.
22/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Color Coding
I
Add colors between 1 and k uniformly at random to the
instance.
I
The probability that a good solution was well colored is a
function of k.
I
Solving the instance is easier with this extra information.
22/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Color Coding
I
Add colors between 1 and k uniformly at random to the
instance.
I
The probability that a good solution was well colored is a
function of k.
I
Solving the instance is easier with this extra information.
I
Do a function of k tries to have a small probability of failure.
22/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Theorem
Weak Grundy Coloring is in FPT.
FPT algorithm: O(f (k)nc ).
23/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Theorem
Weak Grundy Coloring is in FPT.
FPT algorithm: O(f (k)nc ).
I
Idea:
I
I
A witness is of size at most 2k−1 .
This witness is well colored with probability is
1
k 2k−1
.
23/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Theorem
Weak Grundy Coloring is in FPT.
FPT algorithm: O(f (k)nc ).
I
Idea:
I
I
A witness is of size at most 2k−1 .
This witness is well colored with probability is
k−1
I
After log( ε1 )k 2
1 − ε, ∀ε > 0.
1
k 2k−1
.
tries, the probability of success is at least
23/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Guess #1
6
4
1
1
5
3
4
4
6
2
3
2
1
3
2
24/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Guess #1
1
1
4
4
2
3
2
1
2
24/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Guess #1
1
1
2
3
2
1
2
24/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Guess #2
1
2
3
1
5
2
4
1
3
3
4
1
1
6
2
25/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Guess #2
1
3
1
5
2
4
1
3
3
4
1
1
6
2
25/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Guess #2
1
3
1
2
4
1
3
3
4
1
1
6
2
25/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Guess #2
1
3
1
2
4
1
3
3
4
1
1
2
25/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
k
. . . O(k 2 ) unsuccessful guesses later . . .
1
4
6
1
3
2
4
1
1
4
5
3
2
2
1
26/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
k
. . . O(k 2 ) unsuccessful guesses later . . .
1
6
1
3
2
4
1
1
4
5
3
2
2
1
26/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Open Problems
I
Is Grundy Coloring solvable in O(2n ) ?
I
Is Grundy Coloring solvable in O(f (tw )nc ) ?
27/28
Grundy Coloring
florian.sikora@dauphine.fr
Warm Up
Exact algorithms
Weak Grundy Coloring
Open Problems
I
Is Grundy Coloring solvable in O(2n ) ?
I
Is Grundy Coloring solvable in O(f (tw )nc ) ?
Is Grundy Coloring solvable in O(f (k)nc )? Even for
bipartite graph?
I
I
I
True for chordal, claw-free and bounded degree graphs or
graph excluding a fixed graph as a minor.
ETH fails if Grundy Coloring is solvable in
o(k)
O ∗ (22 2o(n+m) ).
27/28
Grundy Coloring
florian.sikora@dauphine.fr
Merci !