Reducing graph coloring to stable set without symmetry
Denis Cornaz (with V. Jost, with P. Meurdesoif)
LAMSADE, Paris-Dauphine
SPOC 11
Cornaz (with Jost and Meurdesoif)
Coloring without symmetry
SPOC 11
1 / 14
Coloring and perfect 0-1 matrices
(P) = {Ax ≤ b} is TDI:
min{y > b : y > A ≥ w > }
= min{y > b : y > A ≥ w > , y integer}
(∀w integer)
P = {x : Ax ≤ b} is integer:
max{w > x : x ∈ P}
= max{w > x : x ∈ P, x integer}
(∀w )
A is 0-1: the clique-matrix {maximal cliques} × {vertices}
α(G ) :=
max{1> x : Ax ≤ 1, x ≥ 0, x integer}
=: ω(G )
>
>
>
χ(G ) := min{y 1 : y A ≥ 1 , y ≥ 0, y integer} =: χ(G )
Cornaz (with Jost and Meurdesoif)
Coloring without symmetry
SPOC 11
2 / 14
Compact LP formulation
χ(G ) = min
P
i
xi
s.t. xi ≥ xiv
P v
i xi = 1
(∀i, ∀v )
(∀v )
xiu + xiv ≤ 1 (∀i, ∀uv )
x ≥0
x integer
where G = (V , E ) is a graph, i ∈ {1, . . . , |V |}, v ∈ V , uv ∈ E .
Ĝ = G1 . . . G|V (G )| add u1 , . . . , u|V (G )| universal
χ(G ) + α(Ĝ ) = 2|V (G )|
Cornaz (with Jost and Meurdesoif)
Coloring without symmetry
SPOC 11
3 / 14
Gallai identities
For all G (without isolated vertex)
ρ(G ) + ν(G ) = |V (G )| = α(G ) + τ (G )
If G has no 3-cycle
χ(G ) + α(L(G )) = |V (G )|
Cornaz (with Jost and Meurdesoif)
Coloring without symmetry
SPOC 11
4 / 14
Gallai identities
For all G (without isolated vertex)
ρ(G ) + ν(G ) = |V (G )| = α(G ) + τ (G )
If G has no 3-cycle
χ(G ) + α(L(G )) = |V (G )| = α(G ) + χ(L(G ))
Cornaz (with Jost and Meurdesoif)
Coloring without symmetry
SPOC 11
5 / 14
The “sandwich” line-graphs S(G ) of G
~ of G without 3-dicycle (cliques are acyclic)
Choose a orientation G
~ ) is the line-graph L(G ) of G minus the edges corresponding to
S(G
simplicial pairs of arcs
Cornaz (with Jost and Meurdesoif)
Coloring without symmetry
SPOC 11
6 / 14
Chromatic Gallai identities
(a)
(b)
(c)
(d)
(e)
(f)
~ is an orientation of G , without 3-dicycle
G
~ )) = |V (G )|
χ(G ) + α(S(G
Cornaz (with Jost and Meurdesoif)
Coloring without symmetry
SPOC 11
7 / 14
Chromatic Gallai identities
(a)
(b)
(c)
(d)
(e)
(f)
~ is an orientation of G , without 3-dicycle
G
~ )) = |V (G )| = α(G ) + χ(S(G
~ ))
χ(G ) + α(S(G
Cornaz (with Jost and Meurdesoif)
Coloring without symmetry
SPOC 11
7 / 14
The operator Φ
Let β(·) be a graph parameter such that α(G ) ≤ β(G ) ≤ χ(G ) (∀G )
~ of G without 3-dicycle and define
Choose an orientation G
~ ))
Φβ (G ) := |V (G )| − β(S(G
Corollary. α(G ) ≤ Φβ (G ) ≤ χ(G )
Proof.
~ )) ≤ |V (G )| − β(S(G
~ )) ≤ |V (G )| − α(S(G
~ ))
|V (G )| − χ(S(G
Cornaz (with Jost and Meurdesoif)
Coloring without symmetry
SPOC 11
8 / 14
Fractional clique-cover (co-chromatic) number

 max 1> x
s.t. x(K ) ≤ 1 (∀clique K )
χf (G ) =

xv ≥ 0
(∀v ∈ V )

>
 min 1
Py
s.t.
(∀v )
=
K 3v yK = 1

yK ≥ 0
(∀clique K )
~)
Lemma. y optimal for G ⇒ x feasible for S(G
xuv :=
X
yK
simplicial stars SK 3uv
Cornaz (with Jost and Meurdesoif)
Coloring without symmetry
SPOC 11
9 / 14
Lovász Theta function

P
2
max

v kxv k


s.t. kxo k2 = 1
ϑ(G ) =
xo> xv = kxv k2
(∀v )



xu> xv = 0
(∀uv ∈ E )

min kyo k2



s.t. kyv k2 = 1
(∀v )
=
>y = 1
y
(∀v )

o v


>
yu yv = 0 (∀uv ∈
/ E)
~ ) ⇒ y ∈ Rnd×n+1 feasible for G
Lemma. x ∈ Rd×m+1 optimal for S(G
X
yo v := xo −
~)
u:uv ∈E (G
Cornaz (with Jost and Meurdesoif)
xuv and yv u

 yo v
:=
x
 uv
0
Coloring without symmetry
if u = v
~)
if uv ∈ E (G
otherwise
SPOC 11
10 / 14
Improving Lovász’s ϑ bound for coloring
The sandwich theorem (Lovász 1979)
α(G ) ≤ ϑ(G ) ≤ χf (G ) ≤ χ(G )
(∀G )
Theorem (C. and Meurdesoif 2014)
α(G ) ≤ Φχf (G ) ≤ χf (G ) and ϑ(G ) ≤ Φϑ (G ) ≤ χ(G )
M3
M4
M5
M6
|V | |E | min ρ := Φϑϑ−ϑ
5
5
−
11 35
26.8%
23 182
26.5%
47 845
26.0%
Cornaz (with Jost and Meurdesoif)
Coloring without symmetry
mean ρ
23.6%
27.3%
27.6%
27.8%
(∀G )
max ρ
−
28.7%
29.5%
29.5%
SPOC 11
11 / 14
Impact of orientation
1/2
1/2
1/4
1/3
1/2
1/6
(x2)
1/6
1/4
1/2
1/2
1/3
1/6
1/3
1/2
=3.75
=2.5
(max)
1/2
(c)
(a)
1/6
1/2
1/2
=3.5
(e)
(min)
2/5
1/2
1
1/5
1/5
1
1 (x2)
1
1
1
3/5 (x2)
1/5
1/2
=2.2
(b)
2 2.2 2.25 2.5
6-wheel
α χ0f
χf
orientation (a) Φχ
Φχf Φχ0f
Φχf
orientation (b) Φχ
Cornaz (with Jost and Meurdesoif)
=3
=3.5
(d)
Coloring without symmetry
(f)
3
χ
Φα
Φχ0f = Φα
SPOC 11
12 / 14
Complexity
Theorem. (Grötschel, Lovász and Schrijver 1981)
ϑ(G ) can be approximated in polynomial time (for any ε > 0)
Theorem. (Gvozdenović and Laurent 2008)
Computing χf (G ) is NP-hard
In fact, [GL08] proved that,
given any parameter β(G ) such that β ∈ [χf , χ],
then computing β(G ) is NP-hard.
Cornaz (with Jost and Meurdesoif)
Coloring without symmetry
SPOC 11
13 / 14
Finding α with Φχf (another proof for GL08)
~ k = G orient and add s1 , . . . , sk universal sources
G
~ k ))
Feasible solution x for χf (S(G
xuv :=
(max)
1
α(G )
0
if u = si
otherwise
~ k ) = k.
Let β ∈ [χf , χ]. Start with k := 1 and grow it until Φβ (G
~k) = k
Φβ (G
⇐⇒
α(G ) = k
Proof.
~ k ) ≤ Φχ (G
~ k ) ≤ k + |V (G )| − k |V (G )|
k ≤ α(G ) = α(Gk ) ≤ Φβ (G
f
α(G )
Cornaz (with Jost and Meurdesoif)
Coloring without symmetry
SPOC 11
14 / 14