Stable project problem’s models
Bertrand Le Cun
c de Versailles-Saint-Quentin
Laboratoire PRiSM, UniversitÃ
April 1st 2013
BLC (PRiSM:UVSQ)
PRiSM
April 1st 2013
1 / 7
Max Stable Set Problem
max
s.t.
Pn
i=1 xi
xi + xj ≤ 1
xi ∈ {0, 1}
if (i, j) ∈ E
∀i ∈ [1..n]
xi = 1 if vertex i is in the stable, 0 otherwise.
BLC (PRiSM:UVSQ)
PRiSM
April 1st 2013
2 / 7
Coloring Problem
Pn
u
Min
i=1 xi
u
u
s.t. P
xi + xj ≤ xuu
if (i, j) ∈ E, (u, i)(u, j) ∈
/E
u
x
=
1
∀i,
∈
[1..n]
i
u=1,ui ∈E
/
xui ∈ {0, 1}
∀1 ≤ u ≤ i ≤ n
xui = 1 if vertex i is in the stable set whose representative i u, 0 otherwise.
BLC (PRiSM:UVSQ)
PRiSM
April 1st 2013
3 / 7
Max stable set weighted by subgraph
Max
s.t.
P
h=subgraph wh xh
xi + xP
j ≤1
yh ≤ i∈H xi
xi ∈ {0, 1}
if (i, j) ∈ E
∀h
yh ∈ 0, 1
xi = 1 if vertex i is in the stable set
yh = 1 if h is touched, 0 otherwise.
BLC (PRiSM:UVSQ)
PRiSM
April 1st 2013
4 / 7
K-Partit
Given G(V,E) find a k-partit induced subgraph as large as possible
P P u
Max
i
u xi
u
u
u
s.t.
x
+
x
(u, i), (u, j) ∈
/ E, (i, j) ∈ E
i
j ≤ xu
P
u
x
=
k
∀i,
∈
[1..n]
u=1 u
xui ∈ {0, 1}
∀u, i
BLC (PRiSM:UVSQ)
PRiSM
April 1st 2013
5 / 7
Weighted Coloring problem
Given G(V,E) find a k-partit induced subgraph as large as possible
P P u
Max
i
u xi
u
u
u
s.t.
x
+
x
(u, i), (u, j) ∈
/ E, (i, j) ∈ E
i
j ≤ xu
P
u
x
=
k
∀i,
∈
[1..n]
u=1 u
xui ∈ {0, 1}
∀u, i
BLC (PRiSM:UVSQ)
PRiSM
April 1st 2013
6 / 7
Weight Coloring Problem
Pn
u
Min
i=1 wu xi
u
u
s.t. P
xi + xj ≤ xuu
if (i, j) ∈ E, (u, i), (u, j) ∈
/E
u
x
=
1
∀i,
∈
[1..n]
i
u=1,ui ∈E
/
xui ∈ {0, 1}
∀1 ≤ u ≤ i ≤ n
xui = 1 if vertex i is in the stable set whose representative i u, 0 otherwise.
vertices are ordered in decreasing order of wu
BLC (PRiSM:UVSQ)
PRiSM
April 1st 2013
7 / 7