Multicoloring Unit Disk Graphs
on Triangular Lattice Points
Yuichiro MIYAMOTO
Sophia University
Tomomi MATSUI
min .
University of Tokyo
s.t.
max c(v)
vV
cv   wv , v V ,
cu   cv    , u, v E,
cv   N wv .
Main purpose: Discuss perfectness &
imperfectness of unit disk graphs on triangular
lattice points
Outline
• Definition
– Unit disk graph
– Multicoloring, weighted coloring
– Triangular lattice points
•
•
•
•
Perfectness & imperfectness
Approximation algorithms for multicoloring
Maximum weight independent set
Imperfection ratio
Multicoloring problem
Weight
Assigned colors
{4,5,6}
Output：multicoloring function c: V → 2N
3
{1}
{2,3}
1
Input：simple undirected graph G=(V,E)
vertex weight function w: V →Z+
2
Objective：
minimize required number of colors
Constraints：
2
{2,3}
0
{}
Objective val.= 6
|c(v)|=w(v), ∀v∈V
(Every vertex requires w(v) colors)
c(u)∩c(v)=φ, ∀ {u,v}∈E
(Every adjacent pair of two vertices
doesn’t share a common color)
w(v)∈{0,1}, ∀ v∈V
→ Coloring problem
Unit disk graph
T
Given a set of unit disks
(diameter = T)
on a 2D plain,
a unit disk graph is
an undirected graph
such that centers of two disks
are adjacent
if and only if
the pair of disks has intersection.
Unit disk graph
P: a set of finite points on a 2D plain
T: a non-negative real threshold
unit disk graph (P,T)
vertex set: P
edge set: {{v,w}: v,w∈P,dE(v,w)≦T}
T
dE(v,w): Euclidean distance
between the pair v & w
We restrict centers of disks to triangular lattice points.
Triangular lattice points

e1 : 1,0, e 2 : 1 2 , 3 2

m, n  N
P(m, n) : xe1  ye 2 : x  0,1, , m  1, y  0,1, , n  1
This figure shows triangular lattice points.
e2
(0,0)
e1
(1,0)
Weighted unit disk graph on triangular
lattice points
weight
We deal with finite graphs.
3
Height  4, Threshold  3
3
0
5
4
0
2
1 4
0
1
2
4
1
0
3
1
Height=4
5
3
9
1
2
4
0
0
6
1
0
0
NP-hard [Miyamoto & Matsui (2004)]
• We investigate polynomial time
approximation algorithms for multicoloring
unit disk graphs on triangular lattice points.
• It is important to find well-solvable cases
to develop efficient approximation
algorithms.
• Key property of this talk:
graph perfectness.
Multicoloring problem and perfect graph
Notation
ω(G,w): weighted clique number of (G,w)
(G,w): multicoloring number of (G,w)
For weighted cases, the following theorem is known.
Theorem [Grötschel, Lovász & Schrijver (1988)]
If graph G is perfect, then
ω(G,w)= (G,w), for every w.
An optimal multicoloring of (G,w) is obtained in
(strongly) polynomial time.
An approximation algorithm
We find perfect subgraphs.
We propose a polynomial time approximation algorithm
based on graph perfectness.
We show a simple case.
[Height=3, Threshold=1]perfect
Proof (abstract)
H: (vertex) induced subgraph
When ω(H)=1 or 3, it is trivial.
If ω(H)=2, then H contains no odd-cycle since height = 3
 bipartite graph →χ(H)=2
Given vertex weights,
we proposed a simple polynomial time multicoloring algorithm.
An approximation algorithm for multicoloring
U.D.G. on T.L.P. when threshold=1
For simplicity, w(v) is multiple of 3, for every v
6
6
0
9
3
3
60
69
33
3639
3930
0693
3690
3333
6993
=
1213
1310
0000
1230
1111
2331
layer1
+
1213
1213
0000
1310
0231
0231
+
1230
0000 +
1111
1111
0000
2331
layer2
layer3
0000
1310
0231
1230
0000
2331
layer4
Every layerTheorem
is perfect from previous observation (slide).
Proper weights
Every layer is optimally multicolorable in polynomial time.
The lines Requied
of 0 weights
appear
every
4 lines.
#
of
colors
≦
4/3×χ(G,w)
→The union of multicolored layers implies feasible multicoloring.
Lines of 0 weights cover all the lines.
Every non-zero weight of every layer
Similar to
Multicoloring number of each layer
is 1/3 of original graph.
the shifting strategy
= Weighted clique number of each layer
[Hochbaum (1987)]
≦ 1/3×ω(G,w) ≦ 1/3×χ(G,w)
Approximation algorithm: known results
• When threshold = 1 & w(v) is not multiple of 3,
4/3ω(G,w)+4 [Miyamoto &Matsui (2004)]
4/3ω(G,w)+1/3 [McDiarmid & Reed (2000)]
If there is a polynomial time approximation
algorithm whose ratio < 4/3, then P=NP.
• [McDiarmid & Reed (2000)]
hard to extend to the case threshold > 1.
Our algorithm
easy to extend to the case threshold > 1,
if a perfect subgraph is known
Perfect? Imperfect?
T 1 3  2 7 3 2 3  13  4 19  21
H
1
2
3
4
5
6
…
Perfect (trivial)
←Perfect
Perfect?(already shown)
Imperfect?
Perfect?
Imperfect?
Imperfect? Imperfect?Imperfect?
Perfect?
Perfect?
Perfect?
[Height ＝3, Threshold ＝1]  perfect
[Height ≦2, Threshold ≧1]  perfect
Which is the remainder?
Main result
Main theorem
height ≦ 3, threshold ≧ 1  perfect
height ≧ 4, threshold ≧ 1,
perfect  Threshold  H 2  3H  3
T 1 3  2 7 3 2 3  13  4 19  21
H
1
2
3
4
5
6
…
We show an abstract
of the proof
of the main theorem.
imperfect
perfect
The boundary
is monotone.
First, we show the perfectness
T 1 3  2 7 3 2 3  13  4 19  21
H
1
2
3
4
5
6
…
already shown
perfect
Comparability graph
Definition（comparability graph）
G=(V,E) is a comparability graph
If there is an orientation F of E such that
(a,b)∈F, (b,c)∈F ⇒ (a,c)∈F.
(transitivity)
Theorem
The comparability graph is perfect.
Theorem
The complement of a perfect graph is perfect.
↓
The complement of a comparability graph is perfect.
[Height  3, Threshold  3 ]  Perfect
Theorem
[Height  3, Threshold  3 ]  Co - comparabil ity
Proof（abstract）
If every pair of non-adjacent vertices is connected
by right headed arrow, then the transitivity holds.
Hight = 3  Perfect
T 1 3  2 7 3 2 3  13  4 19  21
H
1
2
3
4
5
6
…
co-comparability  perfectness
Co-comparability  Perfectness
From previous proof,
threshold is large  co-compalability graph
 perfect graph
Perfectness of U.D.G. on T.L.P.
T 1 3  2 7 3 2 3  13  4 19  21
H
1
2
3
4
5
6
…
co-comparability  perfectness
not co-comparability graph
In a similar way, we can show other cases.
Next, we show the inverse implication.
Odd-hole → imperfect
Theorem
If G contains an odd-hole,
then G is imperfect.
Odd-hole: induced subgraph C2k+3, k=1,2,…
[Height  4, 1  Threshold  3 ]  Imperfect
The graph contains C9 as an induced subgraph.
1
Imperfectness (case 1)
T 1 3  2 7 3 2 3  13  4 19  21
1
2
3
4
5
6
…
perfect
imperfect
H
imperfect
Graphs of height 4
are induced subgraphs of height 5
Imperfectness
T 1 3  2 7 3 2 3  13  4 19  21
case 5
case 4
case 3
perfect
case 2
1
2
3
4
5
6
…
imperfect
H
case 6
In the following, we show other cases.
[Height  4, 3  Threshold  2]  Imperfect
The graph contains C7 as an induced subgraph.
case 2
3
Imperfectness (case 2)
T 1 3  2 7 3 2 3  13  4 19  21
case 5
case 4
perfect
case 3
1
2
3
4
5
6
…
imperfect
H
case 6
[Height  4, 2  Threshold  7 ]  Imperfect
The graph contains C5 as an induced subgraph.
case 3
2
Imperfectness (case 3)
T 1 3  2 7 3 2 3  13  4 19  21
case 5
perfect
case 4
1
2
3
4
5
6
…
imperfect
H
case 6
[Height  5, 7  Threshold  3]  Imperfect
case 4
7
3
Imperfectness (case 4)
T 1 3  2 7 3 2 3  13  4 19  21
perfect
case 5
1
2
3
4
5
6
…
imperfect
H
case 6
[Height  5, 3  Threshold  13 ]  Imperfect
13
3
case 5
Imperfectness (case 5)
T 1 3  2 7 3 2 3  13  4 19  21
1
2
3
4
5
6
…
perfect
imperfect
H
case 6
Height  6,


 H 2  5H  7  Threshold  H 2  3H  3   Imperfect


H  7 H  13
2
・・・・・・
・・・・・・
・・・・・・
H  5H  7
2
・・・・・・
case 6
H-3
H-1
Imperfectness (case 6)
T 1 3  2 7 3 2 3  13  4 19  21
H
1
2
3
4
5
6
…
perfect
Imperfect
By the induction, the proof is completed.
Before we describe our approximation algorithms,
we discuss the square lattice case.
Unit disk graphs on square lattice points
T 1
H
The boundary
is not
monotone.
1
2
3
4
2
2
perfect
imperfect
5
An approximation algorithm (again)
6
6
0
9
3
3
3639
3930
0693
3690
3333
6993
60
69
33
arbitrary
weight
00000
arbitrary
weight
00000
arbitrary
3
1213
1213
1213
1310
0000
1310
If
lines
of
weight
0
0000
0231
0231
=
1 2 3 are
0 + removed,
1230 + 0000 +
1 This
1 1 1 component
1111
1is
111
these
components
are
2331
0000
2331
optimally
multicolorable.
independently
multiclorable.
layer1
layer2
layer3
0000
1310
0231
1230
0000
2331
layer4
Key:
This induced subgraph
is optimally multicolorable.
1

The decomposition into 4 layers implies
4/3-approximation algorithm
Approximation algorithm (general threshold)
For given threshold T, the following graph is perfect
(from our main theorem).
arbitrary
weight
0
arbitrary
weight
0
arbitrary
weight
 3  4T 2  3 


2


 2 
 T
 3 
Theorem
This component is
Whenoptimally
TIf>lines
1, multicolorable.
of
 2weight
 0
T

are removed,
3 

1  components are-approx.
these
 3  4T 2  3 
independently
multiclorable.


2


Table of approximation ratios
ratio
not monotone
2.5
2
2
ratio = 1 
 2.15... (T∞)
3
1.5
1
0.5
T
1・・・
ratio
4/3
3  7  2 3  13  19  21
5/3
7/4
2
9/5
2
11/6
When threshold=2,
our (5/3)-approx. ＜ (7/3)-approx.[Feder & Shende (2000)]
29
28
27
26
25
23
22
21
20
19
17
16
15
14
13
11
10
9.
8.
7
5.
4.
3.
2.
1
0
T
Other results
Maximum weight stable set problem
Imperfection ratio
Maximum weight stable set problem
Our main theorem implies polynomial time
approximation algorithms for the problem.
 3  4T 2  3 


2


ratio:
 3  4T 2  3   2 

   T
2

  3 
Details are omitted.
Table of approximation ratios
0.8
ratio
ratio =
0.7
0.6
3
 0.464... (T∞)
32
0.5
0.4
1・・・
ratio
3/4
3  7  2 3  13  19  21
3/5
4/7
1/2
5/9
1/2
23
25
0.2
T
20
0.3
6/11
0.1
29
27
28
26
21
22
17
19
16
14
15
11
13
10
8.
9.
5.
7
4.
2.
3.
1
0
T
Imperfection ratio
Definition
  f G, w
imp G   max 

wZ 
  G, w 
χf(G,w): fractional weighted coloring number
Our main theorem implies the following.
Corollary




2
U.D.G.
on
T.L.P.


1 ≦ imp( of threshold T ) ≦ 1 

2 3 3 
3


T


Thanks for your attention.