Generalized vertex colorings and
their applications to permutation
graphs
Tınaz Ekim
[email protected]
EPFL - ROSE - Switzerland
09/06/2006
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Contents
• Definitions
– Vertex coloring and applications
– Generalized vertex colorings
• State of the art: complexity results
• Applications to permutation graphs
– Car sorting
– Robotics
• Related results
• Future research
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Definitions (1)
• Min Coloring: Partitioning the vertex set of a given
graph into a minimum number of stable sets. The
optimal value is (G)
Stable set
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Applications:
Telecommunications
Timetabling
Scheduling
etc…
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Definitions (2)
Given a graph G = (V,E), G is (p,k)-colorable if
V can be partitioned into p cliques and k stable sets.
• Min Coloring: (G) = min(k : G is (0,k)-colorable)
• Min Cocoloring: z(G) = min (p+k : G is (p,k)colorable) [Lesniak,77]
• Min Split-coloring: S(G) = min (max(p,k) : G is
(p,k)-colorable) = min (k : G is (k,k)-colorable)
[Ekim, de Werra, 05]
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Definitions and examples (3)
G is a split graph if its vertex set can be partitioned
into a stable set and a clique.
z(G)
S(G)==22
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Complexity results
• Min Split-coloring and Min Cocoloring NP-hard
• Min Split-coloring and Min Cocoloring P in
cacti [Ekim, de Werra, 05], cographs [Demange, Ekim,
de Werra, 05], in chordal graphs [Hell et al. 04].
• Min Cocoloring P in L(Bipartite), L(line-perfect
graph), Min Split-coloring NP-hard in
L(Bipartite) [Demange, Ekim, de Werra, 05]
• Min Cocoloring NP-hard in permutation graphs
[Wagner, 84]
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Permutation graphs
Given a permutation (N) where N=1, … ,n the
permutation graph G=(V,E) corresponding to  is
defined as follows:
V= 1, … ,n and ijE iff i < j and (i) > (j)
5 1 3 7 6 2 4
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clique =
decreasing subsequence
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stable set=
increasing subsequence
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Application 1: Sorting cars
443 643 66
1 2 3 4 5 6
31 52
5 2
5
4 3 6 1 5 2
1 2
Number of tracks needed to reorder  is (G()) = 3
In the modified structure, we only need S(G ()) = 2 tracks
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Storage Area
Storage Area
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Storage Area
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NP-hardness results
• Theorem [Demange, Ekim, de Werra, 05]: Let G be
a class of graphs closed under addition of cliques
without link to the rest of the graph and under
addition of stable sets completely linked to the rest
of the graph, then Min Cocoloring reduces to Min
Split-coloring in G.
• Corollary: Min Split-coloring is NP-hard in
permutation graphs.
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Proof
G
Kn+1
G’
z(G)=p+k
S(G’)  k  n
Kn+1
…
l=k-p cliques Kn+1
k-p additional cliques are in any min split-coloring:
S(G’)=max(k-p+p’,k’)  k
p’  p and k’  k
Also, since z(G)=p+k, p’+k’  p+k
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p’+k’ = p+k
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New version
• The robot makes l+1 trips back and forth before
unloading the items l-modal sequence.
label
Position
  -1
Position
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Min l-modal
• Partitioning a permuatation into a minimum
number of l-modal subsequences is NP-hard even
for l=1 [di Stefano, Krause, Lübbecke, Zimmermann, 05]
• Deciding if  can be partitioned into 2 unimodal
subsequences P
O(m+nlogn) [Demange, Ekim, de Werra, 05]
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Max l-modal subsequence
How many items can the robot collect at most durin
l+1 trips along the corridor?  maximum l-modal
subsequence
• It can be found in time O(n logn) if l is fixed and
in time O(n2 logn) if l is arbitrary [Demange,
Ekim, de Werra, 05]
• Hint: in permutation graphs, a maximum stable set
can be found in time O(n logn)
 Polynomial time approximation scheme for Min lmodal
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π3 = 4 2 3 6 5 7 1 8
π2 = 5 7 6 3 4 2 8 1
π1 = 4 2 3 6 5 7 1 8
l=2, max l-modal subsequence = 6
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DPTAS l-modal
input: permutation , an integer p
output: l-modal covering of  with diff. approx. ratio
of (1-1/p)
• while the current  has a maximum l-modal
subseq. of size at least p(l+2) do
color such an l-modal subseq. with a new color;
• complete the solution by an exhaustive search on
the remaining .
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Thank you for your attention
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