Ioan Todinca (LIFO, Université d'Orléans) Permutation Graphs, an Introduction Algorithms & Permutations 2012, Paris. http://igm.univ-mlv.fr/AlgoB/algoperm2012/

1. Introduction Optimization problems Recognition Encoding all realizers Conclusion
Permutation graphs — an introduction
Ioan Todinca
LIFO - Universit´ d’Orl´ans
e e
Algorithms and permutations, february 2012
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2. Introduction Optimization problems Recognition Encoding all realizers Conclusion
Permutation graphs
1 2 3 4
1
0 1
0 1
0 11
00
1
0 1
0 1
0 11
00 2
3 4
1
1
0 1
0 5
1
0 1
0 6
5 6
• Optimisation algorithms
use, as input, the intersection model (realizer)
• Recognition algorithms
output the intersection(s) model(s)
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3. Introduction Optimization problems Recognition Encoding all realizers Conclusion
Plan of the talk
1. Relationship with other graph classes
2. Optimisation problems :
MaxIndependentSet/MaxClique/Coloring ;
Treewidth
3. Recognition algorithm
4. Encoding all realizers via modular decomposition
5. Conclusion
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4. Introduction Optimization problems Recognition Encoding all realizers Conclusion
Definition and basic properties
2 1 6 5 4 3
1 2 3 4 2
1
0 1
0 1
0 11
00
1
0 1
0 1
0 11
00 3 4
1
5 6
1
0 1
0
1
0 1
0
5 6 1 5 6 3 2 4
• Realizer : (π1 , π2 )
• One can ”reverse” the realizer upside-down or right-left :
(π1 , π2 ) ∼ (π2 , π1 ) ∼ (π1 , π2 ) ∼ (π2 , π1 )
• Complements of permutation graphs are permutation graphs.
→ Reverse the ordering of the bottoms of the segments :
(π1 , π2 ) → (π1 , π2 )
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5. Introduction Optimization problems Recognition Encoding all realizers Conclusion
Definition and basic properties
2 1 6 5 4 3
1 2 3 4
1
0 1
0 1
0 11
00
1
0 1
0 1
0 11
00
1
0 1
0
1
0 1
0
5 6 1 5 6 3 2 4
• Realizer : (π1 , π2 )
• One can ”reverse” the realizer upside-down or right-left :
(π1 , π2 ) ∼ (π2 , π1 ) ∼ (π1 , π2 ) ∼ (π2 , π1 )
• Complements of permutation graphs are permutation graphs.
→ Reverse the ordering of the bottoms of the segments :
(π1 , π2 ) → (π1 , π2 )
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6. Introduction Optimization problems Recognition Encoding all realizers Conclusion
Definition and basic properties
2 1 6 5 4 3
1 2 3 4
1
0 1
0 1
0 11
00
1
0 1
0 1
0 11
00
1
0 1
0
1
0 1
0
5 6 4 2 3 6 5 1
• Realizer : (π1 , π2 )
• One can ”reverse” the realizer upside-down or right-left :
(π1 , π2 ) ∼ (π2 , π1 ) ∼ (π1 , π2 ) ∼ (π2 , π1 )
• Complements of permutation graphs are permutation graphs.
→ Reverse the ordering of the bottoms of the segments :
(π1 , π2 ) → (π1 , π2 )
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7. Introduction Optimization problems Recognition Encoding all realizers Conclusion
Definition and basic properties
2 1 6 5 4 3
1 2 3 4
1
0 1
0 1
0 11
00
1
0 1
0 1
0 11
00
1
0 1
0
1
0 1
0
5 6 4 2 3 6 5 1
• Realizer : (π1 , π2 )
• One can ”reverse” the realizer upside-down or right-left :
(π1 , π2 ) ∼ (π2 , π1 ) ∼ (π1 , π2 ) ∼ (π2 , π1 )
• Complements of permutation graphs are permutation graphs.
→ Reverse the ordering of the bottoms of the segments :
(π1 , π2 ) → (π1 , π2 )
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8. Introduction Optimization problems Recognition Encoding all realizers Conclusion
More intersection graph classes
Circle graphs Trapezoid graphs
Books on graph classes : [Golumbic ’80 ; Brandst¨dt, Le, Spinrad
a
’99 ; Spinrad 2003]
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9. Introduction Optimization problems Recognition Encoding all realizers Conclusion
MaxIndependentSet via Dynamic Programming
1 2 3 4 5 6
2 4 3 6 1 5
• Dynamic programming from left to right :
MIS[i] = 1 + max MIS[j]
j left to i
• MaxIndependentSet corresponds to the longest increasing
sequence in a permutation — O(n log n)
• MaxClique : longest decreasing sequence
• Coloring : chromatic number = max clique (perfect graphs)
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10. Introduction Optimization problems Recognition Encoding all realizers Conclusion
Treewidth via dynamic programming on scanlines
1 2 3 4
1
0
1
0 1
0
1
0 1
0
1
0 11
00
11
00 2 1 6 5 4 3
1
0 1
0
1
0 1
0
5 6
12 256 23 34
1 5 6 3 2 4
• Minimal separators correspond to scanlines
• Bags correspond to areas between two scanlines
• Treewidth can be solved in polynomial time [Bodlaender,
Kloks, Kratsch 95 ; Meister 2010]
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11. Introduction Optimization problems Recognition Encoding all realizers Conclusion
Treewidth via dynamic programming on scanlines
1 2 3 4
1
0
1
0 1
0
1
0 1
0
1
0 11
00
11
00 2 1 6 5 4 3
1
0 1
0
1
0 1
0
5 6
12 256 23 34
1 5 6 3 2 4
• Minimal separators correspond to scanlines
• Bags correspond to areas between two scanlines
• Treewidth can be solved in polynomial time [Bodlaender,
Kloks, Kratsch 95 ; Meister 2010]
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12. Introduction Optimization problems Recognition Encoding all realizers Conclusion
Treewidth via dynamic programming on scanlines
1 2 3 4
1
0
1
0 1
0
1
0 1
0
1
0 11
00
11
00 2 1 6 5 4 3
1
0 1
0
1
0 1
0
5 6
12 256 23 34
1 5 6 3 2 4
• Minimal separators correspond to scanlines
• Bags correspond to areas between two scanlines
• Treewidth can be solved in polynomial time [Bodlaender,
Kloks, Kratsch 95 ; Meister 2010]
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13. Introduction Optimization problems Recognition Encoding all realizers Conclusion
Recognition algorithm
Theorem ([Pnueli, Lempel, Even ’71], see also [Golumbic ’80])
G is a permutation graph if and only if G and G are comparability
graphs.
Algorithm
1. Find a transitive orientation of G and one of G
2. Construct an intersection model for G
In O(n + m) time by [McConnell, Spinrad ’99]
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14. Introduction Optimization problems Recognition Encoding all realizers Conclusion
permutation ⊆ comparability ∩ co-comparability
Transitive orientation of a permutation graph G : orient edges
according to the top endpoints of the segments.
2 1 6 5 4 3
1 2 3 4
1
0 1
0 1
0 11
00
1
0 1
0 1
0 11
00
1
0
1
0 1
0
1
0
5 6 1 5 6 3 2 4
If xy , yz ∈ E and π1 (x) < π1 (y ) < π1 (z) then xz ∈ E .
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15. Introduction Optimization problems Recognition Encoding all realizers Conclusion
permutation ⊆ comparability ∩ co-comparability
Transitive orientation of a permutation graph G : orient edges
according to the top endpoints of the segments.
2 1 6 5 4 3
1 2 3 4
1
0 1
0 1
0 11
00
1
0 1
0 1
0 11
00
1
0
1
0 1
0
1
0
5 6 1 5 6 3 2 4
If xy , yz ∈ E and π1 (x) < π1 (y ) < π1 (z) then xz ∈ E .
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16. Introduction Optimization problems Recognition Encoding all realizers Conclusion
comparability ∩ co-comparability ⊆ permutation
Let Etr be a transitive orientation of G and Ftr a transitive
orientation of its complement.
1 2 3 4
1
0
1
0 1
0
1
0 1
0
1
0 11
00
11
00
1
0 1
0
1
0 1
0
5 6
Lemma
Etr ∪ Ftr induces a total ordering π1 (Etr ∪ Ftr ) on the vertex set.
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17. Introduction Optimization problems Recognition Encoding all realizers Conclusion
comparability ∩ co-comparability ⊆ permutation
Let Etr be a transitive orientation of G and Ftr a transitive
orientation of its complement.
1 2 3 4
1
0
1
0 1
0
1
0 1
0
1
0 11
00
11
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1
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1
0 1
0
5 6
Lemma
Etr ∪ Ftr induces a total ordering π1 (Etr ∪ Ftr ) on the vertex set.
rev (Etr ) ∪ Ftr induces another total ordering π2 (rev (Etr ) ∪ Ftr ).
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18. Introduction Optimization problems Recognition Encoding all realizers Conclusion
A realizer of G
Permutations π1 (Etr ∪ Ftr ) and π2 (rev (Etr ) ∪ Ftr ) form a realizer
of G .
2 1 6 5 4 3
1 2 3 4
1
0 1
0 1
0 11
00
1
0 1
0 1
0 11
00
1
0
1
0 1
0
1
0
1 5 6 3 2 4 5 6
Segments x and y intersect iff (xy ) ∈ Etr and (yx) ∈ rev (Etr ) or
vice-versa ; equivalently, iff xy ∈ E .
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19. Introduction Optimization problems Recognition Encoding all realizers Conclusion
Modules and common intervals
• Substituting a segment (vertex) by the realizer of a
permutation graph (module) produces a new permutation
graph.
• A common interval of π1 and π2 forms a module in G
• Strong modules correspond exactly to strong common
intervals [de Mongolfier 2003]
• A graph is a permutation graphs iff all prime nodes in the
modular decomposition are permutation graphs.
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20. Introduction Optimization problems Recognition Encoding all realizers Conclusion
Modules and common intervals
• Substituting a segment (vertex) by the realizer of a
permutation graph (module) produces a new permutation
graph.
• A common interval of π1 and π2 forms a module in G
• Strong modules correspond exactly to strong common
intervals [de Mongolfier 2003]
• A graph is a permutation graphs iff all prime nodes in the
modular decomposition are permutation graphs.
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21. Introduction Optimization problems Recognition Encoding all realizers Conclusion
Encoding realizers
1 2 3 4
1
0 1
0 1
0 11
00
1
0 1
0 1
0 11
00
1
0 1
0
Theorem ([Gallai ’67]) 1
0 1
0
5 6
A prime permutation graph has a
unique realizer, up to reversals. prime
1 2 4
3
The modular decomposition tree
+ realizers of prime nodes
parallel 2 3 4
encode all possible realizers of G ,
cf. [Crespelle, Paul 2010].
series 1
5 6
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22. Introduction Optimization problems Recognition Encoding all realizers Conclusion
Conclusion
Summary
• Many optimization problems become polynomial on
permutation graphs
• Representations (intersection models) based on modular
decompositions
Some questions
• Is Bandwidth polynomial or NP-complete on permutation
graphs ?
• What about subgraph isomorphism from parametrized point
of view ?
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23. Introduction Optimization problems Recognition Encoding all realizers Conclusion
Conclusion
Summary
• Many optimization problems become polynomial on
permutation graphs
• Representations (intersection models) based on modular
decompositions
Some questions
• Is Bandwidth polynomial or NP-complete on permutation
graphs ?
• What about subgraph isomorphism from parametrized point
of view ?
Thank you ! Your questions ?
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