algoperm2012 - 05 ioan todinca
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Introduction Optimization problems Recognition Encoding all realizers Conclusion
Permutation graphs — an introduction
Ioan Todinca
LIFO - Universite d’Orleans
Algorithms and permutations, february 2012
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Introduction Optimization problems Recognition Encoding all realizers Conclusion
Permutation graphs
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• Optimisation algorithmsuse, as input, the intersection model (realizer)
• Recognition algorithmsoutput the intersection(s) model(s)
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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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Introduction Optimization problems Recognition Encoding all realizers Conclusion
Definition and basic properties
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• 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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Introduction Optimization problems Recognition Encoding all realizers Conclusion
Definition and basic properties
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• 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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Introduction Optimization problems Recognition Encoding all realizers Conclusion
Definition and basic properties
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• 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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Introduction Optimization problems Recognition Encoding all realizers Conclusion
Definition and basic properties
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• 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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Introduction Optimization problems Recognition Encoding all realizers Conclusion
More intersection graph classes
Circle graphs Trapezoid graphs
Books on graph classes : [Golumbic ’80 ; Brandstadt, Le, Spinrad’99 ; Spinrad 2003]
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Introduction Optimization problems Recognition Encoding all realizers Conclusion
MaxIndependentSet via Dynamic Programming
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• Dynamic programming from left to right :
MIS [i ] = 1 + maxj left to i
MIS [j ]
• MaxIndependentSet corresponds to the longest increasingsequence in a permutation — O(n log n)
• MaxClique : longest decreasing sequence
• Coloring : chromatic number = max clique (perfect graphs)
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Introduction Optimization problems Recognition Encoding all realizers Conclusion
Treewidth via dynamic programming on scanlines
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• 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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Introduction Optimization problems Recognition Encoding all realizers Conclusion
Treewidth via dynamic programming on scanlines
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• 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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Introduction Optimization problems Recognition Encoding all realizers Conclusion
Treewidth via dynamic programming on scanlines
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• 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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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 comparabilitygraphs.
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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Introduction Optimization problems Recognition Encoding all realizers Conclusion
permutation ⊆ comparability ∩ co-comparability
Transitive orientation of a permutation graph G : orient edgesaccording to the top endpoints of the segments.
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If xy , yz ∈ E and π1(x) < π1(y) < π1(z) then xz ∈ E .
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Introduction Optimization problems Recognition Encoding all realizers Conclusion
permutation ⊆ comparability ∩ co-comparability
Transitive orientation of a permutation graph G : orient edgesaccording to the top endpoints of the segments.
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If xy , yz ∈ E and π1(x) < π1(y) < π1(z) then xz ∈ E .
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Introduction Optimization problems Recognition Encoding all realizers Conclusion
comparability ∩ co-comparability ⊆ permutation
Let Etr be a transitive orientation of G and Ftr a transitiveorientation of its complement.
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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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Introduction Optimization problems Recognition Encoding all realizers Conclusion
comparability ∩ co-comparability ⊆ permutation
Let Etr be a transitive orientation of G and Ftr a transitiveorientation of its complement.
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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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Introduction Optimization problems Recognition Encoding all realizers Conclusion
A realizer of G
Permutations π1(Etr ∪ Ftr ) and π2(rev(Etr ) ∪ Ftr ) form a realizerof G .
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Segments x and y intersect iff (xy) ∈ Etr and (yx) ∈ rev(Etr ) orvice-versa ; equivalently, iff xy ∈ E .
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Introduction Optimization problems Recognition Encoding all realizers Conclusion
Modules and common intervals
• Substituting a segment (vertex) by the realizer of apermutation graph (module) produces a new permutationgraph.
• 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 themodular decomposition are permutation graphs.
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Introduction Optimization problems Recognition Encoding all realizers Conclusion
Modules and common intervals
• Substituting a segment (vertex) by the realizer of apermutation graph (module) produces a new permutationgraph.
• 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 themodular decomposition are permutation graphs.
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Introduction Optimization problems Recognition Encoding all realizers Conclusion
Encoding realizers
Theorem ([Gallai ’67])
A prime permutation graph has aunique realizer, up to reversals.
The modular decomposition tree+ realizers of prime nodesencode all possible realizers of G ,cf. [Crespelle, Paul 2010].
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series
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Introduction Optimization problems Recognition Encoding all realizers Conclusion
Conclusion
Summary
• Many optimization problems become polynomial onpermutation graphs
• Representations (intersection models) based on modulardecompositions
Some questions
• Is Bandwidth polynomial or NP-complete on permutationgraphs ?
• What about subgraph isomorphism from parametrized pointof view ?
Thank you ! Your questions ?
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Introduction Optimization problems Recognition Encoding all realizers Conclusion
Conclusion
Summary
• Many optimization problems become polynomial onpermutation graphs
• Representations (intersection models) based on modulardecompositions
Some questions
• Is Bandwidth polynomial or NP-complete on permutationgraphs ?
• What about subgraph isomorphism from parametrized pointof view ?
Thank you ! Your questions ?
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