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Permutation Graphs --- an Introduction Introduction Optimization problems Recognition Encoding all realizers Conclusion Permutation graphs | an introduction Ioan Todinca LIFO - Universit´ed'Orl´eans Algorithms and permutations, february 2012 1/14 Introduction Optimization problems Recognition Encoding all realizers Conclusion Permutation graphs 1 23 4 2 3 4 1 5 6 5 6 • Optimisation algorithms use, as input, the intersection model (realizer) • Recognition algorithms output the intersection(s) model(s) 2/14 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 3/14 Introduction Optimization problems Recognition Encoding all realizers Conclusion Definition and basic properties 2 1 6 5 4 3 1 23 4 2 3 4 1 5 6 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) 4/14 Introduction Optimization problems Recognition Encoding all realizers Conclusion Definition and basic properties 2 1 6 5 4 3 1 23 4 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) 4/14 Introduction Optimization problems Recognition Encoding all realizers Conclusion Definition and basic properties 2 1 6 5 4 3 1 23 4 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) 4/14 Introduction Optimization problems Recognition Encoding all realizers Conclusion Definition and basic properties 2 1 6 5 4 3 1 23 4 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) 4/14 Introduction Optimization problems Recognition Encoding all realizers Conclusion More intersection graph classes Circle graphs Trapezoid graphs Books on graph classes : [Golumbic '80 ; Brandst¨adt,Le, Spinrad '99 ; Spinrad 2003] 5/14 Introduction Optimization problems Recognition Encoding all realizers Conclusion MaxIndependentSet via Dynamic Programming 12 34 5 6 24 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) 6/14 Introduction Optimization problems Recognition Encoding all realizers Conclusion Treewidth via dynamic programming on scanlines 2 1 6 5 4 3 1 23 4 5 6 1 5 6 3 2 4 12256 23 34 • 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] 7/14 Introduction Optimization problems Recognition Encoding all realizers Conclusion Treewidth via dynamic programming on scanlines 1 23 4 2 1 6 5 4 3 5 6 12256 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] 7/14 Introduction Optimization problems Recognition Encoding all realizers Conclusion Treewidth via dynamic programming on scanlines 1 2 3 4 2 1 6 5 4 3 5 6 12256 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] 7/14 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] 8/14 Introduction Optimization problems Recognition Encoding all realizers Conclusion permutation ⊆ comparability \ co-comparability 2 1 6 5 4 3 Transitive orientation of a permutation graph G : orient edges according1 2 to the top3 endpoints 4 of the segments. 5 6 1 5 6 3 2 4 If xy; yz 2 E and π1(x) < π1(y) < π1(z) then xz 2 E. 9/14 Introduction Optimization problems Recognition Encoding all realizers Conclusion permutation ⊆ comparability \ co-comparability 2 1 6 5 4 3 Transitive orientation of a permutation graph G : orient edges according to the top endpoints of the segments. 1 5 6 3 2 4 1 23 4 5 6 If xy; yz 2 E and π1(x) < π1(y) < π1(z) then xz 2 E. 9/14 rev(Etr ) [ Ftr induces another total ordering π2(rev(Etr ) [ Ftr ). 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 23 4 5 6 Lemma Etr [ Ftr induces a total ordering π1(Etr [ Ftr ) on the vertex set. 10/14 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 23 4 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 ). 10/14 Introduction Optimization problems Recognition Encoding all realizers Conclusion A realizer of G 2 1 6 5 4 3 Permutations π1(Etr [ Ftr ) and π2(rev(Etr ) [ Ftr ) form a realizer of1G. 23 4 5 6 1 5 6 3 2 4 Segments x and y intersect iff (xy) 2 Etr and (yx) 2 rev(Etr ) or vice-versa ; equivalently, iff xy 2 E. 11/14 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. 12/14 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. 12/14 Introduction Optimization problems Recognition Encoding all realizers Conclusion Encoding realizers 1 23 4 Theorem ([Gallai '67] ) A prime5 permutation 6 graph has a unique realizer, up to reversals. prime 1 2 4 The modular decomposition tree 3 + realizers of prime nodes encode all possible realizers of G, parallel 23 4 cf. [Crespelle, Paul 2010]. series 1 5 6 13/14 Thank you ! Your questions ? 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 ? 14/14 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 ? 14/14.
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