Θεωρία Γραφημάτων Θεμελιώσεις-Αλγόριθμοι-Εφαρμογές

1. Θεωρία ΓραφημάτωνΘεμελιώσεις-Αλγόριθμοι-Εφαρμογές Ενότητα 8 ΤΕΛΕΙΑ ΓΡΑΦΗΜΑΤΑ

2. Εισαγωγή Βασικοί Αλγόριθμοι Γραφημάτων Πολυπλοκότητα χώρου και χρόνου: Ο και Ω Τέλεια Γραφήματα Κλάσεις Ιδιότητες Προβλήματα Αλγοριθμικές Τεχνικές … Αλγόριθμοι Προβλημάτων Αναγνώρισης και Βελτιστοποίησης

3. Αλγόριθμοι Θεωρίας Γραφημάτων Πολυωνυμικοί Αλγόριθμοι… (Γραμμικοί) Προβλήματα: NP-Πλήρη Επιλογές Προσέγγιση Λύσης Περιορισμοί Ιδιοτήτων Τέλεια Γραφήματα, …

4. Κλάσεις Τέλειων Γραφημάτων

5. Κλάσεις Τέλειων Γραφημάτων

6. Μοντελοποίηση Προβλήματος → → http://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg

7. Intersection Graphs • Let F be a family of nonempty sets. • The intersection graph of F is obtained be representing each set in F by a vertex: x y SX ∩ SY ≠ 

8. Intersection Graphs(Interval) • The intersection graph of a family of intervals on a linearly ordered set (like the real line) is called an interval graph 2 I1 I3 I51 3 I2 I4 I6 6 4 I7 5 • unit internal graph • proper internal graph - no internal property contains another 7

9. Intersection Graphs(Circular-arc) • Circular-arc graphs properly contain the internal graphs. • proper circular - arc graphs 5 8 6 9 1 4 1 7 3 2

10. Intersection Graphs (Permutation) • A permutation diagram consists of n points on each of two parallel lines and n straight line segments matching the points. Π= [4,1,3,2] G[Π]

11. Intersection Graphs (Permutation)

12. Intersection Graphs (Chords-of-circle) • Intersecting chords of a circle • Proposition. An induced subgraph of an interval graph is an interval graph.

13. Triangulated Property • Triangulated Graph Property every simple cycle of length l > 3 possesses a chord • Triangulated graphs (or chord graphs)

14. Transitive Orientation Property • Transitive Orientation PropertyEach edge can be assigned a one-way direction in such a way that the resulting oriented graph (V, F): ab F and bc F  ac  F ( a, b, c V) • Graphs which satisfy the transitive orientation property are called comparability graph.

15. Transitive Orientation Property • Transitive Orientation Property ab F and bc F  ac  F ( a, b, c V)

16. Graph Theoretic Foundations (1) • Clique number ω(G)the number of vertices in a maximum clique of G • Stability number α(G) the number of vertices in a stable set of max cardinality Max stable set of G α(G) = 3 Max κλίκα του G ω(G) = 4 a e a b e b  d f c c d f c

17. Graph Theoretic Foundations (2) • Aclique cover of size k is a partition V = C1+C2+…+Ck such that Ci is a clique. • Aproper coloring of size c (proper c-coloring) is a partition V = X1+X2+…+Xc such that Xi is a stable set.

18. Graph Theoretic Foundations (3) • Clique cover number κ(G)the size of the smallest possible clique cover of G • Chromatic number χ(G) the smallest possible c for which there exists a proper c-coloring of G. χ(G) = 2κ(G) = 3clique cover V={2,5}+{3,4}+{1}c-coloring V={1,3,5}+{2,4}κ(G)=3 χ(G)=2

19. Graph Theoretic Foundations (4) • For any graph G: ω(G) ≤ χ(G)α(G) ≤ κ(G) • Obriasly : α(G) = ω(Ğ) and κ(G)= χ(Ğ)

20. Perfect Graphs -Definition • Let G=(V,E) be an undirected graph: (P1) ω(GA)= χ(GA) A V (P2) α(GA)= κ(GA)  A V G is called Perfect

21. Perfect Graphs -Properties • χ-Perfect property For each induced subgraph GAof G χ(GA)= ω(GA) • α-Perfect property For each induced subgraph GAof G α(GA)= κ(GA)

22. Κλάσεις Τέλειων Γραφημάτων

23. Κλάσεις Τέλειων Γραφημάτων

24. Triangulated Graphs • G triangulated G has the triangulated graph property Every simple cycle of length l > 3 possesses a chord. • Triangulated graphs, or Chordal graphs, or Perfect Elimination graphs

25. Triangulated Graphs • Dirac showed that: every chordal graph has a simplicial node, a node all of whose neighbors form a clique. a a, c, f simplicial nodes b, d, e non siplicial e b d f c

26. Triangulated Graphs • It follows easily from the triangulated property that deleting nodes of a chordal graph yields another chordal graph. a a a e b b b   d d d f f c f c

27. Triangulated Graphs • Recognition Algorithm This observation leads to the following easy and simple recognition algorithm: • Find a simplicial node of G • Delete it from G, resulting G’ • Recourse on the resulting graph G’, until no node remain

28. Triangulated Graphs • node-ordering : perfect elimination ordering, or perfect elimination scheme • Rose establishes a connection between chordal graphs and symmetric linear systems. a e b (a, c, b, e, d) (c, d, e, a, b) (c, a, b, d, e) … d c

29. Triangulated Graphs • Let σ = v1,v2, ...,vnbe an ordering of the vertices of a graph G = (V, E). • σ = peoif each vi is a simplicial node to graph Gvi,vi+1, …,vn a a e e b b σ = (c, d, e, a, b) d d c

30. Triangulated Graphs • Example: • σ =1, 7, 2, 6, 3, 5, 4 no simplicial vertex • G1 has 96 different peo.

31. Algorithms for computing peo Algorithm LexBFS 1. forall vV do label(v) :  () ; 2. for i :  V down to 1 do 2.1choose vV with lexmax label (v); 2.1σ (i)  v; 2.3 for all u V ∩N(v) do label (u)  label (u) || i 2.4 V  V \ v; end

32. Algorithms for Computing PEO

33. Algorithms for Computing PEO Algorithm MCS 1. for i :  V down to 1 do 1.1 choose vV with max number of numbered neighbours; 1.2 number v by i; 1.3σ (i)  v; 1.4 V  V \ v; end

34. Algorithms for Computing PEO

35. Minimum Vertex Separators p q z y x r

36. Minimum Vertex Separators

37. Minimum Vertex Separators

38. Perfect Graphs– Optimization Problems Βασικοί Αλγόριθμοι Γραφημάτων Πολυπλοκότητα χώρου και χρόνου: Ο και Ω Τέλεια Γραφήματα Κλάσεις Ιδιότητες Προβλήματα Τεχνικές Διάσπασης (modular decomposition, …) Αλγόριθμοι Προβλημάτων Αναγνώρισης και Βελτιστοποίησης Triangulated Comparability Interval Permutation Split Cographs Threshold graphs QT graphs … Coloring Max Clique Max Stable Set Clique Cover Matching Hamiltonian Path Hamiltonian Cycle …

39. Minimum Vertex Separators