1. Algorithmic Graph Theory1 Algorithmic Graph Theory and its Applications Martin Charles Golumbic

2. Algorithmic Graph Theory2 Introduction Intersection Graphs Interval Graphs Greedy Coloring The Berge Mystery Story Other Structure Families of Graphs Graph Sandwich Problems Probe Graphs and Tolerance Graphs

3. Algorithmic Graph Theory3 The concept of an intersection graph applications in computation operations research molecular biology scheduling designing circuits rich mathematical problems

4. Algorithmic Graph Theory4 Defining some terms graph: a collection of vertices and edges coloring a graph: assigning a color to every vertex, such that adjacent vertices have different colors

5. Algorithmic Graph Theory5 independent set: a collection of vertices NO two of which are connected Example: { d, e, f } or the green set clique (or complete set): EVERY two of which are connected Example: { a, b, d } or { c, e }

6. Algorithmic Graph Theory6 complement of a graph: interchanging the edges and the non-edges The complement GThe original graph G __

7. Algorithmic Graph Theory7 directed graph: edges have directions (possibly both directions) orientation:exactly ONE direction per edge cyclic orientationacyclic orientation

8. Interval Graphs The intersection graphs of intervals on a line: - create a vertex for each interval - connect vertices when their intervals intersect JanFebMarAprMayJunJulySepOctNovDec Phase 1 Phase 2 Phase 3 Task 4 Task 5 123 45 The interval graph G

9. Algorithmic Graph Theory9 History of Interval Graphs Hajos 1957: Combinatorics (scheduling) Benzer 1959: Biology (genetics) Gilmore & Hoffman 1964: Characterization Booth & Lueker 1976: First linear time recognition algorithm Many other applications: mobile radio frequency assignment VLSI design temporal reasoning in AI computer storage allocation

10. Scheduling Example Lectures need to be assigned classrooms at the University.  Lecture #a: 9:00-10:15  Lecture #b: 10:00-12:00  etc. Conflicting lectures  Different rooms How many rooms?

11. Scheduling Example (cont.)

12. Scheduling Example (graphs) (a) The interval graph (b) Its complement (disjointness)

13. Algorithmic Graph Theory13 Coloring Interval Graphs interval graphs have special properties used to color them efficiently coloring algorithm sweeps across from left to right assigning colors in a “greedy manner” This is optimal !

14. Algorithmic Graph Theory14 Coloring Interval Graphs

15. Algorithmic Graph Theory15 Coloring Intervals (greedy)

16. Algorithmic Graph Theory16 Is greedy the best we can do? Can we prove optimality? Yes: It uses the smallest # colors. Proof: Let k be the number of colors used. Look at the point P, when color k was used first. At P all the colors 1 to k-1 were busy! We are forced to use k colors at P. And, they form a clique of size k in the interval graph.

17. Algorithmic Graph Theory17 Coloring Intervals (greedy) P (needs 4 colors)

18. Algorithmic Graph Theory18 Coloring Interval Graphs The clique at point P

19. Algorithmic Graph Theory19 Greedy the best we can do ! Formally, (1) at least k colors are required (because of the clique) (2) greedy succeeded using k colors. Therefore, the solution is optimal. Q.E.D.

20. Algorithmic Graph Theory20 Characterizing Interval Graphs Properties of interval graphs How to recognize them Their mathematical structure

21. Algorithmic Graph Theory21 Characterizing Interval Graphs Properties of interval graphs How to recognize them Their mathematical structure Two properties together characterize interval graphs: - The Chordal Graph Property - The co-TRO Property

22. Algorithmic Graph Theory22 The Chordal Graph Property chordal graph: every cycle of length > 4 has a chord (connecting two vertices that are not consecutive) i.e., they may not contain chordless cycles!

23. Algorithmic Graph Theory23 Interval Graphs are Chordal Interval graphs may not contain chordless cycles! - i.e., they are chordal. Why?

24. Algorithmic Graph Theory24 Interval Graphs are Chordal Interval graphs may not contain chordless cycles! - i.e., they are chordal. Why?

25. Algorithmic Graph Theory25 The co-TRO Property The transitive orientation (TRO) of the complement i.e., the complement must have a TRO Not transitive !Transitive !

26. Algorithmic Graph Theory26 Interval Graphs are co-TRO The complement of an Interval graph has a transitive orientation! - Why? The complement is the disjointness graph. So, orient from the earlier interval to the later interval.

27. Algorithmic Graph Theory27 Gilmore and Hoffman (1964) Theorem: A graph G is an interval graph if and only if G Is chordal and its complement G is transitively orientable. __ This provides the basis for the first set of recognition algorithms in the early 1970’s.

28. A Mystery in the Library The Berge Mystery Story: Six professors had been to the library on the day that the rare tractate was stolen. Each had entered once, stayed for some time and then left. If two were in the library at the same time, then at least one of them saw the other. Detectives questioned the professors and gathered the following testimony:

29. Abe said that he saw Burt and Eddie Burt reported that he saw Abe and Ida Charlotte claimed to have seen Desmond and Ida Desmond said that he saw Abe and Ida Eddie testified to seeing Burt and Charlotte Ida said that she saw Charlotte and Eddie One of the Professor LIED !! Who was it? The Facts:

30. Solving the Mystery The Testimony Graph Clue #1: Double arrows imply TRUTH

31. Solving the Mystery Undirected Testimony Graph We know there is a lie, since {A, B, I, D} is a chordless 4-cycle. cycle

32. Intersecting Intervals cannot form Chordless Cycles BurtDesmond Abe No place for Ida’s interval: It must hit both B and D but cannot hit A. Impossible!

33. Solving the Mystery There are three chordless 4-cycles: {A, B, I, D}{A, D, I, E} {A, E, C, D} Burt is NOT a liar: He is missing from the second cycle. Ida is NOT a liar: She is missing from the third cycle. Charlotte is NOT a liar: She is missing from the second. Eddie is NOT a liar: He is missing from the first cycle. WHO IS THE LIAR? Abe or Desmond ? One professor from the chordless 4-cycle must be a liar.

34. Solving the Mystery (cont.) WHO IS THE LIAR? Abe or Desmond ? If Abe were the liar and Desmond truthful, then {A, B, I, D} would remain a chordless 4-cycle, since B and I are truthful. Therefore: Desmond is the liar.

35. Algorithmic Graph Theory35 Was Desmond Stupid or Just Ignorant? If Desmond had studied algorithmic graph theory, he would have known that his testimony to the police would not hold up.

36. Algorithmic Graph Theory36 Many other Families of Intersection Graphs Victor Klee, in a paper in 1969: ``What are the intersection graphs of arcs in a circle?’’

37. Algorithmic Graph Theory37 Many other Families of Intersection Graphs Victor Klee, in a paper in 1969: ``What are the intersection graphs of arcs in a circle?“

38. Algorithmic Graph Theory38 Many other Families of Intersection Graphs Victor Klee, in a paper in 1969: ``What are the intersection graphs of arcs in a circle?“ Klee’s paper was an implicit challenge - consider a whole variety of problems - on many kinds of intersection graphs.

39. Algorithmic Graph Theory39 Families of Intersection Graphs boxes in the plane paths in a tree chords of a circle spheres in 3-space trapezoids, parallelograms, curves of functions many other geometrical and topological bodies

40. Algorithmic Graph Theory40 Families of Intersection Graphs boxes in the plane paths in a tree chords of a circle spheres in 3-space trapezoids, parallelograms, curves of functions many other geometrical and topological bodies The Algorithmic Problems: → recognize them → color them → find maximum cliques → find maximum independent sets

41. Algorithmic Graph Theory41 A small hierarchy co-TRO

42. Algorithmic Graph Theory42 Bell Labs in New Jersey (Spring 1981) John Klincewicz: Suppose you are routing phone calls in a tree network. Two calls interfere if they share an edge of the tree. How can you optimally schedule the calls? The Story Begins An Olive Tree Network

43. Algorithmic Graph Theory43 Bell Labs in New Jersey (Spring 1981) John Klincewicz: Suppose you are routing phone calls in a tree network. Two calls interfere if they share an edge of the tree. How can you optimally schedule the calls? The Story Begins A call is a path between a pair of nodes. A typical example of a type of intersection graph. Intersection here means “share an edge”. Coloring this intersection graph is scheduling the calls. An Olive Tree Network

44. Algorithmic Graph Theory44 Edge Intersection Graphs of Paths in a Tree (EPT graphs) tree communication network connecting different places if two of these paths overlap, they conflict and cannot use the same resource at the same time. Two types of intersections share an edge vs share a node Two types of intersections – share an edge vs share a node

45. Algorithmic Graph Theory45 EPT graphs EPT graph share an edge

46. Algorithmic Graph Theory46 VPT graphs VPT graph share a node

47. Algorithmic Graph Theory47 Some Interesting Theorems VPT graphs are chordal EPT graphs are NOT chordal

48. Algorithmic Graph Theory48 Some Interesting Theorems VPT graphs are chordal Buneman, Gavril, Wallace (early 1970's) G is the vertex intersection graph of subtrees of a tree if and only if it is a chordal graph. Remark. Since paths are a special species of trees, VPT graphs are chordal. In the literature, VPT graphs are also called “path graphs”.

49. Algorithmic Graph Theory49 Some Interesting Theorems EPT graphs are NOT chordal An EPT representation of C 6 called a “6-pie”. 6 3 2 1 4 5 Theorem (Golumbic & Jamison, 1985). Chordless cycles have a unique EPT representation.

50. Algorithmic Graph Theory50 Algorithmic Complexity Results

51. Algorithmic Graph Theory51 More Interesting Theorems Folklore (1970’s) Every graph G is the edge intersection graph of distinct subtrees of a star. Proof: Exercise. McMorris & Shier (1983) A graph G is a vertex intersection graph of distinct subtrees of a star if and only if both G and its complement G are chordal.

52. Algorithmic Graph Theory52 Degree 3 host trees Theorem (1985): All four classes are equivalent: chordal  EPT  deg3 EPT  VPT  EPT  deg3 VPT What about degree 4?

53. Algorithmic Graph Theory53 Degree 3 host trees (continued) Theorem (1985): All four classes are equivalent: chordal  EPT  deg3 EPT  VPT  EPT  deg3 VPT Theorem (2005) [Golumbic, Lipshteyn, Stern]: weakly chordal  EPT  deg4 EPT Degree 4 host trees

54. Algorithmic Graph Theory54 Definition Weakly Chordal Graph No induced C m for m  5, and no induced C m for m  5. Weakly Chordal Graphs

55. Algorithmic Graph Theory55 The Story Continues

56. Algorithmic Graph Theory56 The Interval Graph Sandwich Problem Interval problems with missing edges Benzer’s original problem  partial intersection data  Is it consistent ? Complete data would be recognition interval graphs (polynomial) Partial data needs a different model and is NP-complete

57. Algorithmic Graph Theory57 Interval Graph Sandwich Problem given a partially specified graph  E 1 required edges  E 2 optional edges  E 3 forbidden edges Can you fill-in some of the optional edges, so that the result will be an interval graph? Golumbic & Shamir (1993): NP-Complete

58. Other Graph Sandwich Problems NP-complete for chordal graphs, permutation graphs, comparability graphs, and many others … Polynomial-time for split graphs, threshold graphs, chain graphs, … Algorithmic Graph Theory58

59. Algorithmic Graph Theory59 Interval Probe Graphs A special tractable case of interval sandwich Computational biology motivated Interval probe graph: vertices are partitioned  P probes & N non-probes (independent set)  can we fill-in some of the missing N x N edges, so that the result will be an interval graph? Motivation

60. Algorithmic Graph Theory60 Example: Interval Probe Graphs Non-Probes are white Probe graphNOT a Probe graph no matter how you partition vertices!

61. Algorithmic Graph Theory61 Tolerance Graphs What if you only have 3 classrooms? Cancel a Lecture? or show Tolerance? More Graph Problems on Intervals

62. Algorithmic Graph Theory62 Tolerance Graphs Measured intersection: a small, or “tolerable’’ amount of overlap, may be ignored does NOT produce an edge at least one of them has to be “bothered’’ Assignment of positive numbers {t v } (v  V) such that vw  E if and only if | I v  I w |  min {t v, t w }

63. Algorithmic Graph Theory63 Tolerance Graphs: Example c and f will no longer conflict | I c  I f | < 60 = min {t c, t f }

65. Summary What have we seen today? Interval graphs – coloring, characterizing, applications, recognition Other families of intersection graphs – paths in trees, subtrees of trees, and more to come… Other graph models using intervals – sandwich and probe problems, tolerance graphs, etc. Algorithmic Graph Theory65

66. Where to find this lecture: Algorithmic Graph Theory66 http://www.cs.haifa.ac.il/~golumbic /courses/algorithmic-graph-theory/

67. What’s coming next? Algorithmic Graph Theory67 May 14: String Graphs and Paths on a Grid May 21: Old and New Graph Dimension Parameters

68. More on Algorithmic Graph Theory