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Algorithmic Graph Theory and its Applications

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Title: Algorithmic Graph Theory and its Applications


1
Algorithmic Graph Theory and its Applications
  • Martin Charles Golumbic

2
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
Theconcept of an intersection graph
  • applications in computation
  • operations research
  • molecular biology
  • scheduling
  • designing circuits
  • rich mathematical problems

4
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
  • 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
  • complement of a graph
  • interchanging the edges and the non-edges

__
The complement G
The original graph G
7
  • directed graph edges have directions
  • (possibly both directions)
  • orientation exactly ONE direction per edge

cyclic orientation
acyclic orientation
8
Interval Graphs
  • The intersection graphs of intervals on a line
  • - create a vertex for each interval
  • - connect vertices when their intervals
    intersect

Task 5
Task 4
1
2
3
The interval graph G
4
5
9
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 900-1015
  • Lecture b 1000-1200
  • etc.
  • Conflicting lectures ? Different rooms
  • How many rooms?

11
Scheduling Example (cont.)
12
Scheduling Example (graphs)
  • The interval graph
  • Its complement (disjointness)

13
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
Coloring Interval Graphs
15
Coloring Intervals (greedy)
16
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
Coloring Intervals (greedy)
P (needs 4 colors)
18
Coloring Interval Graphs
The clique at point P
19
Greedy the best we can do !
  • Formally,
  • 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
Characterizing Interval Graphs
  • Properties of interval graphs
  • How to recognize them
  • Their mathematical structure

21
Characterizing Interval Graphs
  • Properties of interval graphs
  • How to recognize them
  • Their mathematical structure

Two properties characterize interval graphs -
The Chordal Graph Property - The co-TRO Property
22
The Chordal Graph Property
  • chordal graph
  • every cycle of length gt 4 has a chord
  • (connecting two vertices that are not
    consecutive)
  • i.e., they may not contain chordless cycles!

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

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

25
The co-TRO Property
  • The transitive orientation (TRO) of the
    complement
  • i.e., the complement must have a TRO

Not transitive !
Transitive !
26
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
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 1970s.
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
The Facts
  • 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?
30
Solving the Mystery
The Testimony Graph
Clue 1 Double arrows imply TRUTH
31
Solving the Mystery
Undirected Testimony Graph
cycle
We know there is a lie, since A, B, I, D is a
chordless 4-cycle.
32
Intersecting Intervals cannot form Chordless
Cycles
Burt
Desmond
Abe
No place for Idas interval It must hit both
B and D but cannot hit A. Impossible!
33
Solving the Mystery
One professor from the chordless 4-cycle must be
a liar.
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 ?
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
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
Many other Families of Intersection Graphs
  • Victor Klee, in a paper in 1969
  • What are the intersection graphs of arcs in a
    circle?

37
Many other Families of Intersection Graphs
  • Victor Klee, in a paper in 1969
  • What are the intersection graphs of arcs in a
    circle?

38
Many other Families of Intersection Graphs
  • Victor Klee, in a paper in 1969
  • What are the intersection graphs of arcs in a
    circle?
  • Klees paper was an implicit challenge
  • - consider a whole variety of problems
  • - on many kinds of intersection graphs.

39
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
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
A small hierarchy
42
The Story Begins
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?
43
The Story Begins
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?
44
The Story Begins
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?
An Olive Tree Network
  • 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.

45
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
46
EPT graphs
EPT graph share an edge
47
VPT graphs
VPT graph share a node
48
Some Interesting Theorems
  • VPT graphs are chordal
  • EPT graphs are NOT chordal

49
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.
  • 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 are chordal.

50
Some Interesting Theorems
  • EPT graphs are NOT chordal

An EPT representation of C6 called a 6-pie.
1
6
2
5
3
4
Chordless cycles have a unique EPT representation.
51
Algorithmic Complexity Results
52
Some Interesting Theorems
  • Folklore (1970s)
  • Every graph G is the edge intersection graph of
    distinct subtrees of a star.

53
Degree 3 host trees (continued)
Theorem (1985) All four classes are
equivalent chordal ? EPT ? deg3 EPT ?
VPT ? EPT ? deg3 VPT
What about degree 4?
54
Degree 3 host trees (continued)
Theorem (1985) All four classes are
equivalent chordal ? EPT ? deg3 EPT ?
VPT ? EPT ? deg3 VPT
Degree 4 host trees
Theorem (2005) Golumbic, Lipshteyn, Stern
weakly chordal ? EPT ? deg4 EPT
55
Weakly Chordal Graphs
  • Definition Weakly Chordal Graph
  • No induced Cm for m ? 5,
  • and
  • no induced Cm for m ? 5.

56
The Story Continues
57
The Interval Graph Sandwich Problem
  • Interval problems with missing edges
  • Benzers 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

58
Interval Graph Sandwich Problem
  • given a partially specified graph
  • E1 required edges
  • E2 optional edges
  • E3 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

59
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 fill-in some of the N x N edges,
  • so that the result will be an interval graph
  • Motivation

60
Example Interval Probe Graphs
Non-Probes are white
Probe graph
NOT a Probe graph no matter how you partition
vertices!
61
Tolerance Graphs
  • What if you only have 3 classrooms?
  • Cancel a Lecture? or show Tolerance?

62
Tolerance Graphs
Measured intersection small, or tolerable
amount of overlap, may be ignored does NOT
produce an edge at least one of them has to be
bothered
63
Tolerance Graphs
Measured intersection 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
  • tv (v ? V) such that
  • vw ? E if and only if Iv ? Iw ? min
    tv , tw

64
Tolerance Graphs Example
c and f will no longer conflict Ic ? If lt
60 min tc , tf
65
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