Algorithmic Graph Theory and its Applications

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```					Algorithmic Graph Theory
and its Applications
Martin Charles Golumbic

Algorithmic Graph Theory   1
Introduction
   Intersection Graphs
   Interval Graphs
   Greedy Coloring
   The Berge Mystery Story
   Other Structure Families of Graphs
   Graph Sandwich Problems
   Probe Graphs and Tolerance Graphs

Algorithmic Graph Theory    2
The
concept of an
intersection graph
   applications in computation
   operations research
   molecular biology
   scheduling
   designing circuits
   rich mathematical problems

Algorithmic Graph Theory   3
Defining some terms
   graph: a collection of vertices and edges
   coloring a graph:
assigning a color to every vertex,
such that

Algorithmic Graph Theory       4
   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 }

Algorithmic Graph Theory   5
   complement of a graph:
interchanging the edges and the non-edges

__
The complement G                 The original graph G

Algorithmic Graph Theory            6
   directed graph:     edges have directions
(possibly both directions)

   orientation:      exactly ONE direction per edge

cyclic orientation               acyclic orientation
Algorithmic Graph Theory          7
Interval Graphs
The intersection graphs of intervals on a line:
- create a vertex for each interval
- connect vertices when their intervals intersect
Phase 2

Jan   Feb    Mar      Apr       May   Jun    July    Sep     Oct     Nov     Dec

1                     2               3
The interval graph G
4                     5
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:
VLSI design
temporal reasoning in AI
computer storage allocation
Algorithmic Graph Theory   9
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?
Scheduling Example (cont.)
Scheduling Example (graphs)

(a) The interval graph
(b) Its complement (disjointness)
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 !

Algorithmic Graph Theory       13
Coloring Interval Graphs

Algorithmic Graph Theory   14
Coloring Intervals (greedy)

Algorithmic Graph Theory   15
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.
Algorithmic Graph Theory        16
Coloring Intervals (greedy)
P (needs 4 colors)

Algorithmic Graph Theory    17
Coloring Interval Graphs

The clique
at point P

Algorithmic Graph Theory                18
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.

Algorithmic Graph Theory            19
Characterizing Interval Graphs
   Properties of interval graphs
   How to recognize them
   Their mathematical structure

Algorithmic Graph Theory   20
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

Algorithmic Graph Theory   21
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!

Algorithmic Graph Theory             22
Interval Graphs are Chordal
Interval graphs may not contain chordless cycles!

- i.e., they are chordal.     Why?

Algorithmic Graph Theory   23
Interval Graphs are Chordal
Interval graphs may not contain chordless cycles!

- i.e., they are chordal.     Why?

Algorithmic Graph Theory   24
The co-TRO Property
The transitive orientation (TRO) of the complement

i.e., the complement must have a TRO

Not transitive !                              Transitive !
Algorithmic Graph Theory                  25
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.

Algorithmic Graph Theory   26
Gilmore and Hoffman (1964)
Theorem:
A graph G is an interval graph
G
if and only if__ 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.

Algorithmic Graph Theory                  27
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:
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?
Solving the Mystery
The Testimony Graph

Clue #1:
Double arrows imply TRUTH
Solving the Mystery
Undirected Testimony Graph

cycle

We know there is a lie, since {A, B, I, D} is a chordless 4-cycle.
Intersecting Intervals cannot
form Chordless Cycles

Burt                  Desmond

Abe
No place for Ida’s interval:
It must hit both B and D but cannot hit A.
Impossible!
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 ?
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.
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.

Algorithmic Graph Theory    35
Many other Families of
Intersection Graphs
Victor Klee, in a paper in 1969:

``What are the intersection graphs of arcs
in a circle?’’

Algorithmic Graph Theory     36
Many other Families of
Intersection Graphs
Victor Klee, in a paper in 1969:

``What are the intersection graphs of arcs
in a circle?“

Algorithmic Graph Theory     37
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.
Algorithmic Graph Theory        38
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

Algorithmic Graph Theory    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

The Algorithmic Problems:
–   recognize them
–   color them
–   find maximum cliques
–   find maximum independent sets
Algorithmic Graph Theory    40
A small hierarchy

Algorithmic Graph Theory   41
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?

Algorithmic Graph Theory                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?

Algorithmic Graph Theory                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?
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.

Algorithmic Graph Theory                          44
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
Algorithmic Graph Theory             45
EPT graphs

EPT graph
share an edge

Algorithmic Graph Theory             46
VPT graphs

VPT graph
share a node

Algorithmic Graph Theory            47
Some Interesting Theorems
   VPT graphs are chordal
   EPT graphs are NOT chordal

Algorithmic Graph Theory   48
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.

Algorithmic Graph Theory                  49
Some Interesting Theorems
EPT graphs are NOT chordal

An EPT representation of C6                                  1
6
called a “6-pie”.
2
5
4       3

Chordless cycles have a unique EPT representation.

Algorithmic Graph Theory                       50
Algorithmic Complexity Results

Algorithmic Graph Theory   51
Some Interesting Theorems

   Folklore (1970’s)
Every graph G is the edge intersection
graph of distinct subtrees of a star.

Algorithmic Graph Theory         52
Degree 3 host trees (continued)
Theorem (1985): All four classes are equivalent:

chordal  EPT  deg3 EPT
 VPT  EPT  deg3 VPT

Algorithmic Graph Theory     53
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
Algorithmic Graph Theory     54
Weakly Chordal Graphs

Definition Weakly Chordal Graph
No induced Cm for m  5,
and
no induced Cm for m  5.

Algorithmic Graph Theory   55
The Story Continues

Algorithmic Graph Theory   56
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

Algorithmic Graph Theory     57
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

Algorithmic Graph Theory        58
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
Algorithmic Graph Theory        59
Example: Interval Probe Graphs
Non-Probes are white

Probe graph                    NOT a Probe graph
no matter how you
partition vertices!
Algorithmic Graph Theory            60
Tolerance Graphs
   What if you only have 3 classrooms?
   Cancel a Lecture? or show Tolerance?

Algorithmic Graph Theory        61
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’’

Algorithmic Graph Theory         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’’

   Assignment of positive numbers
{tv} (v  V) such that
vw  E if and only if | Iv  Iw |  min {tv , tw}

Algorithmic Graph Theory         63
Tolerance Graphs: Example

c and f will no longer conflict

| Ic  If | < 60 = min {tc , tf}

Algorithmic Graph Theory   64
More on Algorithmic Graph Theory

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