non-hierarchical sequencing graphs
DESCRIPTION
Non-Hierarchical Sequencing Graphs. example. Example. Algorithmic Graph Theory and its Applications. Martin Charles Golumbic. Algorithmic Graph Theory and its Applications. Martin Charles Golumbic. Algorithmic Graph Theory and its Applications. Martin Charles Golumbic. Introduction. - PowerPoint PPT PresentationTRANSCRIPT
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Non-Hierarchical Sequencing Graphs
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Algorithmic Graph Theory 2
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example
Algorithmic Graph Theory 3
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Example
Algorithmic Graph Theory 4
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Algorithmic Graph Theory 5
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Algorithmic Graph Theory 6
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Algorithmic Graph Theory 7
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Algorithmic Graph Theory 8
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Algorithmic Graph Theory 9
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Algorithmic Graph Theory 10
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Algorithmic Graph Theory 11
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Algorithmic Graph Theory 12
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Algorithmic Graph Theory 13
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Algorithmic Graph Theory 14
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Algorithmic Graph Theory 15
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Algorithmic Graph Theory 16
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Algorithmic Graph Theory 17
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Algorithmic Graph Theory 18
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Algorithmic Graph Theory 19
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Algorithmic Graph Theory 20
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Algorithmic Graph Theory 21
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Algorithmic Graph Theory 22
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Algorithmic Graph Theory 23
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Algorithmic Graph Theory 24
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Algorithmic Graph Theory 25
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Algorithmic Graph Theory 26
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Algorithmic Graph Theory 32
Algorithmic Graph Theory and its Applications
Martin Charles Golumbic
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Algorithmic Graph Theory 33
Algorithmic Graph Theory and its Applications
Martin Charles Golumbic
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Algorithmic Graph Theory 34
Algorithmic Graph Theory and its Applications
Martin Charles Golumbic
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Algorithmic Graph Theory 35
Introduction
Intersection Graphs Interval Graphs Greedy Coloring The Berge Mystery Story Other Structure Families of Graphs Graph Sandwich Problems Probe Graphs and Tolerance Graphs
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Algorithmic Graph Theory 36
Theconcept of an intersection graph applications in computation operations research molecular biology scheduling designing circuits rich mathematical problems
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Algorithmic Graph Theory 37
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
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Algorithmic Graph Theory 38
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 }
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Algorithmic Graph Theory 39
complement of a graph:
interchanging the edges and the non-edges
The complement G The original graph G__
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Algorithmic Graph Theory 40
directed graph: edges have directions
(possibly both directions)
orientation: exactly ONE direction per edge
cyclic orientation acyclic orientation
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Interval GraphsThe intersection graphs of intervals on a line:
- create a vertex for each interval
- connect vertices when their intervals intersect
Jan Feb Mar Apr May Jun July Sep Oct Nov Dec
Phase 1Phase 1
Phase 2Phase 2
Phase 3Phase 3Task 4
Task 5
1 2 3
4 5The interval graph G
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Algorithmic Graph Theory 42
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 assignmentVLSI designtemporal reasoning in AIcomputer storage allocation
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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?
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Scheduling Example (cont.)
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Scheduling Example (graphs)
(a) The interval graph (b) Its complement (disjointness)
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Algorithmic Graph Theory 46
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 !
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Algorithmic Graph Theory 47
Coloring Interval Graphs
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Algorithmic Graph Theory 48
Coloring Intervals (greedy)
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Algorithmic Graph Theory 49
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.
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Algorithmic Graph Theory 50
Coloring Intervals (greedy)P (needs 4 colors)
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Algorithmic Graph Theory 51
Coloring Interval Graphs
The clique at point P
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Algorithmic Graph Theory 52
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.
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Algorithmic Graph Theory 53
Characterizing Interval Graphs Properties of interval graphs How to recognize them Their mathematical structure
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Algorithmic Graph Theory 54
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
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Algorithmic Graph Theory 55
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!
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Algorithmic Graph Theory 56
Interval Graphs are Chordal
Interval graphs may not contain chordless cycles!
- i.e., they are chordal. Why?
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Algorithmic Graph Theory 57
Interval Graphs are Chordal
Interval graphs may not contain chordless cycles!
- i.e., they are chordal. Why?
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Algorithmic Graph Theory 58
The co-TRO Property
The transitive orientation (TRO) of the complement
i.e., the complement must have a TRO
Not transitive ! Transitive !
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Algorithmic Graph Theory 59
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.
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Algorithmic Graph Theory 60
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.
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A Mystery in the LibraryThe Berge Mystery Story: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:
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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?One of the Professor LIED !! Who was it?
The Facts:The Facts:
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Solving the Mystery
The Testimony Graph
Clue #1:
Double arrows imply TRUTH
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Solving the Mystery
Undirected Testimony Graph
We know there is a lie, since {A, B, I, D} is a chordless 4-cycle.
cycle
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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!
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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.One professor from the chordless 4-cycle must be a liar.
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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.
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Algorithmic Graph Theory 68
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.
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Algorithmic Graph Theory 69
Many other Families of Intersection Graphs
Victor Klee, in a paper in 1969:
``What are the intersection graphs of arcs in a circle?’’
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Algorithmic Graph Theory 70
Many other Families of Intersection Graphs
Victor Klee, in a paper in 1969:
``What are the intersection graphs of arcs in a circle?“
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Algorithmic Graph Theory 71
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.
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Algorithmic Graph Theory 72
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
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Algorithmic Graph Theory 73
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
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Algorithmic Graph Theory 74
A small hierarchy
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Algorithmic Graph Theory 75
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
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Algorithmic Graph Theory 76
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
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Algorithmic Graph Theory 77
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
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Algorithmic Graph Theory 78
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
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Algorithmic Graph Theory 79
EPT graphs
EPT graph
share an edge
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Algorithmic Graph Theory 80
VPT graphs
VPT graph
share a node
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Algorithmic Graph Theory 81
Some Interesting Theorems VPT graphs are chordal EPT graphs are NOT chordal
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Algorithmic Graph Theory 82
Some Interesting TheoremsVPT 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.
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Algorithmic Graph Theory 83
Some Interesting TheoremsEPT graphs are NOT chordal
An EPT representation of C6
called a “6-pie”.6
3
2
1
4
5
Chordless cycles have a unique EPT representation.
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Algorithmic Graph Theory 84
Algorithmic Complexity Results
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Algorithmic Graph Theory 85
Some Interesting Theorems
Folklore (1970’s)
Every graph G is the edge intersection graph of distinct subtrees of a star.
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Algorithmic Graph Theory 86
Degree 3 host trees (continued)
Theorem (1985): All four classes are equivalent:
chordal EPT deg3 EPT
VPT EPT deg3 VPT
What about degree 4?
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Algorithmic Graph Theory 87
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
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Algorithmic Graph Theory 88
Definition Weakly Chordal Graph
No induced Cm for m 5, and
no induced Cm for m 5.
Weakly Chordal Graphs
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Algorithmic Graph Theory 89
The Story Continues
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Algorithmic Graph Theory 90
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
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Algorithmic Graph Theory 91
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
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Algorithmic Graph Theory 92
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
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Algorithmic Graph Theory 93
Example: Interval Probe GraphsNon-Probes are white
Probe graph NOT a Probe graph no matter how you partition vertices!
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Algorithmic Graph Theory 94
Tolerance Graphs What if you only have 3 classrooms? Cancel a Lecture? or show Tolerance?
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Algorithmic Graph Theory 95
Tolerance GraphsMeasured intersection:
small, or ``tolerable’’ amount of overlap, may be ignored does NOT produce an edge
at least one of them has to be ``bothered’’
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Algorithmic Graph Theory 96
Tolerance Graphs
Assignment of positive numbers
{tv} (v V) such that
vw E if and only if | Iv Iw | min {tv , tw}
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’’
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Algorithmic Graph Theory 97
Tolerance Graphs: Example
c and f will no longer conflict
| Ic If | < 60 = min {tc , tf}
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