Non-Hierarchical Sequencing Graphs. Algorithmic Graph Theory2.

Non-Hierarchical Sequencing Graphs

Algorithmic Graph Theory 2

example


Example



Algorithmic Graph Theory and its Applications

Martin Charles Golumbic


Introduction

Intersection Graphs Interval Graphs Greedy Coloring The Berge Mystery Story Other Structure Families of Graphs Graph Sandwich Problems Probe Graphs and Tolerance Graphs


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


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


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 }


complement of a graph:

interchanging the edges and the non-edges

The complement G The original graph G__


directed graph: edges have directions

(possibly both directions)

orientation: exactly ONE direction per edge

cyclic orientation acyclic orientation

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


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

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 !


Coloring Interval Graphs


Coloring Intervals (greedy)


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.


Coloring Intervals (greedy)P (needs 4 colors)


Coloring Interval Graphs

The clique at point P


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.


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


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


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!


Interval Graphs are Chordal

Interval graphs may not contain chordless cycles!

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


The co-TRO Property

The transitive orientation (TRO) of the complement

i.e., the complement must have a TRO

Not transitive ! Transitive !


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.


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.

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:

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:

Solving the Mystery

The Testimony Graph

Clue #1:

Double arrows imply TRUTH

Solving the Mystery

Undirected Testimony Graph

We know there is a lie, since {A, B, I, D} is a chordless 4-cycle.

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

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.

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.


Many other Families of Intersection Graphs

Victor Klee, in a paper in 1969:

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




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




``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.


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


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


A small hierarchy


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




The Story Begins




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


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


EPT graphs

EPT graph

share an edge


VPT graphs

VPT graph

share a node


Some Interesting Theorems VPT graphs are chordal EPT graphs are NOT chordal


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.


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.


Algorithmic Complexity Results


Some Interesting Theorems

Folklore (1970’s)

Every graph G is the edge intersection graph of distinct subtrees of a star.


Degree 3 host trees (continued)

Theorem (1985): All four classes are equivalent:

chordal EPT deg3 EPT

VPT EPT deg3 VPT

What about degree 4?


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


Definition Weakly Chordal Graph

No induced Cm for m 5, and

no induced Cm for m 5.

Weakly Chordal Graphs


The Story Continues


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


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


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


Example: Interval Probe GraphsNon-Probes are white

Probe graph NOT a Probe graph no matter how you partition vertices!


Tolerance Graphs What if you only have 3 classrooms? Cancel a Lecture? or show Tolerance?


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’’


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’’


Tolerance Graphs: Example

c and f will no longer conflict

| Ic If | < 60 = min {tc , tf}

Non-Hierarchical Sequencing Graphs. Algorithmic Graph Theory2.

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