BySairam sista

Contents

Problem statement

Graph construction

Relation to Graph theory

Graph definition

Special property

Graph solution

References

Problem Statement

Suppose that C1, C2, · · · , Cn are chemical compounds which must be refrigerated under closely monitored conditions.

If a compound Ci (i=1 to n) must be kept at a constant temperature between ti and ti’ degrees then how many refrigerators will be needed to store all the compounds?

Our task here is to arrange the chemical compounds in minimum number of refrigerators.

Problem Statement

Chemical compound Temperatures

A 0-25

B 5-15

C 10-50

D 20-45

E 30-40

F 35-55

G 55-60

Graph construction

If two different chemical compounds

occurred together in the same refrigerator,

then their temperatures must have

overlapped.• To model the situation, we simply need to

construct a graph in which :each vertex represents one of the

chemical compounds each edge connects vertices where their

respective temperatures overlap.

Graph construction

• Let us consider a graph G and the vertices from A-G which represent chemical compounds and their edges which represent the overlap of their temperature intervals.

Graph construction

Relation to Graph theory

This Real world problem is converted to “interval

graph problem”.

An “interval graph” is the graph showing intersecting

intervals on a line. So, we associate a set of intervals

I={I1,…,In} on a line with the interval graph

G=(V,E),where V={1,…,n} and two vertices, x and y,

are linked by an edge if and only if their temperature

intervals overlap.The following is the interval graph

formed from the graph .

Relation to graph theory

Graph Definition

Interval graphs:In a Graph theory,an interval graph is the intersection graph of a family of intervals on the real line. It has one vertex for each interval in the family, and an edge between every pair of vertices whenever intervals intersect.

EXAMPLE:

Special Property

Umbrella Free Ordering:

For every interval graph there will be an

Umbrella Free-Ordering it states that,

arranging the vertices in an order such that

if there is an edge between two vertices

then any edge that lies between the two

vertices must be adjacent to the right

vertex in the ordering.

Special Property

Minimum clique cover: To calculate minimum number of cliques in an umbrella free ordering.

• It is chordal and its complement G is a comparability graph.

• AT ( asteroidal triple) free.• It contains no induced C4 and G is

transitively orientable.

Special Property

Graph solution

According to the graph with vertices A-G we draw umbrella free ordering from left to right.

We then find cliques among the vertices.

Umbrella free ordering after assigning clique:

Graph solution

Graph solution

From the umbrella free ordering we get 3 cliques.

A,B,C and D,E,F and G are the 3 cliques .

So here minimum clique cover is 3 Seven chemical compounds (A-G)

can here by arranged in 3 refrigerators.

Graph solution

Chemical compounds

Temperatures Refrigerator temperature

A 0-25 15

B 5-15 15

C 10-50 15

D 20-45 35

E 30-40 35

F 35-55 35

G 55-60 57

NP-complete

In computational complexity theory, finding a minimum clique cover is a graph-theoretical NP-complete problem. The problem was one of Richard Karp's original 21 problems shown NP-complete in his 1972 paper "Reducibility Among Combinatorial Problems".

References

http://www.csee.umbc.edu/~stephens/NUM/PROJECTS/refrigerator.html

https://halshs.archives-ouvertes.fr/halshs-00123607/document

http://www.researchgate.net/publication/228574261_On-line_algorithm_for_the_minimal_b-clique_cover_problem_in_interval_graphs

http://infoscience.epfl.ch/record/118668/files/EPFL_TH4090.pdf

http://worldwidescience.org/topicpages/m/minimum-diameter+clique+trees.html

Thank you