instructor neelima gupta [email protected]. table of contents five representative problems
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Table of Contents
Five representative problems
Five Representative ProblemsInterval Scheduling : can be solved by a greedy approach.Weighted Interval Scheduling : Natural greedy doesn’t
work, no other greedy is known, more sophisticated technique DP solves the problem.
Maximum Bipartite MatchingIndependent SetCompetitive Facility Location
Thanks to Neha (16)
Interval Scheduling Problem
Time0
P(1)=10
P(3)=4
P(4)=20
P(2)=3
1 2 3 4 5 6 7 8 9
P(5)=2
Examples Jobs submitted to an operating system,
Resource: CPU.
An HR of a company needs to schedule meetings of some committees in a meeting room, resource is meeting room
Scheduling classes in a room, resource is class-room
Greedy Approach :Increasing Finishing Times
Thanks to Neha (16)
Time0
P(1)=10
P(4)=20
P(2)=3
1 2 3 4 5 6 7 8 9
P(5)=2
.
back
Weighted Interval Scheduling
Time0
P(1)=10
P(3)=4
P(4)=20
P(2)=3
1 2 3 4 5 6 7 8 9
P(5)=2
Thanks to Neha (16)
Examples Jobs submitted to an operating system,
Resource: CPU.Weights: profit by executing the job
Greedy Approach
Time0
P(1)=10
P(4)=20
P(2)=3
1 2 3 4 5 6 7 8 9
P(5)=2
.
Thanks to Neha (16)
Greedy does not work
Time0
P(1)=10
P(4)=20
P(2)=3
1 2 3 4 5 6 7 8 9
P(5)=2
Optimal schedule
Schedule chosen by greedy app
Greedy approach takes job 2, 3 and 5 as best schedule and makes profit of 7. While optimal schedule is job 1 and job4 making profit of 30 (10+20). Hence greedy will not work
Thanks to Neha (16)
u1
u2
u3
u4
v1
v2
v3
Figure 1
Example of a Bipartite graph :
u1
u2
Edge like this is not acceptable in Bipartite GraphV1
V2
(Thanks to Aditya(04),Abhishek(03)-Msc 2014)
ExamplesThere is a set T of teachers with a set C of courses.
A teacher can teach only some set of courses represented by the edges in the bipartite graph.
Thus, bipartite graphs are used to represent relationships between two distinct sets of objects…teachers and courses here.
Jobs/Employers and Applicants: An employer receives several applications but only few of them qualify for the interview. Similarly an applicant applies for many jobs but qualify only for few for them for interview. An edge (a, e) in the bipartite graph represents that the applicant ‘a’ qualifies for the interview for job ‘e’.
Maximum Matching is a matching of maximum Cardinality.
u1
u2
u3
u4
v1
v2
v3
V
1
V2
By Applying the definition of matching, If we choose the edge (u1,v1) first
And then (u2, v2)
So no more edge can be included, hence matching in this case is :
(u1,v1) , (u2, v2)
(Thanks to Aditya(04),Abhishek(03)-Msc 2014)
But instead of picking (u2,v2) , if we pick
• (u2 ,v3) after (u1 , v1) then
•( u4 ,v2) .
u1
u2
u3
u4
v1
v2
v3
V
1
V2
So, the problem is to find the Matching with MAXIMUM CARDINALITY in a given Bipartite graph.
Hence the Maximum Matching is :
(u1,v1) , (u2, v3 ) , (u4 , v2)
(Thanks to Aditya(04),Abhishek(03)-Msc 2014)
ExamplesThere is a set T of teachers with a set C of courses.
A teacher can teach only some set of courses represented by the edges in the bipartite graph.A teacher needs to be assigned at most one course
and one course must be taught by only one teacher.
Suppose a number of committee meetings are to be scheduled in various meeting rooms during the time 3pm - 4pm. A committee meeting can be held only in few rooms (may be because other rooms are smaller in size than the committee size etc). An edge (c, r) represents that committee ‘c’ can be scheduled in room ‘r’.A committee needs to be assigned one room and one
room must be assigned to only one committee.
Maximum Bipartite Matching from Abhishek Aditya
Maximum Bipartite Matching from Abhishek Aditya
Maximum Bipartite Matching from Abhishek Aditya
Independent Set Given a graph G = (V, E), a subset S of V
is said to be independent if no two nodes in S are joined by an edge in G.
Thanks to: Sonia Verma (25, MCS '09)
Maximal Independent set of size 2
Thanks to: Sonia Verma (25, MCS '09)
Maximal Independent set of size 3 …also Maximum
More on Independent Set and Comp. FLP from Anurag