1. Instructor Neelima Gupta ngupta@cs.du.ac.in

2. Table of Contents Five representative problems

3. Five Representative Problems Interval Scheduling : can be solved by a greedy approach. Interval Scheduling Weighted Interval Scheduling : Natural greedy doesn’t work, no other greedy is known, more sophisticated technique DP solves the problem. Weighted Interval Scheduling Maximum Bipartite Matching Independent Set Competitive Facility Location

4. Thanks to Neha (16) Interval Scheduling Problem Time 0 P(1)=10 P(3)=4 P(4)=20 P(2)=3 1 23456 7 89 P(5)=2

5. 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

6. Greedy Approach :Increasing Finishing Times Thanks to Neha (16) Time 0 P(1)=10 P(4)=20 P(2)=3 1 23456 7 89 P(5)=2. back

7. Weighted Interval Scheduling Time 0 P(1)=10 P(3)=4 P(4)=20 P(2)=3 1 23456 7 89 P(5)=2 Thanks to Neha (16)

8. Examples Jobs submitted to an operating system, Resource: CPU. Weights: profit by executing the job

9. Greedy Approach Time 0 P(1)=10 P(4)=20 P(2)=3 1 23456 7 89 P(5)=2. Thanks to Neha (16)

10. Greedy does not work Time 0 P(1)=10 P(4)=20 P(2)=3 1 23456 7 89 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)

11. u1u1 u2u2 u3u3 u4u4 v1v1 v2v2 v3 Figure 1 Example of a Bipartite graph : u1u1 u2u2 Edge like this is not acceptable in Bipartite Graph V1V1 V2V2 (Thanks to Aditya(04),Abhishek(03)-Msc 2014)

12. Examples There 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’.

13. Maximum Matching is a matching of maximum Cardinality. u1u1 u2u2 u3u3 u4u4 v1v1 v2v2 v3v3 V1V1 V2V2 By Applying the definition of matching, If we choose the edge (u 1,v 1 ) first And then (u 2, v 2 ) So no more edge can be included, hence matching in this case is : (u 1,v 1 ), (u 2, v 2 ) (Thanks to Aditya(04),Abhishek(03)-Msc 2014)

14. But instead of picking (u 2,v 2 ), if we pick (u 2,v 3 ) after (u 1, v 1 ) then ( u 4,v 2 ). u1u1 u2u2 u3u3 u4u4 v1v1 v2v2 v3v3 V1V1 V2V2 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)

15. Examples There 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.

16. Maximum Bipartite Matching from Abhishek Aditya

19. 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.

20. Thanks to: Sonia Verma (25, MCS '09) Maximal Independent set of size 2

21. Thanks to: Sonia Verma (25, MCS '09) Maximal Independent set of size 3 …also Maximum

22. More on Independent Set and Comp. FLP from Anurag