1. Hon Wai Leong, NUS (CS6234, Spring 2009) Page 1 Copyright © 2009 by Leong Hon Wai CS6234 Lecture 1 -- (14-Jan-09) “Introduction”  Combinatorial Optimization  Topics covered in course  Emphasis of the Course

2. Hon Wai Leong, NUS (CS6234, Spring 2009) Page 2 Copyright © 2009 by Leong Hon Wai Combinatorial Optimization Combinatorial Optimization Problem:  Consists of (R, C), where uR is a set of configuration uC : R  , is a cost function  Given (R, C), find s*  R, such that C(s*) = min s  R { C(s) } Example 1: Travelling Salesman Problem (TSP) Given n cities, and distance matrix [d ij ] To find: shortest tour of n cities (visit each city exactly once) R = { all cyclic permutations  of the n cities }

3. Hon Wai Leong, NUS (CS6234, Spring 2009) Page 3 Copyright © 2009 by Leong Hon Wai Combinatorial Optimization Example 3: Linear Programming (LP) Example 2: Minimum Spanning Tree Problem (MST) Given: G = (V, E), and symmetric distance matrix [d ij ] To find: spanning tree T of G with minimum total edge cost R = { T : T =(V, E’) is a spanning tree of G }

4. Hon Wai Leong, NUS (CS6234, Spring 2009) Page 4 Copyright © 2009 by Leong Hon Wai Readings and exercises…  Exercises:  Formulate the following problems as linear programming problems: u shortest path from s to t in a graph G=(V,E) u vertex cover problem  Now, formulate the above as Comb Opt instances (similar to examples in lectures).  Reading:  [PS82] Chapter 1.

5. Hon Wai Leong, NUS (CS6234, Spring 2009) Page 5 Copyright © 2009 by Leong Hon Wai Topics Covered  Matching in Graph  Linear Programming  Approximation Algorithm  Online Algorithms  Randomized Algorithms  Topics in Data Engineering

6. Hon Wai Leong, NUS (CS6234, Spring 2009) Page 6 Copyright © 2009 by Leong Hon Wai Emphasis of the Course  Cover classic results in the area  Key techniques and insights  not necessarily the most recent  Emphasis practical algorithmic results  Efficient solutions, wherever possible  We also want algorithms to be implementable as well  Lectures will skip some details A good understanding of what is in the polynomial-time tool box is essential also for the NP-hard problem solver Alexander Schrijver, 2003

7. Hon Wai Leong, NUS (CS6234, Spring 2009) Page 7 Copyright © 2009 by Leong Hon Wai  Matching in Graph  Matching in Bipartite Graph  Matching in General Graphs  Weighted Matching in Bipartite Graph  Additional topics: uReading/Presentation by students

8. Hon Wai Leong, NUS (CS6234, Spring 2009) Page 8 Copyright © 2009 by Leong Hon Wai

9. Hon Wai Leong, NUS (CS6234, Spring 2009) Page 9 Copyright © 2009 by Leong Hon Wai Thank you. Q & A