CS6234 Lecture 1 -- (14-Jan-09)
“Introduction”
 Combinatorial Optimization
 Topics covered in course
 Emphasis of the Course
Hon Wai Leong, NUS
(CS6234, Spring 2009) Page 1
Copyright © 2009 by Leong Hon Wai
Combinatorial Optimization
Combinatorial Optimization Problem:
 Consists of (R, C), where
R is a set of configuration
C : R  , is a cost function
 Given (R, C), find s*R, such that
C(s*) = minsR { C(s) }
Example 1: Travelling Salesman Problem (TSP)
Given n cities, and distance matrix [dij]
To find: shortest tour of n cities (visit each city exactly once)
R = { all cyclic permutations  of the n cities }
C ( )  nj 1 d j ( j )
Hon Wai Leong, NUS
(CS6234, Spring 2009) Page 2
Copyright © 2009 by Leong Hon Wai
Combinatorial Optimization
Example 2: Minimum Spanning Tree Problem (MST)
Given: G = (V, E), and symmetric distance matrix [dij]
To find: spanning tree T of G with minimum total edge cost
R = { T : T =(V, E’) is a spanning tree of G }
C(T )   (i, j )E '(T ) d ij
Example 3: Linear Programming (LP)

m 
Given an m  n matrix, A  [aij ], aij  , b   , c   n
 
  
n
R  { x | x   , A  x  b , x  0}
T 
C  x  c x
Hon Wai Leong, NUS
(CS6234, Spring 2009) Page 3
Copyright © 2009 by Leong Hon Wai
Readings and exercises…
 Exercises:
 Formulate the following problems as linear
programming problems:
 shortest path from s to t in a graph G=(V,E)
 vertex cover problem
 Now, formulate the above as Comb Opt
instances (similar to examples in lectures).
 Reading:
 [PS82] Chapter 1.
Hon Wai Leong, NUS
(CS6234, Spring 2009) Page 4
Copyright © 2009 by Leong Hon Wai
Topics Covered
 Matching in Graph
 Linear Programming
 Approximation Algorithm
 Online Algorithms
 Randomized Algorithms
 Topics in Data Engineering
Hon Wai Leong, NUS
(CS6234, Spring 2009) Page 5
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
Hon Wai Leong, NUS
(CS6234, Spring 2009) Page 6
Copyright © 2009 by Leong Hon Wai
 Matching in Graph
 Matching in Bipartite Graph
 Matching in General Graphs
 Weighted Matching in Bipartite Graph
 Additional topics:
Reading/Presentation by students
Hon Wai Leong, NUS
(CS6234, Spring 2009) Page 7
Copyright © 2009 by Leong Hon Wai
Hon Wai Leong, NUS
(CS6234, Spring 2009) Page 8
Copyright © 2009 by Leong Hon Wai
Thank you.
Q &A
Hon Wai Leong, NUS
(CS6234, Spring 2009) Page 9
Copyright © 2009 by Leong Hon Wai