CS 6234 Lecture 1 14 Jan09 Introduction q

CS 6234 Lecture 1 -- (14 -Jan-09) “Introduction” q Combinatorial Optimization q Topics covered in course q Emphasis of the Course Hon Wai Leong, NUS (CS 6234, Spring 2009) Page 1 Copyright © 2009 by Leong Hon Wai

Combinatorial Optimization Problem: v Consists of (R, C), where u. R is a set of configuration u. C : R , is a cost function v 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 } Hon Wai Leong, NUS (CS 6234, 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 } Example 3: Linear Programming (LP) Hon Wai Leong, NUS (CS 6234, Spring 2009) Page 3 Copyright © 2009 by Leong Hon Wai

Readings and exercises… q Exercises: v 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 v Now, formulate the above as Comb Opt instances (similar to examples in lectures). q Reading: v [PS 82] Chapter 1. Hon Wai Leong, NUS (CS 6234, Spring 2009) Page 4 Copyright © 2009 by Leong Hon Wai

Topics Covered q Matching in Graph q Linear Programming q Approximation Algorithm q Online Algorithms q Randomized Algorithms q Topics in Data Engineering Hon Wai Leong, NUS (CS 6234, Spring 2009) Page 5 Copyright © 2009 by Leong Hon Wai

Emphasis of the Course q Cover classic results in the area v Key techniques and insights v not necessarily the most recent q Emphasis practical algorithmic results v Efficient solutions, wherever possible v We also want algorithms to be implementable as well v 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 (CS 6234, Spring 2009) Page 6 Copyright © 2009 by Leong Hon Wai

q Matching in Graph v Matching in Bipartite Graph v Matching in General Graphs v Weighted Matching in Bipartite Graph v Additional topics: u. Reading/Presentation by students Hon Wai Leong, NUS (CS 6234, Spring 2009) Page 7 Copyright © 2009 by Leong Hon Wai

Hon Wai Leong, NUS (CS 6234, Spring 2009) Page 8 Copyright © 2009 by Leong Hon Wai

Thank you. Q&A Hon Wai Leong, NUS (CS 6234, Spring 2009) Page 9 Copyright © 2009 by Leong Hon Wai