Lecture 17

Learning objectives

After this class, you should be able to:

1. Define a semi-definite program and show that the set of feasible solutions to it is convex.
2. Show that vector programs and semi-definite programs are equivalent.
3. Use randomized rounding to get an approximate solution to to MAX-CUT, and derive its approximate factor.
4. Given a combinatorial optimization problem, pose it as a strict quadratic program, relax it to a vector program, use a semidefinite program solver to solve it, and then develop a rounding procedure to obtain an approximate solution to the original problem.

Reading assignment

1. Sections 26.3 and 26.4.

Exercises and review questions

• Questions on current lecture's material
  1. Use a graphing software to plot the ratio of theta/pi and (1 - cos(theta)), and confrm the approximation factor that we derived for MAX-CUT.
  2. Give a strict quadratic program for MAX k-CUT (defined in exercise 2.3). Relax it to a vector program and show an equivalent semi-definite program.
  3. Give an application for MAX-CUT. Post your answer on the discussion board

• Questions on next lecture's material
  1. None.