Lecture 6 : Provable Approximation via Linear Programming

One of the running themes in this course is the notion of approximate solutions. Of course, this notion is tossed around a lot in applied work: whenever the exact solution seems hard to achieve, you do your best and call the resulting solution an approximation. In theoretical work, approximation has a more precise meaning whereby you prove that the computed solution is close to the exact or optimum solution in some precise metric. We saw some earlier examples of approximation in sampling-based algorithms; for instance our hashing-based estimator for set size. It produces an answer that is whp within (1+ ) of the true answer. Today we will see many other examples that rely upon linear programming (LP). Recall that most NP-hard optimization problems involve finding 0/1 solutions. Using LP one can find fractional solutions, where the relevant variables are constrained to take real values in [0, 1]. Recall the example of the assignment problem from last time, which is also a 0/1 problem (a job is either assigned to a particular factory or it is not) but the LP relaxation magically produces a 0/1 solution (although we didn’t prove this in class). Whenever the LP produces a solution in which all variables are 0/1, then this must be the optimum 0/1 solution as well since it is the best fractional solution, and the class of fractional solutions contains every 0/1 solution. Thus the assignment problem is solvable in polynomial time. Needless to say, we don’t expect this magic to repeat for NP-hard problems. So the LP relaxation yields a fractional solution in general. Then we give a way to round the fractional solutions to 0/1 solutions. This is accompanied by a mathematical proof that the new solution is provably approximate.