CS264: Beyond Worst-Case Analysis Lecture #8: Exact Recovery in Stable Cut Instances

This lecture makes explicit a theme introduced surreptitiously last lecture: exact recovery. The genre of question in this sequence of lectures is: for an algorithm A of interest, for which inputs does A solve the problem exactly? For example, last lecture, we proved that the singlelink++ algorithm recovers the optimal solution in every 3-stable k-median instance. (See Homework #4 for an improvement.) We generally think of the algorithm A above as either: (i) an already-implemented algorithm that is available to us, such as a linear programming solver; or (ii) a relatively simple algorithm that we have a good understanding of and could code up easily. The reason is that, in practice, designing a new algorithm from scratch is often a last resort. It is usually far preferable if something in the current algorithmic toolbox is good enough, at least for the problem instances relevant to the application. The exact recovery results in this sequence of lectures develop theoretical guarantees that correspond well to this style of thinking. This lecture covers another case study on exact recovery, for a graph cut problem. We’ll retain from last lecture the definition of γ-stable instances and the flavor of the problems studied, as cut problems are a type of clustering problem. What will be different is the algorithms studied — linear programming instead of more elementary algorithms. The broader theme, to be elaborated on in future lectures, is to explain when NP -hard problems can be solved exactly via linear programming. (Since linear programs can be solved in polynomial time, we do not expect this to be true for worst-case instances.) Thus, even