Implicit Hitting Set Problems

ACO Student Seminar
Wednesday, April 28, 2010 - 13:30
1 hour (actually 50 minutes)
ISyE Executive Classroom
Abstract: A hitting set for a collection of sets T is a set that has a non-empty intersection with eachset in T; the hitting set problem is to find a hitting set of minimum cardinality. Motivated bythe fact that there are instances of the hitting set problem where the number of subsets to behit is large, we introduce the notion of implicit hitting set problems. In an implicit hitting setproblem the collection of sets to be hit is typically too large to list explicitly; instead, an oracleis provided which, given a set H, either determines that H is a hitting set or returns a set inT that H does not hit. I will show a number of examples of classic implicit hitting set problems,and give a generic algorithm for solving such problems exactly in an online model.I will also show how this framework is valuable in developing approximation algorithms by presenting a simple on-line algorithm for the minimum feedback vertex set problem. In particular, our algorithm gives an approximation factor of 1+ 2 log(np)/(np) for the random graph G_{n,p}.Joint work with Richard Karp, Erick Moreno-Centeno (UC, Berkeley) and Santosh Vempala (Georgia Tech).