Algorithms for implicit hitting set problems

Karthekeyan Chandrasekaran, Richard Karp, Erick Moreno-Centeno, Santosh Vempala

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

A hitting set for a collection of sets is a set that has a nonempty intersection with each set in the collection; the hitting set problem is to find a hitting set of minimum cardinality. Motivated by instances of the hitting set problem where the number of sets to be hit is large, we introduce the notion of implicit hitting set problems. In an implicit hitting set problem the collection of sets to be hit is typically too large to list explicitly; instead, an oracle is provided which, given a set H, either determines that H is a hitting set or returns a set that H does not hit. We show a number of examples of classic implicit hitting set problems, and give a generic algorithm for solving such problems optimally. The main contribution of this paper is to show that this framework is valuable in developing approximation algorithms. We illustrate this methodology by presenting a simple on-line algorithm for the minimum feedback vertex set problem on random graphs. In particular our algorithm gives a feedback vertex set of size n - ( 1/p) log np( 1 - o(1)) with probability at least 3/4 for the random graph Gn,p (the smallest feedback vertex set is of size n - (2/p) log np(l + o(l))). We also consider a planted model for the feedback vertex set in directed random graphs. Here we show that a hitting set for a polynomial-sized subset of cycles is a hitting set for the planted raindom graph and this allows us to exactly recover the planted feedback vertex set.

Original languageEnglish (US)
Title of host publicationProceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011
PublisherAssociation for Computing Machinery
Pages614-629
Number of pages16
ISBN (Print)9780898719932
DOIs
StatePublished - 2011
Externally publishedYes

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

ASJC Scopus subject areas

  • Software
  • Mathematics(all)

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