Algorithms for Implicit Hitting Set Problems

TL;DR

This paper introduces implicit hitting set problems, presents a generic algorithm for solving them, and demonstrates its effectiveness through approximation algorithms for feedback vertex set problems in random graphs.

Contribution

It formalizes implicit hitting set problems, provides a generic optimal solution algorithm, and applies it to develop approximation algorithms for feedback vertex set problems.

Findings

The online algorithm achieves a feedback vertex set size close to optimal in random graphs.

The framework enables exact recovery of planted feedback vertex sets in directed random graphs.

Approximation bounds are established for feedback vertex sets in G_{n,p}.

Abstract

A hitting set for a collection of sets is a set that has a non-empty 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 G_{n,p} (the smallest feedback vertex set is of size n-(2/p)\log{np}(1+o(1))). 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 random graph and this allows us to exactly recover the planted feedback vertex set.