On the independence number of random graphs
Discrete Mathematics
The largest induced tree in a sparse random graph
Proceedings of the seventh international conference on Random structures and algorithms
Finding a large hidden clique in a random graph
Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms
Exploiting the power of MIP solvers in MAXSAT
SAT'13 Proceedings of the 16th international conference on Theory and Applications of Satisfiability Testing
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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 Gn,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.