The Lovász Local Lemma [5] (LLL) is a powerful result in probability theory that states that the probability that none of a set of bad events happens is nonzero if the probability of each event is small compared to the number of events that depend on it. It is often used in combination with the probabilistic method for non-constructive existence proofs. A prominent application is to k-CNF formulas, where LLL implies that, if every clause in the formula shares variables with at most d ≤ 2 k /e other clauses then such a formula has a satisfying assignment. Recently, a randomized algorithm to efficiently construct a satisfying assignment was given by Moser [12]. Subsequently Moser and Tardos [13] gave a randomized algorithm to construct the structures guaranteed by the LLL in a very general algorithmic framework. We address the main problem left open by Moser and Tardos of derandomizing these algorithms efficiently. Specifically, for a k-CNF formula with m clauses and d ≤ 2 k/(1+ǫ) /e for some ǫ ∈ (0, 1), we give an algorithm that finds a satisfying assignment in timeÕ(m 2(1+1/ǫ) ). This improves upon the deterministic algorithms of Moser and of Moser-Tardos with running time m Ω(k 2 ) which is superpolynomial for k = ω(1) and upon other previous algorithms which work only for d ≤ 2 k/16 /e. Our algorithm works efficiently for a general version of LLL under the algorithmic framework of Moser and Tardos [13], and is also parallelizable, i.e., has polylogarithmic running time using polynomially many processors.
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.
An undirected graph G = (V, E) is stable if its inessential vertices (those that are exposed by at least one maximum matching) form a stable set. We call a set of edges F ⊆ E a stabilizer if its removal from G yields a stable graph. In this paper we study the following natural edge-deletion question: given a graph G = (V, E), can we find a minimum-cardinality stabilizer? Stable graphs play an important role in cooperative game theory. In the classic matching game introduced by Shapley and Shubik [19] we are given an undirected graph G = (V, E) where vertices represent players, and we define the value of each subset S ⊆ V as the cardinality of a maximum matching in the subgraph induced by S. The core of such a game contains all fair allocations of the value of V among the players, and is well-known to be non-empty iff graph G is stable. The stabilizer problem addresses the question of how to modify the graph to ensure that the core is non-empty. We show that this problem is vertex-cover hard. We then prove that there is a minimum-cardinality stabilizer that avoids some maximum matching of G. We use this insight to give efficient approximation algorithms for sparse graphs and for regular graphs.
We study the problem of answering k-way marginal queries on a database D ∈ ({0, 1} d ) n , while preserving differential privacy. The answer to a k-way marginal query is the fraction of the database's records x ∈ {0, 1} d with a given value in each of a given set of up to k columns. Marginal queries enable a rich class of statistical analyses on a dataset, and designing efficient algorithms for privately answering marginal queries has been identified as an important open problem in private data analysis. For any k, we give a differentially private online algorithm that runs in time poly n, 2 o(d) per query and answers any sequence of poly(n) many k-way marginal queries with error at most ±0.01 on every query, provided n d 0.51 . To the best of our knowledge, this is the first algorithm capable of privately answering marginal queries with a non-trivial worst-case accuracy guarantee for databases containing poly(d, k) records in time exp(o(d)). Our algorithm runs the private multiplicative weights algorithm (Hardt and Rothblum, FOCS '10) on a new approximate polynomial representation of the database.We derive our representation for the database by approximating the OR function restricted to low Hamming weight * inputs using low-degree polynomials with coefficients of bounded L1-norm. In doing so, we show new upper and lower bounds on the degree of such polynomials, which may be of independent approximation-theoretic interest.
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