Estimation of the Probit Model from Anonymized Micro Data

The demand of scientists for confidential micro data from official sources has created discussion of how to anonymize these data in such a way that they can be given to the scientific community. We report results from a German project which exploits various options of anonymization for producing such ”scientific-use- files”. The main concern in the project however is whether estimation of stochastic models from these perturbed data is possible and – more importantly – leads to reliable results. In this paper we concentrate on estimation of the probit model under the assumption that only anonymized data are available. In particular we assume that the binary dependent variable has undergone post-randomization (PRAM) and that the set of explanatory variables has been perturbed by addition of noise. We employ a maximum likelihood estimator which is consistent if only the dependent variable has been anonymized by PRAM. The errors-in-variables structure of the regressors then is handled by the simulation extrapolation (SIMEX) estimation procedure where we compare performance of quadratic and nonlinear (rational) extrapolation.anonymization, misclassification, noise addition, post-randomization, SIMEX procedure, statistical disclosure.

Distributed Symmetry Breaking in Hypergraphs

Fundamental local symmetry breaking problems such as Maximal Independent Set (MIS) and coloring have been recognized as important by the community, and studied extensively in (standard) graphs. In particular, fast (i.e., logarithmic run time) randomized algorithms are well-established for MIS and $\Delta +1$-coloring in both the LOCAL and CONGEST distributed computing models. On the other hand, comparatively much less is known on the complexity of distributed symmetry breaking in {\em hypergraphs}. In particular, a key question is whether a fast (randomized) algorithm for MIS exists for hypergraphs. In this paper, we study the distributed complexity of symmetry breaking in hypergraphs by presenting distributed randomized algorithms for a variety of fundamental problems under a natural distributed computing model for hypergraphs. We first show that MIS in hypergraphs (of arbitrary dimension) can be solved in $O(\log^2 n)$ rounds ($n$ is the number of nodes of the hypergraph) in the LOCAL model. We then present a key result of this paper --- an $O(\Delta^{\epsilon}\text{polylog}(n))$-round hypergraph MIS algorithm in the CONGEST model where $\Delta$ is the maximum node degree of the hypergraph and $\epsilon > 0$ is any arbitrarily small constant. To demonstrate the usefulness of hypergraph MIS, we present applications of our hypergraph algorithm to solving problems in (standard) graphs. In particular, the hypergraph MIS yields fast distributed algorithms for the {\em balanced minimal dominating set} problem (left open in Harris et al. [ICALP 2013]) and the {\em minimal connected dominating set problem}. We also present distributed algorithms for coloring, maximal matching, and maximal clique in hypergraphs.Comment: Changes from the previous version: More references adde

A Faster Distributed Single-Source Shortest Paths Algorithm

We devise new algorithms for the single-source shortest paths (SSSP) problem with non-negative edge weights in the CONGEST model of distributed computing. While close-to-optimal solutions, in terms of the number of rounds spent by the algorithm, have recently been developed for computing SSSP approximately, the fastest known exact algorithms are still far away from matching the lower bound of $\tilde \Omega (\sqrt{n} + D)$ rounds by Peleg and Rubinovich [SIAM Journal on Computing 2000], where $n$ is the number of nodes in the network and $D$ is its diameter. The state of the art is Elkin's randomized algorithm [STOC 2017] that performs $\tilde O(n^{2/3} D^{1/3} + n^{5/6})$ rounds. We significantly improve upon this upper bound with our two new randomized algorithms for polynomially bounded integer edge weights, the first performing $\tilde O (\sqrt{n D})$ rounds and the second performing $\tilde O (\sqrt{n} D^{1/4} + n^{3/5} + D)$ rounds. Our bounds also compare favorably to the independent result by Ghaffari and Li [STOC 2018]. As side results, we obtain a $(1 + \epsilon)$-approximation $\tilde O ((\sqrt{n} D^{1/4} + D) / \epsilon)$-round algorithm for directed SSSP and a new work/depth trade-off for exact SSSP on directed graphs in the PRAM model.Comment: Presented at the the 59th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2018

Parallel algorithms and concentration bounds for the Lovasz Local Lemma via witness DAGs

The Lov\'{a}sz Local Lemma (LLL) is a cornerstone principle in the probabilistic method of combinatorics, and a seminal algorithm of Moser & Tardos (2010) provides an efficient randomized algorithm to implement it. This can be parallelized to give an algorithm that uses polynomially many processors and runs in $O(\log^3 n)$ time on an EREW PRAM, stemming from $O(\log n)$ adaptive computations of a maximal independent set (MIS). Chung et al. (2014) developed faster local and parallel algorithms, potentially running in time $O(\log^2 n)$, but these algorithms require more stringent conditions than the LLL. We give a new parallel algorithm that works under essentially the same conditions as the original algorithm of Moser & Tardos but uses only a single MIS computation, thus running in $O(\log^2 n)$ time on an EREW PRAM. This can be derandomized to give an NC algorithm running in time $O(\log^2 n)$ as well, speeding up a previous NC LLL algorithm of Chandrasekaran et al. (2013). We also provide improved and tighter bounds on the run-times of the sequential and parallel resampling-based algorithms originally developed by Moser & Tardos. These apply to any problem instance in which the tighter Shearer LLL criterion is satisfied

Massively Parallel Algorithms for Distance Approximation and Spanners

Over the past decade, there has been increasing interest in distributed/parallel algorithms for processing large-scale graphs. By now, we have quite fast algorithms -- usually sublogarithmic-time and often $poly(\log\log n)$-time, or even faster -- for a number of fundamental graph problems in the massively parallel computation (MPC) model. This model is a widely-adopted theoretical abstraction of MapReduce style settings, where a number of machines communicate in an all-to-all manner to process large-scale data. Contributing to this line of work on MPC graph algorithms, we present $poly(\log k) \in poly(\log\log n)$ round MPC algorithms for computing $O(k^{1+{o(1)}})$-spanners in the strongly sublinear regime of local memory. To the best of our knowledge, these are the first sublogarithmic-time MPC algorithms for spanner construction. As primary applications of our spanners, we get two important implications, as follows: -For the MPC setting, we get an $O(\log^2\log n)$-round algorithm for $O(\log^{1+o(1)} n)$ approximation of all pairs shortest paths (APSP) in the near-linear regime of local memory. To the best of our knowledge, this is the first sublogarithmic-time MPC algorithm for distance approximations. -Our result above also extends to the Congested Clique model of distributed computing, with the same round complexity and approximation guarantee. This gives the first sub-logarithmic algorithm for approximating APSP in weighted graphs in the Congested Clique model

Round Compression for Parallel Matching Algorithms

For over a decade now we have been witnessing the success of {\em massive parallel computation} (MPC) frameworks, such as MapReduce, Hadoop, Dryad, or Spark. One of the reasons for their success is the fact that these frameworks are able to accurately capture the nature of large-scale computation. In particular, compared to the classic distributed algorithms or PRAM models, these frameworks allow for much more local computation. The fundamental question that arises in this context is though: can we leverage this additional power to obtain even faster parallel algorithms? A prominent example here is the {\em maximum matching} problem---one of the most classic graph problems. It is well known that in the PRAM model one can compute a 2-approximate maximum matching in $O(\log{n})$ rounds. However, the exact complexity of this problem in the MPC framework is still far from understood. Lattanzi et al. showed that if each machine has $n^{1+\Omega(1)}$ memory, this problem can also be solved $2$-approximately in a constant number of rounds. These techniques, as well as the approaches developed in the follow up work, seem though to get stuck in a fundamental way at roughly $O(\log{n})$ rounds once we enter the near-linear memory regime. It is thus entirely possible that in this regime, which captures in particular the case of sparse graph computations, the best MPC round complexity matches what one can already get in the PRAM model, without the need to take advantage of the extra local computation power. In this paper, we finally refute that perplexing possibility. That is, we break the above $O(\log n)$ round complexity bound even in the case of {\em slightly sublinear} memory per machine. In fact, our improvement here is {\em almost exponential}: we are able to deliver a $(2+\epsilon)$-approximation to maximum matching, for any fixed constant $\epsilon>0$, in $O((\log \log n)^2)$ rounds