O(log⁡2k/log⁡log⁡k)O(\log^2k/\log\log{k})-Approximation Algorithm for Directed Steiner Tree: A Tight Quasi-Polynomial-Time Algorithm

In the Directed Steiner Tree (DST) problem we are given an $n$-vertex directed edge-weighted graph, a root $r$, and a collection of $k$ terminal nodes. Our goal is to find a minimum-cost arborescence that contains a directed path from $r$ to every terminal. We present an $O(\log^2 k/\log\log{k})$-approximation algorithm for DST that runs in quasi-polynomial-time. By adjusting the parameters in the hardness result of Halperin and Krauthgamer, we show the matching lower bound of $\Omega(\log^2{k}/\log\log{k})$ for the class of quasi-polynomial-time algorithms. This is the first improvement on the DST problem since the classical quasi-polynomial-time $O(\log^3 k)$ approximation algorithm by Charikar et al. (The paper erroneously claims an $O(\log^2k)$ approximation due to a mistake in prior work.) Our approach is based on two main ingredients. First, we derive an approximation preserving reduction to the Label-Consistent Subtree (LCST) problem. The LCST instance has quasi-polynomial size and logarithmic height. We remark that, in contrast, Zelikovsky's heigh-reduction theorem used in all prior work on DST achieves a reduction to a tree instance of the related Group Steiner Tree (GST) problem of similar height, however losing a logarithmic factor in the approximation ratio. Our second ingredient is an LP-rounding algorithm to approximately solve LCST instances, which is inspired by the framework developed by Rothvo{\ss}. We consider a Sherali-Adams lifting of a proper LP relaxation of LCST. Our rounding algorithm proceeds level by level from the root to the leaves, rounding and conditioning each time on a proper subset of label variables. A small enough (namely, polylogarithmic) number of Sherali-Adams lifting levels is sufficient to condition up to the leaves

Duplicate Detection in Probabilistic Data

Collected data often contains uncertainties. Probabilistic databases have been proposed to manage uncertain data. To combine data from multiple autonomous probabilistic databases, an integration of probabilistic data has to be performed. Until now, however, data integration approaches have focused on the integration of certain source data (relational or XML). There is no work on the integration of uncertain (esp. probabilistic) source data so far. In this paper, we present a first step towards a concise consolidation of probabilistic data. We focus on duplicate detection as a representative and essential step in an integration process. We present techniques for identifying multiple probabilistic representations of the same real-world entities. Furthermore, for increasing the efficiency of the duplicate detection process we introduce search space reduction methods adapted to probabilistic data

Proving uniformity and independence by self-composition and coupling

Proof by coupling is a classical proof technique for establishing probabilistic properties of two probabilistic processes, like stochastic dominance and rapid mixing of Markov chains. More recently, couplings have been investigated as a useful abstraction for formal reasoning about relational properties of probabilistic programs, in particular for modeling reduction-based cryptographic proofs and for verifying differential privacy. In this paper, we demonstrate that probabilistic couplings can be used for verifying non-relational probabilistic properties. Specifically, we show that the program logic pRHL---whose proofs are formal versions of proofs by coupling---can be used for formalizing uniformity and probabilistic independence. We formally verify our main examples using the EasyCrypt proof assistant

Model-checking branching-time properties of probabilistic automata and probabilistic one-counter automata

This paper studies the problem of model-checking of probabilistic automaton and probabilistic one-counter automata against probabilistic branching-time temporal logics (PCTL and PCTL$^*$). We show that it is undecidable for these problems. We first show, by reducing to emptiness problem of probabilistic automata, that the model-checking of probabilistic finite automata against branching-time temporal logics are undecidable. And then, for each probabilistic automata, by constructing a probabilistic one-counter automaton with the same behavior as questioned probabilistic automata the undecidability of model-checking problems against branching-time temporal logics are derived, herein.Comment: Comments are welcom

Pushing undecidability of the isolation problem for probabilistic automata

This short note aims at proving that the isolation problem is undecidable for probabilistic automata with only one probabilistic transition. This problem is known to be undecidable for general probabilistic automata, without restriction on the number of probabilistic transitions. In this note, we develop a simulation technique that allows to simulate any probabilistic automaton with one having only one probabilistic transition

Disjunctive Probabilistic Modal Logic is Enough for Bisimilarity on Reactive Probabilistic Systems

Larsen and Skou characterized probabilistic bisimilarity over reactive probabilistic systems with a logic including true, negation, conjunction, and a diamond modality decorated with a probabilistic lower bound. Later on, Desharnais, Edalat, and Panangaden showed that negation is not necessary to characterize the same equivalence. In this paper, we prove that the logical characterization holds also when conjunction is replaced by disjunction, with negation still being not necessary. To this end, we introduce reactive probabilistic trees, a fully abstract model for reactive probabilistic systems that allows us to demonstrate expressiveness of the disjunctive probabilistic modal logic, as well as of the previously mentioned logics, by means of a compactness argument.Comment: Aligned content with version accepted at ICTCS 2016: fixed minor typos, added reference, improved definitions in Section 3. Still 10 pages in sigplanconf forma