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This paper introduces Farkas certificates for lower and upper bounds on minimal and maximal reachability probabilities in Markov decision processes (MDP), which we derive using an MDP-variant of Farkas' Lemma. The set of all such certificates is shown to form a polytope whose points correspond to witnessing subsystems of the model and the property. This allows translating the problem of finding minimal witnesses to the problem of finding vertices with a maximal number of zeros. While computing such vertices is computationally hard in general, we derive new heuristics from our formulations that exhibit competitive performance compared to state-of-the-art techniques and apply to more situations. As an argument that asymptotically better algorithms cannot be hoped for, we show that the decision version of finding minimal witnesses is NP-complete even for acyclic Markov chains.

We investigate Friedl-Lück's universal L 2 -torsion for descending HNN extensions of finitely generated free groups, and so in particular for Fnby-Z groups. This invariant induces a semi-norm on the first cohomology of the group which is an analogue of the Thurston norm for 3-manifold groups.We prove that this Thurston semi-norm is an upper bound for the Alexander semi-norm defined by McMullen, as well as for the higher Alexander seminorms defined by Harvey. The same inequalities are known to hold for 3manifold groups.We also prove that the Newton polytopes of the universal L 2 -torsion of a descending HNN extension of F 2 locally determine the Bieri-Neumann-Strebel invariant of the group. We give an explicit means of computing the BNS invariant for such groups. As a corollary, we prove that the Bieri-Neumann-Strebel invariant of a descending HNN extension of F 2 has finitely many connected components.When the HNN extension is taken over Fn along a polynomially growing automorphism with unipotent image in GL(n, Z), we show that the Newton polytope of the universal L 2 -torsion and the BNS invariant completely determine one another. We also show that in this case the Alexander norm, its higher incarnations, and the Thurston norm all coincide.

This work aims at reducing the main-memory footprint in high performance hybrid OLTP & OLAP databases, while retaining high query performance and transactional throughput. For this purpose, an innovative compressed columnar storage format for cold data, called Data Block s is introduced. Data Blocks further incorporate a new lightweight index structure called Positional SMA that narrows scan ranges within Data Blocks even if the entire block cannot be ruled out. To achieve highest OLTP performance, the compression schemes of Data Blocks are very lightweight , such that OLTP transactions can still quickly access individual tuples. This sets our storage scheme apart from those used in specialized analytical databases where data must usually be bit-unpacked. Up to now, high-performance analytical systems use either vectorized query execution or "just-in-time" (JIT) query compilation. The fine-grained adaptivity of Data Blocks necessitates the integration of the best features of each approach by an interpreted vectorized scan subsystem feeding into JIT-compiled query pipelines. Experimental evaluation of HyPer, our full-fledged hybrid OLTP & OLAP database system, shows that Data Blocks accelerate performance on a variety of query workloads while retaining high transaction throughput.