arXiv subjects

Ernst Althaus

Publications and source records attributed to Ernst Althaus.

7 recordsLinked to original sources

arXiv2026

A Parameterized Algorithm for Testing whether the Limit of a Diagram is Empty

A limit of a (small) diagram $d : J \to E$ in a complete category $E$ can be thought of as specifying a set of equations involving the objects of $E$. To motivate this intuitively, one can think of each object $d(j)$ as a "variable" and each morphism in $J$ as a "constraint" connecting these variables. If $E$ has an initial object, a natural question arises: does our set of equations have any solution at all? Equivalently, we can ask: is the limit of $d$ initial? In this paper we consider the computational problem that, given finite diagram $d$ in a finitely complete category $E$, asks whether its limit is empty. We construct a fast algorithm (in the sense of parameterized complexity theory) that solves this problem when $E$ is of the form $\mathbf{FinSet}^{J}$ for a finite category $J$ and $d$ is a structured co-decomposition, i.e. a diagram arising from the opposite of the Grothendieck construction of a simple graph.

math.CT↗

arXiv2025

N-Parties Private Structure and Parameter Learning for Sum-Product Networks

A sum-product network (SPN) is a graphical model that allows several types of probabilistic inference to be performed efficiently. In this paper, we propose a privacy-preserving protocol which tackles structure generation and parameter learning of SPNs. Additionally, we provide a protocol for private inference on SPNs, subsequent to training. To preserve the privacy of the participants, we derive our protocol based on secret sharing, which guarantees privacy in the honest-but-curious setting even when at most half of the parties cooperate to disclose the data. The protocol makes use of a forest of randomly generated SPNs, which is trained and weighted privately and can then be used for private inference on data points. Our experiments indicate that preserving the privacy of all participants does not decrease log-likelihood performance on both homogeneously and heterogeneously partitioned data. We furthermore show that our protocol's performance is comparable to current state-of-the-art SPN learners in homogeneously partitioned data settings. In terms of runtime and memory usage, we demonstrate that our implementation scales well when increasing the number of parties, comparing favorably to protocols for neural networks, when they are trained to reproduce the input-output behavior of SPNs.

cs.CR↗

arXiv2025

A Random Walk Approach to Broadcasting on Random Recursive Trees

In the broadcasting problem on trees, a $\{-1,1\}$-message originating in an unknown node is passed along the tree with a certain error probability $q$. The goal is to estimate the original message without knowing the order in which the nodes were informed. We show a connection to random walks with memory effects and use this to develop a novel approach to analyse the majority estimator on random recursive trees. With this powerful approach, we study the entire group of very simple increasing trees as well as shape exchangeable trees together. This also extends Addario-Berry et al. (2022) who investigated this estimator for uniform and linear preferential attachment random recursive trees.

math.PR↗

arXiv2023

Compositional Algorithms on Compositional Data: Deciding Sheaves on Presheaves

Algorithmicists are well-aware that fast dynamic programming algorithms are very often the correct choice when computing on compositional (or even recursive) graphs. Here we initiate the study of how to generalize this folklore intuition to mathematical structures writ large. We achieve this horizontal generality by adopting a categorial perspective which allows us to show that: (1) structured decompositions (a recent, abstract generalization of many graph decompositions) define Grothendieck topologies on categories of data (adhesive categories) and that (2) any computational problem which can be represented as a sheaf with respect to these topologies can be decided in linear time on classes of inputs which admit decompositions of bounded width and whose decomposition shapes have bounded feedback vertex number. This immediately leads to algorithms on objects of any C-set category; these include -- to name but a few examples -- structures such as: symmetric graphs, directed graphs, directed multigraphs, hypergraphs, directed hypergraphs, databases, simplicial complexes, circular port graphs and half-edge graphs. Thus we initiate the bridging of tools from sheaf theory, structural graph theory and parameterized complexity theory; we believe this to be a very fruitful approach for a general, algebraic theory of dynamic programming algorithms. Finally we pair our theoretical results with concrete implementations of our main algorithmic contribution in the AlgebraicJulia ecosystem.

cs.CC↗

arXiv2021

Fast Private Parameter Learning and Inference for Sum-Product Networks

A sum-product network (SPN) is a graphical model that allows several types of inferences to be drawn efficiently. There are two types of learning for SPNs: Learning the architecture of the model, and learning the parameters. In this paper, we tackle the second problem: We show how to learn the weights for the sum nodes, assuming the architecture is fixed, and the data is horizontally partitioned between multiple parties. The computations will preserve the privacy of each participant. Furthermore, we will use secret sharing instead of (homomorphic) encryption, which allows fast computations and requires little computational resources. To this end, we use a novel integer division to compute approximate real divisions. We also show how simple and private inferences can be performed using the learned SPN.

cs.LG↗

arXiv2020

Optimal Tree Decompositions Revisited: A Simpler Linear-Time FPT Algorithm

In 1996, Bodlaender showed the celebrated result that an optimal tree decomposition of a graph of bounded treewidth can be found in linear time. The algorithm is based on an algorithm of Bodlaender and Kloks that computes an optimal tree decomposition given a non-optimal tree decomposition of bounded width. Both algorithms, in particular the second, are hardly accessible. In our review, we present them in a much simpler way than the original presentations. In our description of the second algorithm, we start by explaining how all tree decompositions of subtrees defined by the nodes of the given tree decomposition can be enumerated. We group tree decompositions into equivalence classes depending on the current node of the given tree decomposition, such that it suffices to enumerate one tree decomposition per equivalence class and, for each node of the given tree decomposition, there are only a constant number of classes which can be represented in constant space. Our description of the first algorithm further simplifies Perkovic and Reed's simplification.

cs.DS↗

arXiv2010

Fully Automatic Trunk Packing with Free Placements

We present a new algorithm to compute the volume of a trunk according to the SAE J1100 standard. Our new algorithm uses state-of-the-art methods from computational geometry and from combinatorial optimization. It finds better solutions than previous approaches for small trunks.

cs.DS↗