In particular, we introduce the notion of a hierarchical route distribution and show how they can be leveraged to construct tractable PSDDs over route distributions, allowing them to scale to larger maps.

We investigate the problem of computing the top-$k$ simple shortest paths in weighted digraphs. While the single-pair variant -- finding the top-$k$ simple shortest paths between two specified vertices -- has been extensively studied over the past decades, with Yen's algorithm and its heuristic improvements emerging as the most effective solving strategies, relatively little attention has been devoted to the more general single-source version, where the goal is determining top-$k$ simple shortest paths from a source vertex to all other vertices. Motivated by the numerous practical applications of ranked shortest paths, in this paper we provide new insights and algorithmic contributions to this problem. In particular, we first present a theoretical characterization of the structural properties of its solutions. Then, we introduce the first polynomial-time algorithm specifically designed to handle it. On the one hand, we prove our new algorithm is on par, in terms of time complexity, with the best (and only) polynomial-time approach known in the literature to solve the problem, that is applying the fastest single-pair algorithm independently to each vertex pair formed by the source and the remaining vertices. On the other hand, through an extensive experimental evaluation on both real-world and synthetic graphs, we demonstrate that our algorithm consistently and significantly outperforms the latter baseline in terms of running time, achieving speed-ups of up to several orders of magnitude. These results establish our new algorithm as the solution to be preferred for computing $k$ simple shortest paths from a single source in practical settings.

Here we proved that (1) and (2) are equivalent; (1) and (3) are equivalent. Proposition 3. 14 Lemma 2. Given With the lemma above, we now present the proof of Proposition 3. B.1 Example Implementation We provide an example implementation of Algorithm 2 in Listing 1. 17 1 Based on exponentiation by squaring. Best results are in bold. Based on the observation, Wei et al. Our method identified a different source of gradient vanishing caused by the small coefficients for higher-order terms in DAG constraints.

In this paper, we propose a model for building natural language explanations for Bayesian Network Reasoning in terms of factor arguments, which are argumentation graphs of flowing evidence, relating the observed evidence to a target variable we want to learn about. We introduce the notion of factor argument independence to address the outstanding question of defining when arguments should be presented jointly or separately and present an algorithm that, starting from the evidence nodes and a target node, produces a list of all independent factor arguments ordered by their strength. Finally, we implemented a scheme to build natural language explanations of Bayesian Reasoning using this approach. Our proposal has been validated in the medical domain through a human-driven evaluation study where we compare the Bayesian Network Reasoning explanations obtained using factor arguments with an alternative explanation method. Evaluation results indicate that our proposed explanation approach is deemed by users as significantly more useful for understanding Bayesian Network Reasoning than another existing explanation method it is compared to.

Recently, the Probabilistic Sentential Decision Diagram (PSDD) has been proposed as a framework for systematically inducing and learning distributions over structured objects, including combinatorial objects such as permutations and rankings, paths and matchings on a graph, etc. In this paper, we study the scalability of such models in the context of representing and learning distributions over routes on a map. In particular, we introduce the notion of a hierarchical route distribution and show how they can be leveraged to construct tractable PSDDs over route distributions, allowing them to scale to larger maps. We illustrate the utility of our model empirically, in a route prediction task, showing how accuracy can be increased significantly compared to Markov models.

The rapid progress of AI-powered programming assistants, such as GitHub Copilot, has facilitated the development of software applications. These assistants rely on large language models (LLMs), which are foundation models (FMs) that support a wide range of tasks related to understanding and generating language. LLMs have demonstrated their ability to express UML model specifications using formal languages like the Object Constraint Language (OCL). However, the context size of the prompt is limited by the number of tokens an LLM can process. This limitation becomes significant as the size of UML class models increases. In this study, we introduce PathOCL, a novel path-based prompt augmentation technique designed to facilitate OCL generation. PathOCL addresses the limitations of LLMs, specifically their token processing limit and the challenges posed by large UML class models. PathOCL is based on the concept of chunking, which selectively augments the prompts with a subset of UML classes relevant to the English specification. Our findings demonstrate that PathOCL, compared to augmenting the complete UML class model (UML-Augmentation), generates a higher number of valid and correct OCL constraints using the GPT-4 model. Moreover, the average prompt size crafted using PathOCL significantly decreases when scaling the size of the UML class models.

Recently, Their goal is to learn or search for optimal graph objects that heterogeneous graph neural networks (HGNs) have maximize mutual information with the predictions. While become one of the standard paradigms for modeling rich such explanations answer the question "what is salient to semantics of heterogeneous graphs in various application the prediction", they fail to unveil "how the salient objects domains such as e-commerce, finance, and healthcare (Lv affect the prediction". In particular, there may exist multiple et al. 2021; Wang et al. 2022). In parallel with the proliferation paths in the graph to propagate the information of the salient of HGNs, understanding the reasons behind the objects to the target object and affect its prediction. Without predictions from HGNs is urgently demanded in order to distinguishing these different influential paths, the answer to build trust and confidence in the models for both users and the "how" question remains unclear, which could compromise stakeholders. For example, a customer would be satisfied if the utility of the explanation. This issue becomes more an HGN-based recommender system accompanies recommended prominent when it comes to explaining HGNs due to the items with explanations; a bank manager may want complex semantics of heterogeneous graphs.

Graph neural networks (GNNs) have recently become the standard approach for learning with graph-structured data. Prior work has shed light into their potential, but also their limitations. Unfortunately, it was shown that standard GNNs are limited in their expressive power. These models are no more powerful than the 1-dimensional Weisfeiler-Leman (1-WL) algorithm in terms of distinguishing non-isomorphic graphs. In this paper, we propose Path Neural Networks (PathNNs), a model that updates node representations by aggregating paths emanating from nodes. We derive three different variants of the PathNN model that aggregate single shortest paths, all shortest paths and all simple paths of length up to K. We prove that two of these variants are strictly more powerful than the 1-WL algorithm, and we experimentally validate our theoretical results. We find that PathNNs can distinguish pairs of non-isomorphic graphs that are indistinguishable by 1-WL, while our most expressive PathNN variant can even distinguish between 3-WL indistinguishable graphs. The different PathNN variants are also evaluated on graph classification and graph regression datasets, where in most cases, they outperform the baseline methods.

Active Directory is the default security management system for Windows domain networks. We study the shortest path edge interdiction problem for defending Active Directory style attack graphs. The problem is formulated as a Stackelberg game between one defender and one attacker. The attack graph contains one destination node and multiple entry nodes. The attacker's entry node is chosen by nature. The defender chooses to block a set of edges limited by his budget. The attacker then picks the shortest unblocked attack path. The defender aims to maximize the expected shortest path length for the attacker, where the expectation is taken over entry nodes. We observe that practical Active Directory attack graphs have small maximum attack path lengths and are structurally close to trees. We first show that even if the maximum attack path length is a constant, the problem is still $W[1]$-hard with respect to the defender's budget. Having a small maximum attack path length and a small budget is not enough to design fixed-parameter algorithms. If we further assume that the number of entry nodes is small, then we derive a fixed-parameter tractable algorithm. We then propose two other fixed-parameter algorithms by exploiting the tree-like features. One is based on tree decomposition and requires a small tree width. The other assumes a small number of splitting nodes (nodes with multiple out-going edges). Finally, the last algorithm is converted into a graph convolutional neural network based heuristic, which scales to larger graphs with more splitting nodes.