For example, theleader aims tooptimize some traffic-networkparameters (e.g., road width values) so that players can spend less traveling time at equilibrium.

We address Stackelberg models of combinatorial congestion games (CCGs); we aim to optimize the parameters of CCGs so that the selfish behavior of non-atomic players attains desirable equilibria. This model is essential for designing such social infrastructures as traffic and communication networks. Nevertheless, computational approaches to the model have not been thoroughly studied due to two difficulties: (I) bilevel-programming structures and (II) the combinatorial nature of CCGs. We tackle them by carefully combining (I) the idea of \textit{differentiable} optimization and (II) data structures called \textit{zero-suppressed binary decision diagrams} (ZDDs), which can compactly represent sets of combinatorial strategies. Our algorithm numerically approximates the equilibria of CCGs, which we can differentiate with respect to parameters of CCGs by automatic differentiation. With the resulting derivatives, we can apply gradient-based methods to Stackelberg models of CCGs. Our method is tailored to induce Nesterov's acceleration and can fully utilize the empirical compactness of ZDDs. These technical advantages enable us to deal with CCGs with a vast number of combinatorial strategies. Experiments on real-world network design instances demonstrate the practicality of our method.

It has been long argued that, because of inherent ambiguity and noise, the brain needs to represent uncertainty in the form of probability distributions. The neural encoding of such distributions remains however highly controversial. Here we present a novel circuit model for representing multidimensional real-valued distributions using a spike based spatio-temporal code. Our model combines the computational advantages of the currently competing models for probabilistic codes and exhibits realistic neural responses along a variety of classic measures. Furthermore, the model highlights the challenges associated with interpreting neural activity in relation to behavioral uncertainty and points to alternative population-level approaches for the experimental validation of distributed representations. Core brain computations, such as sensory perception, have been successfully characterized as probabilistic inference, whereby sensory stimuli are interpreted in terms of the objects or features that gave rise to them [1, 2].

This model is essential for designing such social infrastructures as traffic and communication networks.

This model is essential for designing such social infrastructures as traffic and communication networks.

It has been long argued that, because of inherent ambiguity and noise, the brain needs to represent uncertainty in the form of probability distributions. The neural encoding of such distributions remains however highly controversial. Here we present a novel circuit model for representing multidimensional real-valued distributions using a spike based spatio-temporal code. Our model combines the computational advantages of the currently competing models for probabilistic codes and exhibits realistic neural responses along a variety of classic measures. Furthermore, the model highlights the challenges associated with interpreting neural activity in relation to behavioral uncertainty and points to alternative population-level approaches for the experimental validation of distributed representations. Core brain computations, such as sensory perception, have been successfully characterized as probabilistic inference, whereby sensory stimuli are interpreted in terms of the objects or features that gave rise to them [1, 2].

--Software requirements expressed in natural language (NL) frequently suffer from verbosity, ambiguity, and inconsistency. This creates a range of challenges, including selecting an appropriate architecture for a system and assessing different architectural alternatives. Relying on human expertise to accomplish the task of mapping NL requirements to architecture is time-consuming and error-prone. This paper proposes ARLO, an approach that automates this task by leveraging (1) a set of NL requirements for a system, (2) an existing standard that specifies architecturally relevant software quality attributes, and (3) a readily available Large Language Model (LLM). Specifically, ARLO determines the subset of NL requirements for a given system that is architecturally relevant and maps that subset to a tailorable matrix of architectural choices. ARLO applies integer linear programming on the architectural-choice matrix to determine the optimal architecture for the current requirements. We demonstrate ARLO's efficacy using a set of real-world examples. We highlight ARLO's ability (1) to trace the selected architectural choices to the requirements and (2) to isolate NL requirements that exert a particular influence on a system's architecture. This allows the identification, comparative assessment, and exploration of alternative architectural choices based on the requirements and constraints expressed therein. I NTRODUCTION Software systems are ever-evolving, with increases in size and complexity that call for careful elicitation of requirements and architectural choices [1]. While it has been long recognized that requirements and architecture co-evolve [2], [3], understanding their interactions, especially early in the software development process, is still an open challenge. More specifically, there is a scarcity of knowledge regarding their alignment, architecture-to-requirements traceability, and conserving architectural knowledge [4]. Most software requirements are still captured using natural language (NL) [5]-[7]. The informal nature of NLs is a significant obstacle in machine processing of such requirements [8]. In the past, researchers have proposed approaches to classify the NL requirements into different categories of functional and non-functional requirements [5], [6], to identify quality attributes from NL requirements, to facilitate architectural decisions by leveraging machine learning [7], and so on. While promising, these approaches require large, curated datasets and high model-building effort [9]. Translating NL requirements to architecture-related design decisions is a cumbersome task that has long been recognized in the research community [1].

It has been long argued that, because of inherent ambiguity and noise, the brain needs to represent uncertainty in the form of probability distributions. The neural encoding of such distributions remains however highly controversial. Here we present a novel circuit model for representing multidimensional real-valued distributions using a spike based spatio-temporal code. Our model combines the computational advantages of the currently competing models for probabilistic codes and exhibits realistic neural responses along a variety of classic measures. Furthermore, the model highlights the challenges associated with interpreting neural activity in relation to behavioral uncertainty and points to alternative populationlevel approaches for the experimental validation of distributed representations. Core brain computations, such as sensory perception, have been successfully characterized as probabilistic inference, whereby sensory stimuli are interpreted in terms of the objects or features that gave rise to them [1, 2].

We address Stackelberg models of combinatorial congestion games (CCGs); we aim to optimize the parameters of CCGs so that the selfish behavior of non-atomic players attains desirable equilibria. This model is essential for designing such social infrastructures as traffic and communication networks. Nevertheless, computational approaches to the model have not been thoroughly studied due to two difficulties: (I) bilevel-programming structures and (II) the combinatorial nature of CCGs. We tackle them by carefully combining (I) the idea of \textit{differentiable} optimization and (II) data structures called \textit{zero-suppressed binary decision diagrams} (ZDDs), which can compactly represent sets of combinatorial strategies. Our algorithm numerically approximates the equilibria of CCGs, which we can differentiate with respect to parameters of CCGs by automatic differentiation.