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Applying Graph Theory to Problems in Air Traffic Management

Graph theory is used to investigate three different problems arising in air traffic management. First, using a polynomial reduction from a graph partitioning problem, it is shown that both the airspace sectorization problem and its incremental counterpart, the sector combination problem are NP-hard, in general, under several simple workload models. Second, using a polynomial time reduction from maximum independent set in graphs, it is shown that for any fixed e, the problem of finding a solution to the minimum delay scheduling problem in traffic flow management that is guaranteed to be within n1-e of the optimal, where n is the number of aircraft in the problem instance, is NP-hard. Finally, a problem arising in precision arrival scheduling is formulated and solved using graph reachability. These results demonstrate that graph theory provides a powerful framework for modeling, reasoning about, and devising algorithmic solutions to diverse problems arising in air traffic management.

precision arrival scheduling

Applying Graph Theory to Problems in Air Traffic Management

Graph theory is used to investigate three different problems arising in air traffic management. First, using a polynomial reduction from a graph partitioning problem, it isshown that both the airspace sectorization problem and its incremental counterpart, the sector combination problem are NP-hard, in general, under several simple workload models. Second, using a polynomial time reduction from maximum independent set in graphs, it is shown that for any fixed e, the problem of finding a solution to the minimum delay scheduling problem in traffic flow management that is guaranteed to be within n1-e of the optimal, where n is the number of aircraft in the problem instance, is NP-hard. Finally, a problem arising in precision arrival scheduling is formulated and solved using graph reachability. These results demonstrate that graph theory provides a powerful framework for modeling, reasoning about, and devising algorithmic solutions to diverse problems arising in air traffic management.

computational complexity

A PVS Graph Theory Library

This paper documents the NASA Langley PVS graph theory library. The library provides fundamental definitions for graphs, subgraphs, walks, paths, subgraphs generated by walks, trees, cycles, degree, separating sets, and four notions of connectedness. Theorems provided include Ramsey's and Menger's and the equivalence of all four notions of connectedness.

Butler, Ricky W.

Graph theory approach to the eigenvalue problem of large space structures

Graph theory is used to obtain numerical solutions to eigenvalue problems of large space structures (LSS) characterized by a state vector of large dimensions. The LSS are considered as large, flexible systems requiring both orientation and surface shape control. Graphic interpretation of the determinant of a matrix is employed to reduce a higher dimensional matrix into combinations of smaller dimensional sub-matrices. The reduction is implemented by means of a Boolean equivalent of the original matrices formulated to obtain smaller dimensional equivalents of the original numerical matrix. Computation time becomes less and more accurate solutions are possible. An example is provided in the form of a free-free square plate. Linearized system equations and numerical values of a stiffness matrix are presented, featuring a state vector with 16 components.

Reddy, A. S. S. R.

Graph Theory Roots of Spatial Operators for Kinematics and Dynamics

Spatial operators have been used to analyze the dynamics of robotic multibody systems and to develop novel computational dynamics algorithms. Mass matrix factorization, inversion, diagonalization, and linearization are among several new insights obtained using such operators. While initially developed for serial rigid body manipulators, the spatial operators and the related mathematical analysis have been shown to extend very broadly including to tree and closed topology systems, to systems with flexible joints, links, etc. This work uses concepts from graph theory to explore the mathematical foundations of spatial operators. The goal is to study and characterize the properties of the spatial operators at an abstract level so that they can be applied to a broader range of dynamics problems. The rich mathematical properties of the kinematics and dynamics of robotic multibody systems has been an area of strong research interest for several decades. These properties are important to understand the inherent physical behavior of systems, for stability and control analysis, for the development of computational algorithms, and for model development of faithful models. Recurring patterns in spatial operators leads one to ask the more abstract question about the properties and characteristics of spatial operators that make them so broadly applicable. The idea is to step back from the specific application systems, and understand more deeply the generic requirements and properties of spatial operators, so that the insights and techniques are readily available across different kinematics and dynamics problems. In this work, techniques from graph theory were used to explore the abstract basis for the spatial operators. The close relationship between the mathematical properties of adjacency matrices for graphs and those of spatial operators and their kernels were established. The connections hold across very basic requirements on the system topology, the nature of the component bodies, the indexing schemes, etc. The relationship of the underlying structure is intimately connected with efficient, recursive computational algorithms. The results provide the foundational groundwork for a much broader look at the key problems in kinematics and dynamics. The properties of general graphs and trees of nodes and edge were examined, as well as the properties of adjacency matrices that are used to describe graph connectivity. The nilpotency property of such matrices for directed trees was reviewed, and the adjacency matrices were generalized to the notion of block weighted adjacency matrices that support block matrix elements. This leads us to the development of the notion of Spatial Kernel Operator SKO kernels. These kernels provide the basis for the development of SKO resolvent operators.

Jain, Abhinandan

Microstructure Validation of Graph Theory Model-Derived Cooling Rates in the Wire Arc Additive Manufacturing of ER70S-6 Steel

Wire arc additive manufacturing (WAAM) enables high-rate fabrication of large metallic components, but spatial variations in thermal history can lead to microstructural heterogeneity that requires efficient process models to evaluate. This study evaluates whether cooling rates extracted from a graph theory model (GTM)-based thermal simulation are consistent with the microstructural evolution observed in an ER70S-6 WAAM wall. Thermal histories from the model were analyzed at selected build heights, and cooling rates were extracted from the final thermal excursion through the austenite phase field. Microstructures at corresponding locations were characterized using electron backscatter diffraction (EBSD) to quantify grain size distributions, and pearlite interlamellar spacing was used as an additional indicator of cooling behavior. The modeled cooling rates were highest near the substrate and generally decreased with build height, consistent with the observed reduction in the fine grain fraction and the progressive shift in the grain size distribution as build height increased. Pearlite spacing trends also supported the modeled cooling rate variation. These results indicate that GTM-derived thermal histories can be post-processed into metallurgically meaningful cooling rate estimates for WAAM steel builds and linked to dataset specific empirical grain size distribution relationships for process–thermal history–microstructure assessment.

36 MATERIALS SCIENCE

Navigating Large Chemical Spaces Using Graph Theory and Integer Programming

Navigating and analyzing large chemical spaces are necessary to accelerate the design and discovery of new molecules and chemical processes. In this work, we introduce a computational framework that integrates graph theory and integer programming to enable the efficient navigation of large chemical spaces. Our framework represents the chemical space as a graph, wherein nodes represent molecules and edges represent the degree of similarity or connectivity based on domain-specific information. Using the graph representation, we identify representative molecules by computing the so-called minimum dominating set (MDS), which in our context is the minimum set of molecules that is connected to all other molecules. We present a suite of solution strategies for the MDS problem including heuristic and rigorous integer programming (IP) approaches. We show that these approaches allow us to capture physicochemical properties and domain-specific logic and constraints, facilitating the identification of molecules with the target properties. We demonstrate the effectiveness of the proposed approach by navigating the chemical space of per- and polyfluoroalkyl substances (PFAS); this comprises approximately 15,000 molecular structures. We compare our framework against traditional dimensionality reduction and clustering methods such as t-SNE and K-means clustering.

Chemical structure

Augmenting Conceptual Design Trajectory Tradespace Exploration with Graph Theory

Within conceptual design changes occur rapidly due to a combination of uncertainty and shifting requirements. To stay relevant in this fluid time, trade studies must also be performed rapidly. In order to drive down analysis time while improving the information gained by these studies, surrogate models can be created to represent the complex output of a tool or tools within a specified tradespace. In order to create this model however, a large amount of data must be collected in a short amount of time. By this method, the historical approach of relying on subject matter experts to generate the data required is schedule infeasible. However, by implementing automation and distributed analysis the required data can be generated in a fraction of the time. Previous work focused on setting up a tool called multiPOST capable of orchestrating many simultaneous runs of an analysis tool assessing these automated analyses utilizing heuristics gleaned from the best practices of current subject matter experts. In this update to the previous work, elements of graph theory are included to further drive down analysis time by leveraging data previously gathered. It is shown to outperform the previous method in both time required, and the quantity and quality of data produced.

Dees, Patrick D.

Augmenting Conceptual Design Trajectory Tradespace Exploration with Graph Theory

Within conceptual design changes occur rapidly due to a combination of uncertainty and shifting requirements. To stay relevant in this fluid time, trade studies must also be performed rapidly. In order to drive down analysis time while improving the information gained by these studies, surrogate models can be created to represent the complex output of a tool or tools within a specified tradespace. In order to create this model however, a large amount of data must be collected in a short amount of time. By this method, the historical approach of relying on subject matter experts to generate the data required is schedule infeasible. However, by implementing automation and distributed analysis the required data can be generated in a fraction of the time. Previous work focused on setting up a tool called multiPOST capable of orchestrating many simultaneous runs of an analysis tool assessing these automated analyses utilizing heuristics gleaned from the best practices of current subject matter experts. In this update to the previous work, elements of graph theory are included to further drive down analysis time by leveraging data previously gathered. It is shown to outperform the previous method in both time required, and the quantity and quality of data produced.

Dees, Patrick D.

Graph theory inspired anomaly detection at the LHC

Designing model-independent anomaly detection algorithms for analyzing LHC data remains a central challenge in the search for new physics, due to the high dimensionality of collider events. In this work, we develop a graph autoencoder as an unsupervised, model-agnostic tool for anomaly detection, using the LHC Olympics dataset as a benchmark. By representing jet constituents as a graph, we introduce a method to systematically control the information available to the model through sparse graph constructions that serve as physically motivated inductive biases. Specifically, (1) we construct graph autoencoders based on locally rigid Laman graphs and globally rigid unique graphs, and (2) we explore the clustering of jet constituents into subjets to interpolate between high- and low-level input representations. We obtain the best performance, measured in terms of the Significance Improvement Characteristic curve for an intermediate level of subjet clustering and certain sparse unique graph constructions. We further investigate the role of graph connectivity in jet classification tasks. Our results demonstrate the potential of leveraging graph-theoretic insights to refine and increase the interpretability of machine learning tools for collider experiments.

Automation

Augmenting Parametric Optimal Ascent Trajectory Modeling with Graph Theory

It has been well documented that decisions made in the early stages of Conceptual and Pre-Conceptual design commit up to 80% of total Life-Cycle Cost (LCC) while engineers know the least about the product they are designing [1]. Once within Preliminary and Detailed design however, making changes to the design becomes far more difficult to enact in both cost and schedule. Primarily this has been due to a lack of detailed data usually uncovered later during the Preliminary and Detailed design phases. In our current budget-constrained environment, making decisions within Conceptual and Pre-Conceptual design which minimize LCC while meeting requirements is paramount to a program's success. Within the arena of launch vehicle design, optimizing the ascent trajectory is critical for minimizing the costs present within such concerns as propellant, aerodynamic, aeroheating, and acceleration loads while meeting requirements such as payload delivered to a desired orbit. In order to optimize the vehicle design its constraints and requirements must be known, however as the design cycle proceeds it is all but inevitable that the conditions will change. Upon that change, the previously optimized trajectory may no longer be optimal, or meet design requirements. The current paradigm for adjusting to these updates is generating point solutions for every change in the design's requirements [2]. This can be a tedious, time-consuming task as changes in virtually any piece of a launch vehicle's design can have a disproportionately large effect on the ascent trajectory, as the solution space of the trajectory optimization problem is both non-linear and multimodal [3]. In addition, an industry standard tool, Program to Optimize Simulated Trajectories (POST), requires an expert analyst to produce simulated trajectories that are feasible and optimal [4]. In a previous publication the authors presented a method for combatting these challenges [5]. In order to bring more detailed information into Conceptual and Pre-Conceptual design, knowledge of the effects originating from changes to the vehicle must be calculated. In order to do this, a model capable of quantitatively describing any vehicle within the entire design space under consideration must be constructed. This model must be based upon analysis of acceptable fidelity, which in this work comes from POST. Design space interrogation can be achieved with surrogate modeling, a parametric, polynomial equation representing a tool. A surrogate model must be informed by data from the tool with enough points to represent the solution space for the chosen number of variables with an acceptable level of error. Therefore, Design Of Experiments (DOE) is used to select points within the design space to maximize information gained on the design space while minimizing number of data points required. To represent a design space with a non-trivial number of variable parameters the number of points required still represent an amount of work which would take an inordinate amount of time via the current paradigm of manual analysis, and so an automated method was developed. The best practices of expert trajectory analysts working within NASA Marshall's Advanced Concepts Office (ACO) were implemented within a tool called multiPOST. These practices include how to use the output data from a previous run of POST to inform the next, determining whether a trajectory solution is feasible from a real-world perspective, and how to handle program execution errors. The tool was then augmented with multiprocessing capability to enable analysis on multiple trajectories simultaneously, allowing throughput to scale with available computational resources. In this update to the previous work the authors discuss issues with the method and solutions.

Patrick D Dees