Layered Graph Research Articles

Time-Efficient RSA over Large-Scale Multi-Domain EON.

The poor timeliness of routing has always been an urgent problem in practical operator networks, especially in situations with large-scale networks and multiple network domains. In this article, a pruning idea of routing integrated with Dijkstra's shortest path searching is utilized to accelerate the process of routing in large-scale multi-domain elastic optical networks (EONs). The layered-graph approach is adopted in the spectrum allocation stage. To this end, an efficient heuristic algorithm is proposed, called "Branch-and-Bound based Routing and Layered Graph based Spectrum Allocation algorithm (BBR-LGSA)", which is an integrated RSA algorithm. Notably, the significant reduction in algorithm time complexity is not only reflected in the pruning method used in the routing stage but also in the construction of auxiliary graphs during the spectrum allocation stage utilizing the Branch-and-Bound method. Simulation results show that the proposed BBR-LGSA significantly reduces the average running time by nearly 78% with higher spectrum utilization in large-scale multi-domain EONs, compared with benchmark algorithms. In addition, the impact of key parameters on performance comparisons of different algorithms is evaluated.

Layered Graph Models for the Electric Vehicle Routing Problem With Nonlinear Charging Functions

AbstractElectric vehicle routing problems (EVRPs) involve the routing of a fleet of electric vehicles (EVs) to visit a set of customers while typically minimizing the total travel and charging time. Due to their limited autonomy, EVs may need to recharge their batteries en‐route at charging stations (CSs). Thus, routing decisions also include which CSs to visit, and how much energy to charge during those visits. These decisions are compounded by the fact that charging times follow a nonlinear charging function with respect to the EV's state of charge (SoC). We propose a layered graph representation for the EVRP with nonlinear charging functions (EVRP‐NL). Specifically, the layers correspond to discretized SoC values. Therefore, the arcs' energy consumption is approximated to match those values. We develop two compact formulations based on the layered graph. Furthermore, we introduced two charging policies that facilitate aligning charging duration with practical considerations. Computational results demonstrate the effectiveness of our formulations. Our best formulation effectively handles instances with up to 40 customers. On those instances, compared to the state‐of‐the‐art compact formulation, our formulation solves 13 more instances to optimality with less than half of the computational time. Considering instances solved by both formulations to optimally, the approximation entailed by our formulation yields a 0.94% deviation on average. Since our best performing formulation is compact, it may be readily used by a broad audience. Furthermore, as the majority of algorithms for the EVPR and its variants are heuristics, our formulation could be beneficial in evaluating the performance of these methods.

A Graph-Refinement Algorithm to Minimize Squared Delivery Delays Using Parcel Robots

In recent years, parcel volumes have reached record highs, prompting the logistics industry to explore innovative solutions to meet growing demand. In densely populated areas, delivery robots offer a promising alternative to traditional truck-based delivery systems. These autonomous electric robots operate on sidewalks and deliver time-sensitive goods, such as express parcels, medicine and meals. However, their limited cargo capacity and battery life require a return to a depot after each delivery. This challenge can be modeled as an electric vehicle-routing problem with soft time windows and single-unit capacity constraints. The objective is to serve all customers while minimizing the quadratic sum of delivery delays and ensuring each vehicle operates within its battery limitations. To address this problem, we propose a mixed-integer quadratic programming model and introduce an enhanced formulation using a layered graph structure. For this layered graph, we present two solution approaches based on relaxations that reduce the number of nodes and arcs compared to the expanded formulation. The first approach, Iterative Refinement, solves the current relaxation to optimality and refines the graph when the solution is infeasible for the expanded formulation. This process continues until a proven optimal solution is obtained. The second approach, Branch and Refine, integrates graph refinement into a branch-and-bound framework, eliminating the need for restarts. Computational experiments on modified Solomon instances demonstrate the effectiveness of our solution approaches, with Branch and Refine consistently outperforming Iterative Refinement across all tested parameter configurations.

LPulse: An efficient algorithm for service function chain placement and routing with delay guarantee

Modern network services increasingly depend on the effective orchestration of the Service Function Chain (SFC) with stringent end-to-end delay guarantees. To achieve this, the Delay-Constrained Service Function Chain Placement and Routing (DC-SFCPR) problem must be addressed. This problem involves the optimal selection of nodes for placing network functions and routes that adhere to a specific sequence of service functions, to minimize network bandwidth and CPU costs while strictly adhering to stringent end-to-end delay constraints. The DC-SFCPR problem is NP-hard and existing algorithms either fail to guarantee strict delay constraints or are computationally expensive, making them unsuitable for expanding network topologies. We propose the LPulse algorithm, designed to efficiently solve the DC-SFCPR problem. This algorithm utilizes a layered graph to embed the requirements of service functions, transforming the DC-SFCPR problem into a Delay-Constrained Shortest Path (DCSP) problem. The LPulse algorithm then applies Pulse, a depth-first search framework enhanced with efficient pruning strategies, and incorporates two novel acceleration strategies to solve the DCSP problem. We prove that LPulse ensures the optimality of solutions. Evaluations conducted across various topologies, with node scales ranging from 22 to 10,000, show that LPulse surpasses existing algorithms in both solution quality and speed. For instance, the number of cases meeting strict delay constraints with LPulse is 1.9× that of those solved by deep reinforcement learning algorithms; furthermore, its solving efficiency is 4.9× that of the highest-performing existing optimal algorithm, the LagrangianKsp algorithm.

On Graphs Embeddable in a Layer of a Hypercube and Their Extremal Numbers

AbstractA graph is cubical if it is a subgraph of a hypercube. For a cubical graph H and a hypercube $$Q_n$$ Q n , $$\textrm{ex}(Q_n, H)$$ ex ( Q n , H ) is the largest number of edges in an H-free subgraph of $$Q_n$$ Q n . If $$\textrm{ex}(Q_n, H)$$ ex ( Q n , H ) is at least a positive proportion of the number of edges in $$Q_n$$ Q n , then H is said to have positive Turán density in the hypercube; otherwise it has zero Turán density. Determining $$\textrm{ex}(Q_n, H)$$ ex ( Q n , H ) and even identifying whether H has positive or zero Turán density remains a widely open question for general H. In this paper we focus on layered graphs, i.e., graphs that are contained in an edge layer of some hypercube. Graphs H that are not layered have positive Turán density because one can form an H-free subgraph of $$Q_n$$ Q n consisting of edges of every other layer. For example, a 4-cycle is not layered and has positive Turán density. However, in general, it is not obvious what properties layered graphs have. We give a characterization of layered graphs in terms of edge-colorings. We show that most non-trivial subdivisions have zero Turán density, extending known results on zero Turán density of even cycles of length at least 12 and of length 8. However, we prove that there are cubical graphs of girth 8 that are not layered and thus having positive Turán density. The cycle of length 10 remains the only cycle for which it is not known whether its Turán density is positive or not. We prove that $$\textrm{ex}(Q_n, C_{10})= \Omega (n2^n/ \log ^a n)$$ ex ( Q n , C 10 ) = Ω ( n 2 n / log a n ) , for a constant a, showing that the extremal number for a 10-cycle behaves differently from any other cycle of zero Turán density.

Performance Analysis of Multi-Tote Storage and Retrieval Autonomous Mobile Robot Systems

Multi-tote storage and retrieval (MTSR) autonomous mobile robots can carry multiple product totes, store and retrieve them from different shelf rack tiers, and transport them to a workstation where the products are picked to fulfill customer orders. In each robot trip, totes retrieved during the previous trip must be stored. This leads to a mixed storage and retrieval route. We analyze this mixed storage and retrieval route problem and derive the optimal travel route for a multiblock warehouse by a layered graph algorithm, based on storage first-retrieval second and mixed storage and retrieval policies. We also propose an effective heuristic routing policy, the closest retrieval (CR) sequence policy, based on a local shortest path. Numerical results show that the CR policy leads to shorter travel times than the well-known S-shape policy, whereas the gap with the optimal mixed storage and retrieval policy in practical scenarios is small. Based on the CR policy, we model the stochastic behavior of the system using a semiopen queuing network (SOQN). This model can accurately estimate average tote throughput time and system throughput capacity as a function of the number of robots in the system. We use the SOQN and corresponding closed queuing network models to optimize the total annual cost as a function of the warehouse shape, the number of robots, and tote buffer positions on the robots for a given average tote throughput time and throughput capacity. Compared with robots that retrieve a single tote per trip, an MTSR system with at least five buffer positions can achieve lower operational costs while meeting given average tote throughput time and tote throughput capacity constraints. Funding: This work was supported by National Natural Science Foundation of China [Grant 72372088] and the Shenzhen Science and Technology Program [Grant GJHZ20220913143003006]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2023.0397 .

A hybrid metaheuristic to optimize electric first-mile feeder services with charging synchronization constraints and customer rejections

This paper addresses the on-demand meeting-point-based feeder electric bus routing and charging scheduling problem under charging synchronization constraints. The problem considered exhibits the structure of the location routing problem, which is more difficult to solve than many electric vehicle routing problems with capacitated charging stations. We propose to model the problem using a mixed-integer linear programming approach based on a layered graph structure. An efficient hybrid metaheuristic solution algorithm is proposed. A mixture of random and greedy partial charging scheduling strategies is used to find feasible charging schedules under the synchronization constraints. The algorithm is tested on instances with up to 100 customers and 49 bus stops/meeting points. The results show that the proposed algorithm provides near-optimal solutions within less one minute on average compared with the best solutions found by a mixed-integer linear programming solver set with a 4-hour computation time limit. A case study on a larger sized case with 1000 customers and 111 meeting points shows the proposed method is applicable to real-world situations.

МОДЕЛЬ ТА МЕТОД АНАЛІЗУ ПРАЦЕЗДАТНОСТІ ЕЛЕКТРОЕНЕРГЕТИЧНОЇ СИСТЕМИ ПРИ НАНЕСЕННІ НЕПРОЕКТНИХ ЗБУРЕНЬ

Abstract. Information and simulation models are important tools in the modern world for analyzing, designing, and optimizing various systems. One effective approach to their construction is the use of graph theory. In this article, we will examine the fundamental principles of constructing an information model using graphs and the method for analyzing the operability of this model under non-projected disturbances. In amidst increasing demands for reliability in the energy sector, the concept of developing an energy system in the form of a layered graph is becoming increasingly relevant and promising. A layered graph simulation model can provide an effective tool for optimizing energy supply and ensuring the stability and reliability of the system. Using a layered graph, it is possible to visualize the hierarchy of different energy sources, including renewable sources, traditional fuel types, and energy-efficient technologies. This will facilitate understanding the interaction between various components of the system and allow for the improvement of their integration. Keywords: electric power system, operability of the electric power system, assessment of the operability of the power supply system, non-project disturbances in the electric power system, generation, transformer substations, electricity consumers.

Evaluating and extending speedup techniques for optimal crossing minimization in layered graph drawings.

A layered graph is an important category of graph in which every node is assigned to a layer, and layers are drawn as parallel or radial lines. They are commonly used to display temporal data or hierarchical graphs. Previous research has demonstrated that minimizing edge crossings is the most important criterion to consider when looking to improve the readability of such graphs. While heuristic approaches exist for crossing minimization, we are interested in optimal approaches to the problem that prioritize human readability over computational scalability. We aim to improve the usefulness and applicability of such optimal methods by understanding and improving their scalability to larger graphs. This paper categorizes and evaluates the state-of-the-art linear programming formulations for exact crossing minimization and describes nine new and existing techniques that could plausibly accelerate the optimization algorithm. Through a computational evaluation, we explore each technique's effect on calculation time and how the techniques assist or inhibit one another, allowing researchers and practitioners to adapt them to the characteristics of their graphs. Our best-performing techniques yielded a median improvement of 2.5-17× depending on the solver used, giving us the capability to create optimal layouts faster and for larger graphs. We provide an open-source implementation of our methodology in Python, where users can pick which combination of techniques to enable according to their use case. A free copy of this paper and all supplemental materials, datasets used, and source code are available at https://osf.io/5vq79.

MONO-HYDRA: REAL-TIME 3D SCENE GRAPH CONSTRUCTION FROM MONOCULAR CAMERA INPUT WITH IMU

Abstract. The ability of robots to autonomously navigate through 3D environments depends on their comprehension of spatial concepts, ranging from low-level geometry to high-level semantics, such as objects, places, and buildings. To enable such comprehension, 3D scene graphs have emerged as a robust tool for representing the environment as a layered graph of concepts and their relationships. However, building these representations using monocular vision systems in real-time remains a difficult task that has not been explored in depth.This paper puts forth a real-time spatial perception system Mono-Hydra, combining a monocular camera and an IMU sensor setup, focusing on indoor scenarios. However, the proposed approach is adaptable to outdoor applications, offering flexibility in its potential uses. The system employs a suite of deep learning algorithms to derive depth and semantics. It uses a robocentric visual-inertial odometry (VIO) algorithm based on square-root information, thereby ensuring consistent visual odometry with an IMU and a monocular camera. This system achieves sub-20 cm error in real-time processing at 15 fps, enabling real-time 3D scene graph construction using a laptop GPU (NVIDIA 3080). This enhances decision-making efficiency and effectiveness in simple camera setups, augmenting robotic system agility. We make Mono-Hydra publicly available at: https://github.com/UAV-Centre-ITC/Mono_Hydra.

CLARIFY: cell-cell interaction and gene regulatory network refinement from spatially resolved transcriptomics.

Gene regulatory networks (GRNs) in a cell provide the tight feedback needed to synchronize cell actions. However, genes in a cell also take input from, and provide signals to other neighboring cells. These cell-cell interactions (CCIs) and the GRNs deeply influence each other. Many computational methods have been developed for GRN inference in cells. More recently, methods were proposed to infer CCIs using single cell gene expression data with or without cell spatial location information. However, in reality, the two processes do not exist in isolation and are subject to spatial constraints. Despite this rationale, no methods currently exist to infer GRNs and CCIs using the same model. We propose CLARIFY, a tool that takes GRNs as input, uses them and spatially resolved gene expression data to infer CCIs, while simultaneously outputting refined cell-specific GRNs. CLARIFY uses a novel multi-level graph autoencoder, which mimics cellular networks at a higher level and cell-specific GRNs at a deeper level. We applied CLARIFY to two real spatial transcriptomic datasets, one using seqFISH and the other using MERFISH, and also tested on simulated datasets from scMultiSim. We compared the quality of predicted GRNs and CCIs with state-of-the-art baseline methods that inferred either only GRNs or only CCIs. The results show that CLARIFY consistently outperforms the baseline in terms of commonly used evaluation metrics. Our results point to the importance of co-inference of CCIs and GRNs and to the use of layered graph neural networks as an inference tool for biological networks. The source code and data is available at https://github.com/MihirBafna/CLARIFY.

Optimization framework for splitting DNN inference jobs over computing networks

Ubiquitous artificial intelligence (AI) is considered one of the key services in 6G systems. AI services typically rely on deep neural network (DNN) requiring heavy computation. Hence, in order to support ubiquitous AI, it is crucial to provide a solution for offloading or distributing computational burden due to DNN, especially at end devices with limited resources. We develop an optimization framework for assigning the tasks of DNN layer computations to computing resources in the network, so as to reduce the inference latency. To this end, we propose a layered graph model with which simple conventional routing jointly solves the problem of selecting nodes for computation and paths for data transfer between nodes. We show that using our model, the existing approaches to splitting DNN layer computations can be equivalently reformulated as a routing problem that possesses better numerical properties. We also apply the proposed framework to derive algorithms for minimizing the end-to-end inference latency. We show through numerical evaluations that our new formulation can find a solution for DNN inference job distribution much faster than the existing formulation, and that our algorithms can select computing nodes and data paths adaptively to the computational attributes of given DNN inference jobs.

DynamicG2B: Dynamic Node Classification with Layered Graph Neural Networks and BiLSTM

Most studies in graph theory assume that graphs are static, but in reality, graph structures and features change over time, leading to the concept of dynamic graphs, which is an under-researched area. Contemporary research in dynamic graph representation learning typically treats different snapshots of the graph as separate entities, disregarding the benefits of incorporating temporal information. While some techniques try to solve this problem using recurrent neural network-based solutions, these approaches still face the challenge of the vanishing or exploding gradient problem and complicated training procedures. To address these issues, we propose DynamicG2B, a BiLSTM-based graph neural architecture that computes node representations guided by attention using neighborhood aggregation. Our method applies relevant attention weights at different time steps to classify nodes in a supervised manner, utilizing dynamic edges and node feature information. Our evaluation of two benchmark datasets shows that DynamicG2B outperforms seven state-of-the-art baseline models in node classification in dynamic graphs. Additionally, our analysis of attention weights opens up opportunities for further research into exploring the importance of relationships among graph nodes.

Crossing edge minimization in radial outerplanar layered graphs using segment paths

Crossing minimization is a natural problem in graph drawing, which consists of drawing it in the plane with a minimum number of crossings on the edges. We present an algorithm to minimize the crossing edges in radial layered graphs. The algorithm determines the optimal location of nodes in their corresponding circle layer and consists of two phases. In the first phase, a counterclockwise rotation of concentric circles is performed; in the next phase, edges are converted into segment paths. Unlike other algorithms with circular drawings, our algorithm does not require the insertion or removal of nodes and edges; only rotation and path construction operations are utilized. Two versions of the algorithm were created; the exhaustive version tests all the possible paths, while the random version tests a few number of paths. The goal is to minimize the crossing edges to obtain a clear visualization using three criteria: rotation, crossing edge detection and segment path construction. The algorithm has been successfully implemented in 30 instances of graphs with 20, 35 and 50 layers, where crossings are minimized in 86–93%, 93–96% and 95–97%, respectively.

The Odd Harmonious Labeling of Layered Graphs

Graphs that have the properties of odd harmonious labeling are odd harmonious graphs. The research objective of this paper is to obtain odd harmonious labeling on layered graph C(x,y) and layered graph D(x,y). The research used in this paper is a qualitative method. The research flow consists of data collection, processing, and analysis. The data collection stage consists of constructing the definition of the new class graph, the data processing stage consists of constructing the vertex labeling and edge labeling, and the data analysis stage consists of constructing the theorem and proving it. The research results show that the layered graph C(x,y) and layered graph D(x,y) fulfill odd harmonious labeling. Such that the layered graph C(x,y) and layered graph D(x,y) are odd harmonious graphs. The benefit of this research is to add new properties of odd harmonious graphs. In addition, it does not rule out the possibility that this research can be developed again both in theory and application.

Optimal Algorithms for Right-sizing Data Centers

Electricity cost is a dominant and rapidly growing expense in data centers. Unfortunately, much of the consumed energy is wasted, because servers are idle for extended periods of time. We study a capacity management problem that dynamically right-sizes a data center, matching the number of active servers with the varying demand for computing capacity. We resort to a data-center optimization problem introduced by Lin, Wierman, Andrew, and Thereska [ 25 , 27 ] that, over a time horizon, minimizes a combined objective function consisting of operating cost, modeled by a sequence of convex functions, and server switching cost. All prior work addresses a continuous setting in which the number of active servers, at any time, may take a fractional value. In this article, we investigate for the first time the discrete data-center optimization problem where the number of active servers, at any time, must be integer valued. Thereby, we seek truly feasible solutions. First, we show that the offline problem can be solved in polynomial time. Our algorithm relies on a new, yet intuitive graph theoretic model of the optimization problem and performs binary search in a layered graph. Second, we study the online problem and extend the algorithm Lazy Capacity Provisioning (LCP) by Lin et al. [ 25 , 27 ] to the discrete setting. We prove that LCP is 3-competitive. Moreover, we show that no deterministic online algorithm can achieve a competitive ratio smaller than 3. Hence, while LCP does not attain an optimal competitiveness in the continuous setting, it does so in the discrete problem examined here. We prove that the lower bound of 3 also holds in a problem variant with more restricted operating cost functions, introduced by Lin et al. [ 25 ]. In addition, we develop a randomized online algorithm that is 2-competitive against an oblivious adversary. It is based on the algorithm of Bansal et al. [ 7 ] (a deterministic, 2-competitive algorithm for the continuous setting) and uses randomized rounding to obtain an integral solution. Moreover, we prove that 2 is a lower bound for the competitive ratio of randomized online algorithms, so our algorithm is optimal. We prove that the lower bound still holds for the more restricted model. Finally, we address the continuous setting and give a lower bound of 2 on the best competitiveness of online algorithms. This matches an upper bound by Bansal et al. [ 7 ]. A lower bound of 2 was also shown by Antoniadis and Schewior [ 4 ]. We develop an independent proof that extends to the scenario with more restricted operating cost.

Static and Dynamic Dependency Visualization in a Layered Software City

A Software City is an established way to visualize metrics such as the test coverage or complexity. As current layouting algorithms are mainly based on the static code structure, dependencies that are orthogonal to this structure often clutter the visualization and are hard to grasp. This paper applies layered graph drawing to layout a Software City in 3D. The proposed layout takes both the dependencies and the static code structure into account. While having the static dependencies encoded in the layout, we can additionally display dynamic dependencies as arcs atop the city in the night view of the Layered Software City. By applying a trace clustering technique we can further reduce the number of shown arcs. We evaluate the advantages of our layout over a classic layouting algorithm in a controlled study on a real-world project and also report on a short study that evaluates the visualization of dynamic dependencies. The source code of the layouting algorithm and the raw data of the quantitative evaluations are available from https://github.com/qaware/holoware-software-city.

Six methods for transforming layered hypergraphs to apply layered graph layout algorithms

AbstractHypergraphs are a generalization of graphs in which edges (hyperedges) can connect more than two vertices—as opposed to ordinary graphs where edges involve only two vertices. Hypergraphs are a fairly common data structure but there is little consensus on how to visualize them. To optimize a hypergraph drawing for readability, we need a layout algorithm. Common graph layout algorithms only consider ordinary graphs and do not take hyperedges into account. We focus on layered hypergraphs, a particular class of hypergraphs that, like layered graphs, assigns every vertex to a layer, and the vertices in a layer are drawn aligned on a linear axis with the axes arranged in parallel. In this paper, we propose a general method to apply layered graph layout algorithms to layered hypergraphs. We introduce six different transformations for layered hypergraphs. The choice of transformation affects the subsequent graph layout algorithm in terms of computational performance and readability of the results. Thus, we perform a comparative evaluation of these transformations in terms of number of crossings, edge length, and impact on performance. We also provide two case studies showing how our transformations can be applied to real‐life use cases. A copy of this paper with all appendices and supplemental material is available at osf.io/grvwu.

Timetable optimization for a moving block system

We present an optimization model which is capable of routing and ordering trains on a microscopic level under a moving block regime. Based on a general timetabling definition (GTTP) that allows the plug in of arbitrarily detailed methods to compute running and headway times, we describe a layered graph approach using velocity expansion, and develop a mixed integer linear programming formulation. Finally, we present promising results for a German corridor scenario with mixed traffic, indicating that applying branch-and-cut to our model can solve reasonably sized instances with up to a hundred trains to optimality.

Optimal Picking Policies in E-Commerce Warehouses

In e-commerce warehouses, online retailers increase their efficiency by using a mixed-shelves (or scattered storage) concept, where unit loads are purposefully broken down into single items, which are individually stored in multiple locations. Irrespective of the stock keeping units a customer jointly orders, this storage strategy increases the likelihood that somewhere in the warehouse the items of the requested stock keeping units will be in close vicinity, which may significantly reduce an order picker’s unproductive walking time. This paper optimizes picker routing through such mixed-shelves warehouses. Specifically, we introduce a generic exact algorithmic framework that covers a multitude of picking policies, independently of the underlying picking zone layout, and is suitable for real-time applications. This framework embeds a bidirectional layered graph algorithm that provides the best known performance for the simple picking problem with a single depot and no further attributes. We compare three different real-world e-commerce warehouse settings that differ slightly in their application of scattered storage and in their picking policies. Based on these, we derive additional layouts and settings that yield further managerial insights. Our results reveal that the right combination of drop-off points, dynamic batching, the utilization of picking carts, and the picking zone layout can greatly improve the picking performance. In particular, some combinations of policies yield efficiency increases of more than 30% compared with standard policies currently used in practice. This paper was accepted by Chung Piaw Teo, optimization.