Graph Optimization Problem Research Articles

Reducing Bandwidth Usage in CVSLAM: A Novel Approach to Map Point Selection and Efficient Data Compression

In the field of collaborative visual simultaneous localization and mapping (CVSLAM), efficient data communication poses a significant challenge, particularly in environments with limited bandwidth. To address this issue, we introduce a method aimed at reducing communication consumption. Our approach starts with a strategic culling of map points, aiming at maximizing pose-visibility and expanding spatial diversity to effectively eliminate redundant data in CVSLAM. We achieve this by formulating the problem of maximizing pose-visibility and spatial diversity as a minimum-cost maximum-flow graph optimization problem. Subsequently, we apply finite state entropy encoding for the compression of visual information, further alleviating bandwidth constraints. To verify the proposed method, we implement it within a centralized collaborative monocular simultaneous localization and mapping (SLAM) system. Our approach has been tested on publicly available datasets and in real-world scene. The results show a prominent reduction in bandwidth usage by 49% while maintaining mapping accuracy and without introducing additional latency, confirming its effectiveness in a multi-agent system setting.

A Graph-Neural-Network-Powered Solver Framework for Graph Optimization Problems.

Backtracking combined with branching heuristics is a prevalent approach for tackling constraint satisfaction problems (CSPs) and combinatorial optimization problems (COPs). While branching heuristics specifically designed for certain problems can be theoretically efficient, they are often complex and difficult to implement in practice. On the other hand, general branching heuristics can be applied across various problems, but at the risk of suboptimality. We introduce a solver framework that leverages the Shannon entropy in branching heuristics to bridge the gap between generality and specificity in branching heuristics. This enables backtracking to follow the path of least uncertainty, based on probability distributions that conform to problem constraints. We employ graph neural network (GNN) models with loss functions derived from the probabilistic method to learn these probability distributions. We have evaluated our approach by its applications to two NP-hard problems: the (minimum) dominating-clique problem and the edge-clique-cover problem. Compared with the state-of-the-art solvers for both problems, our solver framework outputs competitive results. Specifically, for the (minimum) dominating-clique problem, our approach generates fewer branches than the solver presented by Culberson et al. (2005). For the edge-clique-cover problem, our approach produces smaller-sized edge clique covers (ECCs) than the solvers referenced by Conte et al. (2020) and Kellerman (1973).

Fully automated structured light scanning for high-fidelity 3D reconstruction via graph optimization.

Convenient and high-fidelity 3D model reconstruction is crucial for industries like manufacturing, medicine and archaeology. Current scanning approaches struggle with high manual costs and the accumulation of errors in large-scale modeling. This paper is dedicated to achieving industrial-grade seamless and high-fidelity 3D reconstruction with minimal manual intervention. The innovative method proposed transforms the multi-frame registration into a graph optimization problem, addressing the issue of error accumulation encountered in frame-by-frame registration. Initially, a global consistency cost is established based on point cloud cross-multipath registration, followed by using the geometric and color differences of corresponding points as dynamic nonlinear weights. Finally, the iteratively reweighted least squares (IRLS) method is adopted to perform the bundle adjustment (BA) optimization of all poses. Significantly enhances registration accuracy and robustness under the premise of maintaining near real-time efficiency. Additionally, for generating watertight, seamless surface models, a local-to-global transitioning strategy for multiframe fusion is introduced. This method facilitates efficient correction of normal vector consistency, addressing mesh discontinuities in surface reconstruction resulting from normal flips. To validate our algorithm, we designed a 3D reconstruction platform enabling spatial viewpoint transformations. We collected extensive real and simulated model data. These datasets were rigorously evaluated against advanced methods, roving the effectiveness of our approach. Our data and implementation is made available on GitHub for community development.

Foundations of spatial perception for robotics: Hierarchical representations and real-time systems

3D spatial perception is the problem of building and maintaining an actionable and persistent representation of the environment in real-time using sensor data and prior knowledge. Despite the fast-paced progress in robot perception, most existing methods either build purely geometric maps (as in traditional SLAM) or “flat” metric-semantic maps that do not scale to large environments or large dictionaries of semantic labels. The first part of this paper is concerned with representations: we show that scalable representations for spatial perception need to be hierarchical in nature. Hierarchical representations are efficient to store, and lead to layered graphs with small treewidth, which enable provably efficient inference. We then introduce an example of hierarchical representation for indoor environments, namely a 3D scene graph, and discuss its structure and properties. The second part of the paper focuses on algorithms to incrementally construct a 3D scene graph as the robot explores the environment. Our algorithms combine 3D geometry (e.g., to cluster the free space into a graph of places), topology (to cluster the places into rooms), and geometric deep learning (e.g., to classify the type of rooms the robot is moving across). The third part of the paper focuses on algorithms to maintain and correct 3D scene graphs during long-term operation. We propose hierarchical descriptors for loop closure detection and describe how to correct a scene graph in response to loop closures, by solving a 3D scene graph optimization problem. We conclude the paper by combining the proposed perception algorithms into Hydra, a real-time spatial perception system that builds a 3D scene graph from visual-inertial data in real-time. We showcase Hydra’s performance in photo-realistic simulations and real data collected by a Clearpath Jackal robots and a Unitree A1 robot. We release an open-source implementation of Hydra at https://github.com/MIT-SPARK/Hydra.

A precise solution to the shortest path optimization problem in graphs using Z-numbers

<p>Communication networks are exposed to internal or external risks that can affect all or part of the system. The most important components that form the infrastructure of these systems are routers, which act as nodes. In the field of graph theory, there are sophisticated techniques that can be used to optimize the path of a packet as it travels through various routers from its origin to its destination. A notable example of such an algorithm is Dijkstra's algorithm, which is designed to efficiently determine the shortest path. The algorithm works under the assumption that the system operates under ideal conditions. Real-time systems can perform better if risk factors and optimal conditions are taken into account. The relationship between the nodes can be expressed by various metrics such as distance, delay, and bandwidth. The aforementioned metrics facilitate the calculation of the optimal path, with the ultimate objective of achieving low-latency networks characterized by rapid response times. Round-trip time (RTT) can be employed as a metric for measuring enhancements in a range of latency types, including those associated with processing, transmission, queuing, and propagation. The use of Z-numbers was employed in this study to incorporate risk into the optimal path metric. RTT was the preferred metric and reliability was represented by fuzzy linguistic qualifiers. A comparison of several scenarios was shown using a numerical example of a communication network. It is expected that this study will have a significant impact on the evolution from models that consider only ideal conditions to real-time systems that include risks using Z-numbers.</p>

3D Reconstruction with Spherical Cameras

The goal of image-based 3D reconstruction is to establish a high-quality 3D expression from images. In order to achieve a high-resolution and real-time 3D model, inspired by the open source COLMAP, we propose a novel framework (3DMAP) to reconstruct 3D scenes based on spherical cameras. Unlike traditional methods which focus on building a 3D plane by Poisson distribution function, our method illustrates the key processes of 3D reconstruction: locating the camera based on global feature, estimating the scene’s relative depth from monocular panoramic images, and obtaining a high-quality 3D surface reconstruction. In the camera locating part, we use a global descriptor augmentation model to build a labeled panorama dataset GDAP, in which the images are captured by our designed spherical cameras; In the depth estimation part, we propose a new network UMDE that can estimate the depth of both indoor and outdoor scenes; Finally, in the 3D surface reconstruction section, we turn the reconstruction problem to a graph optimization problem, called GraphFit, in which, we optimize the point clouds with s-t graph and smoothing method successively. We conduct experiments on our own dataset to demonstrate the effectiveness of our proposed 3DMAP framework. Experimental results show that our 3DMAP has achieved good evaluation scores and visual effects.

Extension of some edge graph problems: Standard, parameterized and approximation complexity

We consider extension variants of some edge optimization problems in graphs containing the classical Edge Cover, Matching, and Edge Dominating Set problems and generalizations thereof. Given a graph G=(V,E) and an edge set U⊆E, it is asked whether there exists an inclusion-wise minimal (or maximal, respectively) feasible solution E′ which satisfies a given property, for instance, being an edge dominating set (or a matching, respectively) and containing the forced edge set U (or avoiding any edges from the forbidden edge set E∖U, respectively). We present hardness results for these problems, for restricted instances such as bipartite or planar graphs. We counter-balance these negative results with parameterized complexity results. We also consider the price of extension, a natural optimization problem variant of extension problems, leading to some approximation and inapproximability results.

The nilcatenation problem and its application for detecting money laundering activities in cryptocurrency networks

AbstractThis work considers a combinatorial optimization problem in graphs, the nilcatenation problem, and investigates its potential application for detecting money laundering activities in cryptocurrency networks. The nilcatenation problem consists of finding a set of arcs that can be removed from an arc‐weighted directed graph without changing the balance of any vertex. The balance of a vertex is defined as the difference between the sum of the weights of outgoing and incoming arcs. We propose a 0/1 integer linear programming formulation and a local branching algorithm. The approaches are computationally evaluated and compared using three sets of test instances, two of them generated from Bitcoin's testnet and mainnet networks. An experiment on the testnet showed that it is possible to retrieve a nilcatenation artificially introduced with fake bitcoin transactions. Experiments on the mainnet showed that it is possible to find large nilcatenations, possibly indicating money laundering activities.

A Distributed-GPU Deep Reinforcement Learning System for Solving Large Graph Optimization Problems

Graph optimization problems (such as minimum vertex cover, maximum cut, traveling salesman problems) appear in many fields including social sciences, power systems, chemistry, and bioinformatics. Recently, deep reinforcement learning (DRL) has shown success in automatically learning good heuristics to solve graph optimization problems. However, the existing RL systems either do not support graph RL environments or do not support multiple or many GPUs in a distributed setting. This has compromised the ability of reinforcement learning in solving large-scale graph optimization problems due to lack of parallelization and high scalability. To address the challenges of parallelization and scalability, we develop RL4GO , a high-performance distributed-GPU DRL framework for solving graph optimization problems. RL4GO focuses on a class of computationally demanding RL problems, where both the RL environment and policy model are highly computation intensive. Traditional reinforcement learning systems often assume either the RL environment is of low time complexity or the policy model is small. In this work, we distribute large-scale graphs across distributed GPUs and use the spatial parallelism and data parallelism to achieve scalable performance. We compare and analyze the performance of the spatial parallelism and data parallelism and show their differences. To support graph neural network (GNN) layers that take as input data samples partitioned across distributed GPUs, we design parallel mathematical kernels to perform operations on distributed 3D sparse and 3D dense tensors. To handle costly RL environments, we design a parallel graph environment to scale up all RL-environment-related operations. By combining the scalable GNN layers with the scalable RL environment, we are able to develop high-performance RL4GO training and inference algorithms in parallel. Furthermore, we propose two optimization techniques—replay buffer on-the-fly graph generation and adaptive multiple-node selection—to minimize the spatial cost and accelerate reinforcement learning. This work also conducts in-depth analyses of parallel efficiency and memory cost and shows that the designed RL4GO algorithms are scalable on numerous distributed GPUs. Evaluations on large-scale graphs show that (1) RL4GO training and inference can achieve good parallel efficiency on 192 GPUs, (2) its training time can be 18 times faster than the state-of-the-art Gorila distributed RL framework [ 34 ], and (3) its inference performance achieves a 26 times improvement over Gorila.

Minimum gradation in greyscales of graphs

In this paper we present the notion of greyscale of a graph as a colouring of its vertices that uses colours from the real interval [0,1]. Any greyscale induces another colouring by assigning to each edge the non-negative difference between the colours of its vertices. These edge colours are ordered in lexicographical decreasing ordering and give rise to a new element of the graph: the gradation vector. We introduce the notion of minimum gradation vector as a new invariant for the graph and give polynomial algorithms to obtain it. These algorithms also output all greyscales that produce the minimum gradation vector. This way we tackle and solve a novel vectorial optimization problem in graphs that may generate more satisfactory solutions than those generated by known scalar optimization approaches.

MIP formulations for induced graph optimization problems: a tutorial

AbstractGiven a graph and a subset of its vertices , the subgraph induced by in G is that with vertex set and edge set formed by all the edges in E linking two vertices in . Mixed integer programming (MIP) approaches are among the most successful techniques for solving induced graph optimization problems, that is, those related to obtaining maximum or minimum (weighted or not) induced subgraphs with certain properties. In this tutorial, we provide a literature review of some of these problems. Furthermore, we illustrate the use of MIP formulations and techniques for solving combinatorial optimization problems involving induced graphs. We focus on compact formulations and those with an exponential number of constraints that can be effectively solved using branch‐and‐cut procedures. More specifically, we revisit applications of their use for problems of finding induced forests (which correspond to the complement of feedback vertex sets), trees, paths, as well as quasi‐clique partitionings.

Communication Efficient, Distributed Relative State Estimation in UAV Networks

Distributed estimation of 6-DOF relative states, including three-dimensional relative poses and three-dimensional relative positions, is a key problem in UAV (Unmanned Aerial Vehicle) networks, which generally requires vision-involved iterative state estimation. How to achieve communication efficiency is a crucial challenge considering the large volume of vision data. This paper jointly considers the communication efficiency, latency, and accuracy for distributed relative state estimation involving vision data in UAV networks. The key is to solve a distributed graph optimization problem, which includes two key steps: (1) local graph construction and node state initialization in an initialization phase, and (2) iterative state update and communication with neighbors until convergence in online iteration phase. A communication efficient, Locating Then Informing (LTI) initialization scheme is proposed, which is run only once by each node to initialize each node’s local graph and initial states. For online iteration, a RIPPLE-like distributed state iteration scheme is proposed. It inherits the advantages of traditional sequential and parallel methods while avoiding their drawbacks. It enables nodes’ states to converge quickly using fewer rounds of communications. The communication costs for the initialization and online iteration processes are analyzed theoretically. Extensive evaluations use synthetic data generated by AirSim (a widely used UAV network simulation platform) and real-world data are presented. The results show that the proposed method provides accuracy comparable to the centralized graph optimization method and significantly outperforms the other distributed methods in terms of accuracy, communication cost, and latency.

Optimization Problems in Graphs with Locational Uncertainty

Many discrete optimization problems amount to selecting a feasible set of edges of least weight. We consider in this paper the context of spatial graphs where the positions of the vertices are uncertain and belong to known uncertainty sets. The objective is to minimize the sum of the distances of the chosen set of edges for the worst positions of the vertices in their uncertainty sets. We first prove that these problems are [Formula: see text]-hard even when the feasible sets consist either of all spanning trees or of all s – t paths. Given this hardness, we propose an exact solution algorithm combining integer programming formulations with a cutting plane algorithm, identifying the cases where the separation problem can be solved efficiently. We also propose a conservative approximation and show its equivalence to the affine decision rule approximation in the context of Euclidean distances. We compare our algorithms to three deterministic reformulations on instances inspired by the scientific literature for the Steiner tree problem and a facility location problem. History: Accepted by David Alderson, Area Editor for Network Optimization: Algorithms & Applications. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2023.1276 .

Robust Algorithms for TSP and Steiner Tree

Robust optimization is a widely studied area in operations research, where the algorithm takes as input a range of values and outputs a single solution that performs well for the entire range. Specifically, a robust algorithm aims to minimize regret , defined as the maximum difference between the solution’s cost and that of an optimal solution in hindsight once the input has been realized. For graph problems in P , such as shortest path and minimum spanning tree, robust polynomial-time algorithms that obtain a constant approximation on regret are known. In this paper, we study robust algorithms for minimizing regret in NP -hard graph optimization problems, and give constant approximations on regret for the classical traveling salesman and Steiner tree problems.

DAG-Pathwidth: Graph Algorithmic Analyses of DAG-Type Blockchain Networks

This paper analyzes a blockchain network forming a directed acyclic graph (DAG), called a DAG-type blockchain, from the viewpoint of graph algorithm theory. To use a DAG-type blockchain, NP-hard graph optimization problems on the DAG are required to be solved. Although various problems for undirected and directed graphs can be efficiently solved by using the notions of graph parameters, these currently known parameters are meaningless for DAGs, which implies that it is hopeless to design efficient algorithms based on the parameters for such problems. In this work, we propose a novel graph parameter for directed graphs called a DAG-pathwidth, which represents the closeness to a directed path. This is an extension of the pathwidth, a well-known graph parameter for undirected graphs. We analyze the features of the DAG-pathwidth and prove that computing the DAG-pathwidth of a DAG (directed graph in general) is NP-complete. Finally, we propose an efficient algorithm for a variant of the maximum k-independent set problem for the DAG-type blockchain when the DAG-pathwidth of the input graph is small.

Incremental Cycle Bases for Cycle-Based Pose Graph Optimization

Pose graph optimization is a special case of the simultaneous localization and mapping problem where the only variables to be estimated are pose variables and the only measurements are inter-pose constraints. The vast majority of pose graph optimization techniques are vertex based (variables are robot poses), but recent work has parameterized the pose graph optimization problem in a relative fashion (variables are the transformations between poses) that utilizes a minimum cycle basis to maximize the sparsity of the problem. We explore the construction of a cycle basis in an incremental manner while maximizing the sparsity. We validate an algorithm that constructs a sparse cycle basis incrementally and compare its performance with a minimum cycle basis. Additionally, we present an algorithm to approximate the minimum cycle basis of two graphs that are sparsely connected as is common in multi-agent scenarios. Lastly, the relative parameterization of pose graph optimization has been limited to using rigid body transforms on <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$SE(2)$</tex-math></inline-formula> or <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$SE(3)$</tex-math></inline-formula> as the constraints between poses. We introduce a methodology to allow for the use of lower-degree-of-freedom measurements in the relative pose graph optimization problem. We provide extensive validation of our algorithms on standard benchmarks, simulated datasets, and custom hardware.

Regularity Index of Uncertain Random Graph

A graph containing some edges with probability measures and other edges with uncertain measures is referred to as an uncertain random graph. Numerous real-world problems in social networks and transportation networks can be boiled down to optimization problems in uncertain random graphs. Actually, information in optimization problems in uncertain random graphs is always asymmetric. Regularization is a common optimization problem in graph theory, and the regularity index is a fundamentally measurable indicator of graphs. Therefore, this paper investigates the regularity index of an uncertain random graph within the framework of chance theory and information asymmetry theory. The concepts of k-regularity index and regularity index of the uncertain random graph are first presented on the basis of the chance theory. Then, in order to compute the k-regularity index and the regularity index of the uncertain random graph, a simple and straightforward calculating approach is presented and discussed. Furthermore, we discuss the relationship between the regularity index and the k-regularity index of the uncertain random graph. Additionally, an adjacency matrix-based algorithm that can compute the k-regularity index of the uncertain random graph is provided. Some specific examples are given to illustrate the proposed method and algorithm. Finally, we conclude by highlighting some potential applications of uncertain random graphs in social networks and transportation networks, as well as the future vision of its combination with symmetry.

Improvement of Ant Colony Algorithm Performance for the Job-Shop Scheduling Problem Using Evolutionary Adaptation and Software Realization Heuristics

Planning tasks are important in construction, manufacturing, logistics, and education. At the same time, scheduling problems belong to the class of NP-hard optimization problems. Ant colony algorithm optimization is one of the most common swarm intelligence algorithms and is a leader in solving complex optimization problems in graphs. This paper discusses the solution to the job-shop scheduling problem using the ant colony optimization algorithm. An original way of representing the scheduling problem in the form of a graph, which increases the flexibility of the approach and allows for taking into account additional restrictions in the scheduling problems, is proposed. A dynamic evolutionary adaptation of the algorithm to the conditions of the problem is proposed based on the genetic algorithm. In addition, some heuristic techniques that make it possible to increase the performance of the software implementation of this evolutionary ant colony algorithm are presented. One of these techniques is parallelization; therefore, a study of the algorithm’s parallelization effectiveness was made. The obtained results are compared with the results of other authors on test problems of scheduling. It is shown that the best heuristics coefficients of the ant colony optimization algorithm differ even for similar job-shop scheduling problems.

DeTTO: Dependency-Aware Trustworthy Task Offloading in Vehicular IoT

In this paper, we investigate the dependency-aware trustworthy task offloading problem (DeTTO), especially in an IoT-enabled vehicular network, where a large computation-intensive task offloaded from a vehicle is fragmented into multiple subtasks and then offloaded to multiple trusted nodes. First, we formulate the task offloading problem as a graph optimization problem intending to find an optimal set of trustworthy nodes for offloading the subtasks. We aim to minimize the task completion delay and energy consumption, while satisfying the dependency relations between the subtasks and the trust requirements of the tasks. We consider three types of dependency structures – fully independent task, fully dependent task, and partially dependent task. For a fully independent task with no dependency between the subtasks, we propose a greedy algorithm to get the optimal set of nodes for task offloading. After showing the NP-hardness of solving the dependent task offloading problem, we propose a two-fold efficient heuristic approach for the tasks with all dependent subtasks. We adopt the solution approaches used by the first two types of tasks for a partially dependent task. Through simulation experiments, we analyze the performance of the proposed algorithms for three types of intra-task dependencies. The experimental results show that the proposed algorithms significantly reduce the delay and energy consumption, when compared to the benchmark schemes.

On constant-time quantum annealing and guaranteed approximations for graph optimization problems

Quantum annealing (QA) is a computational framework where a quantum system’s continuous evolution is used to find the global minimum of an objective function over an unstructured search space. It can be seen as a general metaheuristic for optimization problems, including NP-hard ones if we allow an exponentially large running time. While QA is widely studied from a heuristic point of view, little is known about theoretical guarantees on the quality of the solutions obtained in polynomial time. In this paper, we use a technique borrowed from theoretical physics, the Lieb–Robinson (LR) bound, and develop new tools proving that short, constant-time QA guarantees constant factor approximations ratios for some optimization problems when restricted to bounded degree graphs. Informally, on bounded degree graphs the LR bound allows us to retrieve a (relaxed) locality argument, through which the approximation ratio can be deduced by studying subgraphs of bounded radius. We illustrate our tools on problems MaxCut and maximum independent set for cubic graphs, providing explicit approximation ratios and the runtimes needed to obtain them. Our results are of similar flavor to the well-known ones obtained in the different but related QAOA (quantum optimization algorithms) framework.