Edge Weights Research Articles

Identification of risk key factors and prevention strategies for collision accidents between merchant and fishing vessels in China waters based on complex network

In recent years, Collision Accidents between Merchant and Fishing Vessels (CAMF) in China waters have been increasing, resulting in severe casualties and property losses. To effectively prevent such frequent collision accidents and ensure the safety of waterways and navigation routes, this study aims to identify the key risk factors associated with CAMF. Firstly, the Human Factors Analysis and Classification System (HFACS) model is employed as a comprehensive framework to extract the risk factors contributing to CAMF. By introducing edge length as the weight of network edges, a directed weighted network of CAMF is constructed. Secondly, the overall topological characteristics of the network are analyzed, and the network's robustness is studied based on the degree centrality, weighted closeness centrality, and weighted betweenness centrality of the risk factors in the network, and the key factors contributing to the accidents are identified. The research findings indicate that the network exhibits evident scale-free characteristics and small-world network properties. Deliberate attacks on approximately top 15% of the risk factors lead to a 55% degradation in the global efficiency of the accident network. Ultimately, accident prevention and control strategies and recommendations are proposed from the perspectives of both merchant and fishing vessels.

Inference of gene regulatory networks based on directed graph convolutional networks.

Inferring gene regulatory network (GRN) is one of the important challenges in systems biology, and many outstanding computational methods have been proposed; however there remains some challenges especially in real datasets. In this study, we propose Directed Graph Convolutional neural network-based method for GRN inference (DGCGRN). To better understand and process the directed graph structure data of GRN, a directed graph convolutional neural network is conducted which retains the structural information of the directed graph while also making full use of neighbor node features. The local augmentation strategy is adopted in graph neural network to solve the problem of poor prediction accuracy caused by a large number of low-degree nodes in GRN. In addition, for real data such as E.coli, sequence features are obtained by extracting hidden features using Bi-GRU and calculating the statistical physicochemical characteristics of gene sequence. At the training stage, a dynamic update strategy is used to convert the obtained edge prediction scores into edge weights to guide the subsequent training process of the model. The results on synthetic benchmark datasets and real datasets show that the prediction performance of DGCGRN is significantly better than existing models. Furthermore, the case studies on bladder uroepithelial carcinoma and lung cancer cells also illustrate the performance of the proposed model.

Relaxations and Exact Solutions to Quantum Max Cut via the Algebraic Structure of Swap Operators

The Quantum Max Cut (QMC) problem has emerged as a test-problem for designing approximation algorithms for local Hamiltonian problems. In this paper we attack this problem using the algebraic structure of QMC, in particular the relationship between the quantum max cut Hamiltonian and the representation theory of the symmetric group.The first major contribution of this paper is an extension of non-commutative Sum of Squares (ncSoS) optimization techniques to give a new hierarchy of relaxations to Quantum Max Cut. The hierarchy we present is based on optimizations over polynomials in the qubit swap operators. This is in contrast to the "standard" quantum Lasserre Hierarchy, which is based on polynomials expressed in terms of the Pauli matrices. To prove correctness of this hierarchy, we exploit a finite presentation of the algebra generated by the qubit swap operators. This presentation allows for the use of computer algebraic techniques to manipulate and simplify polynomials written in terms of the swap operators, and may be of independent interest. Surprisingly, we find that level-2 of this new hierarchy is numerically exact (up to tolerance 10−7) on all QMC instances with uniform edge weights on graphs with at most 8 vertices.The second major contribution of this paper is a polynomial-time algorithm that computes (in exact arithmetic) the maximum eigenvalue of the QMC Hamiltonian for certain graphs, including graphs that can be "decomposed" as a signed combination of cliques. A special case of the latter are complete bipartite graphs with uniform edge-weights, for which exact solutions are known from the work of Lieb and Mattis \cite{lieb1962ordering}. Our methods, which use representation theory of the symmetric group, can be seen as a generalization of the Lieb-Mattis result.

Heuristic Incident Edge Path Algorithm for Interval-Valued Neutrosophic Transportation Network

Neutrosophic Set is an extension of Fuzzy set theory dealts with uncertain environments in Transportation, Decision-making etc. Finding the shortest path for an uncertain environment is one of the most challenging tasks, many heuristic algorithms play a vital role in releasing this task. This research article an heuristic algorithmic approach using graph theoretical methods helps the researcher to crack the challenge and which type of algorithm can use the type of score functions are discussed. especially, the interval-valued Neutrosophic transportation problem (IVNTP) is considered for finding the shortest path which is a real-world decision-making problem in an ambiguous (uncertain) environment. Providing the shortest and finest way to address this ambiguity leads to the success of researcher to the organization. The algorithm used has been introduced for determination of interval-valued Neutrosophic shortest path (IVNSP) from the origin node to all other nodes by de-neutrosophicated edges using score functions. An illustrative Interval-valued neutrosophic transportation problem explains the algorithm’s efficacy and authenticity to find optimum result; this research article discusses about the nature of score functions, and some score function were taken into consideration. Further, the heuristic algorithm used is applicable for both positive and negative weighted edges; it was elaborated and compared with existing algorithm results also it provided all pair of shortest path. Future study and brief study of this article was disclosed in conclusion.

Elucidating transport dynamics and regional division of PM2.5 and O3 in China using an advanced network model

Air pollution exhibits significant spatial spillover effects, complicating and challenging regional governance models. This study innovatively applied and optimized a statistics-based complex network method in atmospheric environmental field. The methodology was enhanced through improvements in edge weighting and threshold calculations, leading to the development of an advanced pollutant transport network model. This model integrates pollution, meteorological, and geographical data, thereby comprehensively revealing the dynamic characteristics of PM2.5 and O3 transport among various cities in China. Research findings indicated that, throughout the year, the O3 transport network surpassed the PM2.5 network in edge count, average degree, and average weighted degree, showcasing a higher network density, broader city connections, and greater transmission strength. Particularly during the warm period, these characteristics of the O3 network were more pronounced, showcasing significant transport potential. Furthermore, the model successfully identified key influential cities in different periods; it also provided detailed descriptions of the interprovincial spillover flux and pathways of PM2.5 and O3 across various time scales. It pinpointed major pollution spillover and receiving provinces, with primary spillover pathways concentrated in crucial areas such as the Beijing-Tianjin-Hebei (BTH) region and its surrounding areas, the Yangtze River Delta, and the Fen-Wei Plain. Building on this, the model divided the O3, PM2.5, and synergistic pollution transmission regions in China into 6, 7, and 8 zones, respectively, based on network weights and the Girvan Newman (GN) algorithm. Such division offers novel perspectives and strategies for regional joint prevention and control. The validity of the model was further corroborated by source analysis results from the WRF-CAMx model in the BTH area. Overall, this research provides valuable insights for local and regional atmospheric pollution control strategies. Additionally, it offers a robust analytical tool for research in the field of atmospheric pollution.

DMS: Low-overlap Registration of 3D Point Clouds with Double-layer Multi-scale Star-graph.

Registering 3D point clouds with low overlap is challenging in 3D computer vision, primarily due to difficulties in identifying small overlap regions and removing correspondence outliers. We observe that the neighborhood similarity can be utilized to detect point correspondence, and the consistent neighborhood correspondence can be used as a criterion to detect robust overlapping regions. So that a Double-layer Multi-scale Star-graph (DMS) structure is proposed to detect robust correspondences using two different types of multi-scale star-graphs. The first-layer Multi-scale Neighbor Feature Star-graphs (MNFS) takes each point as the center and its multi-scale nearest neighbors as the leaves. The MNFS enables to establish the initial correspondence candidate set between the two point clouds based on multi-scale neighborhood topology and feature similarity. Subsequently, each pair of corresponding points find their nearest neighbors within the correspondence sets to construct a Multi-scale Matching Star-graphs (MMS) on each side, so the mutual correspondence relationships between the MMS vertices are identified. These identified mutual correspondences are treated as vertices to construct the Multi-scale Correspondence Star-graphs (MCS), that indicate the relationships among the correspondences. We design edge weight and vertex weight criterion in MCS to detect only the robust correspondence set that has strong neighborhood consistency, so as to reject the outliers. Finally, the point cloud registration is conducted based on the detected robust correspondence. The experimental results demonstrate clearly that the proposed DMS method exhibits superior robustness when compared to existing state-of-the-art registration algorithms. The code of this study will be available at https://github.com/HualongCao/DMS.

Tight Lower Bounds for Directed Cut Sparsification and Distributed Min-Cut

In this paper, we consider two fundamental cut approximation problems on large graphs. We prove new lower bounds for both problems that are optimal up to logarithmic factors. The first problem is approximating cuts in balanced directed graphs. In this problem, we want to build a data structure that can provide (1 ± ε)-approximation of cut values on a graph with n vertices. For arbitrary directed graphs, such a data structure requires Ω(n 2 ) bits even for constant ε. To circumvent this, recent works study β-balanced graphs, meaning that for every directed cut, the total weight of edges in one direction is at most β times the total weight in the other direction. We consider the for-each model, where the goal is to approximate each cut with constant probability, and the for-all model, where all cuts must be preserved simultaneously. We improve the previous Ømega(n √β/ε) lower bound in the for-each model to ~Ω (n √β /ε) and we improve the previous Ω(n β/ε) lower bound in the for-all model to Ω(n β/ε 2 ). This resolves the main open questions of (Cen et al., ICALP, 2021). The second problem is approximating the global minimum cut in a local query model, where we can only access the graph via degree, edge, and adjacency queries. We prove an ΩL(min m, m/ε 2 k R) lower bound for this problem, which improves the previous ΩL(m/k R) lower bound, where m is the number of edges, k is the minimum cut size, and we seek a (1+ε)-approximation. In addition, we show that existing upper bounds with minor modifications match our lower bound up to logarithmic factors.

Cycle-Star Motifs: Network Response to Link Modifications

Understanding efficient modifications to improve network functionality is a fundamental problem of scientific and industrial interest. We study the response of network dynamics against link modifications on a weakly connected directed graph consisting of two strongly connected components: an undirected star and an undirected cycle. We assume that there are directed edges starting from the cycle and ending at the star (master–slave formalism). We modify the graph by adding directed edges of arbitrarily large weights starting from the star and ending at the cycle (opposite direction of the cutset). We provide criteria (based on the sizes of the star and cycle, the coupling structure, and the weights of cutset and modification edges) that determine how the modification affects the spectral gap of the Laplacian matrix. We apply our approach to understand the modifications that either enhance or hinder synchronization in networks of chaotic Lorenz systems as well as Rössler. Our results show that the hindrance of collective dynamics due to link additions is not atypical as previously anticipated by modification analysis and thus allows for better control of collective properties.

Exploring the differential effects of psychological resilience and social support in mitigating post-traumatic psychiatric symptoms: real-world network analysis of front-line rescuers.

Although both psychological resilience and social support are widely believed to be effective in alleviating post-traumatic psychiatric symptoms in individuals with traumatic events, there has been a lack of comparative analysis of their intervention effects on different post-traumatic psychiatric symptoms. Furthermore, previous studies have mostly failed to control for potential confounding effects caused by different traumatic events. We used the novel network analysis approach to examine the differential moderating effects of psychological resilience and social support on post-traumatic psychiatric symptoms, controlling for the confounding effects of traumatic events. We recruited 264 front-line rescuers who experienced the same traumatic event. Quantified edge weights and bridge expected influence (BEI) were applied to compare the alleviating effects of psychological resilience and social support. Our study revealed distinct correlations in a sample of front-line rescuers: social support negatively correlates more with psychosomatic symptoms, notably fatigue in depressive networks and sleep disturbance in post-traumatic stress disorder (PTSD) networks, whereas psychological resilience shows fewer such correlations. Quantitative analysis using BEI indicated that psychological resilience more effectively suppresses depressive and anxiety symptom networks, whereas social support more significantly inhibits PTSD symptom networks. The current study represents the first attempt to examine the differential effects of psychological resilience and social support on post-traumatic outcomes in real-world emergency rescuers, controlling for the confounding effect of traumatic events. Our results can act as the theoretical reference for future precise and efficient post-trauma psychological interventions.

Multiscale dynamic graph signal analysis

We consider the problem of decomposing a time-varying graph signal into its constituent amplitude- and frequency-modulated components and inferring their dynamic functional connectivity structures in a data-driven manner, referred to as graph mode decomposition (GMD). The currently available method for GMD obtains static connectivity structures, limiting its applicability to real-life data. Further, the performance of the method relies critically on the accurate prior knowledge of the number of components to be extracted. We address those key limitations by presenting two approaches: (1) a two-stage method for obtaining the graph modes as well as corresponding connectivity structures by solving two separate optimization problems, and (2) a joint optimization formulation to achieve that task. The proposed optimization formulations include a prior to account for the time variation of graph edge weights. Moreover, we use a successive scheme to solve the problems which alleviates the limitation of having to specify the number of decomposed components a priori — improving the accuracy and data-driven capability while reducing the computational requirement of the methods. The performance of these methods is validated on a wide range of synthetic and real data sets. We also provide a public toolbox containing the MATLAB codes of the proposed methods.

Network Analysis Revealed the Role of Helplessness as a Central Feature Among Late-Life Depressive Symptoms in Patients With Mild Cognitive Impairment and Early Stage Dementia.

It has been reported that depressive symptoms in older adults are different from those in younger adults, especially when accompanied by cognitive decline. However, few studies have investigated the network structure of depressive symptoms in this population. The participants consisted of 627 older adults (>60 yr) who were diagnosed with mild cognitive impairment (MCI) or early stage dementia. Among them, 36.7% were male and the mean age was 76.20±7.71 years. The Korean form of Geriatric Depression Scale (KGDS) was used to evaluate their depressive symptoms and network analyses were performed using bootnet R-package to identify the central features among depressive symptoms. Of all the KGDS items, we found that KGDS 2 (often feel helpless) had the highest node strength followed by KGDS 21 (in good spirits), KGDS 14 (not confident at all), and KGDS 15 (cheerful and happy). In terms of node betweenness, KGDS 2 also showed the highest value. The edge weights of edges connected to node KGDS 2 were strongest in KGDS 3 (restless and fidgety) and KGDS 28 (easily get tired). In this study, we presented which symptoms are central among the elderly with MCI and early stage dementia. This result not only increases the understanding of depressive symptoms in this group but would also help determine target symptoms in the treatment program.

Optimal hybrid classification model for event recommendation system

There is a growing need for recommender systems and other ML-based systems as an abundance of data is now available across all industries. Various industries are currently using recommender systems in slightly different ways. These programs utilize algorithms to propose appropriate products to consumers based on their prior choices and interactions. Moreover, Systems for recommending events to users suggest pertinent happenings that they might find interesting. As opposed to an object recommender that suggests books or movies; event-based recommender systems typically require distinct algorithms. A developed event recommendation method is introduced which includes two stages: feature extraction and recommendation. In stage, I, a Set of features like personal willingness, community willingness, informative content, edge weight, and node interest degree are extracted. Stage II of the event recommendation system performs a hybrid classification by combining LSTM and CNN. In the LSTM classifier, optimal tuning is done by Improvised Cat and Mouse optimization (ICMO) algorithm. The results of the ICMO technique at an 80% training percentage have the maximum sensitivity value of 95.19%, whereas those of the existing approaches SSA, DINGO, BOA, and CMBO have values of 93.89%, 93.35%, 92.36%, and 92.24%. Finally, the best result is then determined by evaluating the whole performance.

MMGAT: a graph attention network framework for ATAC-seq motifs finding.

Motif finding in Assay for Transposase-Accessible Chromatin using sequencing (ATAC-seq) data is essential to reveal the intricacies of transcription factor binding sites (TFBSs) and their pivotal roles in gene regulation. Deep learning technologies including convolutional neural networks (CNNs) and graph neural networks (GNNs), have achieved success in finding ATAC-seq motifs. However, CNN-based methods are limited by the fixed width of the convolutional kernel, which makes it difficult to find multiple transcription factor binding sites with different lengths. GNN-based methods has the limitation of using the edge weight information directly, makes it difficult to aggregate the neighboring nodes' information more efficiently when representing node embedding. To address this challenge, we developed a novel graph attention network framework named MMGAT, which employs an attention mechanism to adjust the attention coefficients among different nodes. And then MMGAT finds multiple ATAC-seq motifs based on the attention coefficients of sequence nodes and k-mer nodes as well as the coexisting probability of k-mers. Our approach achieved better performance on the human ATAC-seq datasets compared to existing tools, as evidenced the highest scores on the precision, recall, F1_score, ACC, AUC, and PRC metrics, as well as finding 389 higher quality motifs. To validate the performance of MMGAT in predicting TFBSs and finding motifs on more datasets, we enlarged the number of the human ATAC-seq datasets to 180 and newly integrated 80 mouse ATAC-seq datasets for multi-species experimental validation. Specifically on the mouse ATAC-seq dataset, MMGAT also achieved the highest scores on six metrics and found 356 higher-quality motifs. To facilitate researchers in utilizing MMGAT, we have also developed a user-friendly web server named MMGAT-S that hosts the MMGAT method and ATAC-seq motif finding results. The advanced methodology MMGAT provides a robust tool for finding ATAC-seq motifs, and the comprehensive server MMGAT-S makes a significant contribution to genomics research. The open-source code of MMGAT can be found at https://github.com/xiaotianr/MMGAT , and MMGAT-S is freely available at https://www.mmgraphws.com/MMGAT-S/ .

Improving Graph Convolutional Network with Learnable Edge Weights and Edge-Node Co-Embedding for Graph Anomaly Detection.

The era of Industry 4.0 is gradually transforming our society into a data-driven one, which can help us uncover valuable information from accumulated data, thereby improving the level of social governance. The detection of anomalies, is crucial for maintaining societal trust and fairness, yet it poses significant challenges due to the ubiquity of anomalies and the difficulty in identifying them accurately. This paper aims to enhance the performance of the current Graph Convolutional Network (GCN)-based Graph Anomaly Detection (GAD) algorithm on datasets with extremely low proportions of anomalous labels. This goal is achieved through modifying the GCN network structure and conducting feature extraction, thus fully utilizing three types of information in the graph: node label information, node feature information, and edge information. Firstly, we theoretically demonstrate the relationship between label propagation and feature convolution, indicating that the Label Propagation Algorithm (LPA) can serve as a regularization penalty term for GCN, aiding in training and enabling learnable edge weights, providing a basis for incorporating node label information into GCN networks. Secondly, we introduce a method to aggregate node and edge features, thereby incorporating edge information into GCN networks. Finally, we design different GCN trainable weights for node features and co-embedding features. This design allows different features to be projected into different spaces, greatly enhancing model expressiveness. Experimental results on the DGraph dataset demonstrate superior AUC performance compared to baseline models, highlighting the feasibility and efficacy of the proposed approach in addressing GAD tasks in the scene with extremely low proportions of anomalous data.

Transfer-Aware Graph U-Net with Cross-Level Interactions for PolSAR Image Semantic Segmentation

Although graph convolutional networks have found application in polarimetric synthetic aperture radar (PolSAR) image classification tasks, the available approaches cannot operate on multiple graphs, which hinders their potential to generalize effective feature representations across different datasets. To overcome this limitation and achieve robust PolSAR image classification, this paper proposes a novel end-to-end cross-level interaction graph U-Net (CLIGUNet), where weighted max-relative spatial convolution is proposed to enable simultaneous learning of latent features from batch input. Moreover, it integrates weighted adjacency matrices, derived from the symmetric revised Wishart distance, to encode polarimetric similarity into weighted max-relative spatial graph convolution. Employing end-to-end trainable residual transformers with multi-head attention, our proposed cross-level interactions enable the decoder to fuse multi-scale graph feature representations, enhancing effective features from various scales through a deep supervision strategy. Additionally, multi-scale dynamic graphs are introduced to expand the receptive field, enabling trainable adjacency matrices with refined connectivity relationships and edge weights within each resolution. Experiments undertaken on real PolSAR datasets show the superiority of our CLIGUNet with respect to state-of-the-art networks in classification accuracy and robustness in handling unknown imagery with similar land covers.

Shortest Path of a Random Graph and its Application

The goal of this work is to provide an effective method for determining the shortest path in random graphs, which are complicated networks with random connectivity patterns. We have developed an algorithm that can identify the shortest path for both weighted and unweighted random graphs to accomplish our objective. As connectivity in these types of structures is changing, the algorithm adjusts to different edge weights and node configurations to provide fast and precise shortest path searching. The study shows that the suggested method performs more successfully in finding the shortest path throughout random graphs using comprehensive computations. Many networks, including social networks, granular networks, road traffic networks, etc., include nodes that can connect to one another and create random graphs in the present-day computational era. The outcomes demonstrate how flexible it is, which makes it a useful tool for practical uses in domains where random graph structures are common, like transportation networks, communication systems, and social networks. For illustration, we have taken into consideration an actual case study of communication road networks here. We have determined the shortest path of the road networks using our proposed algorithm, and the results have been presented. Better decision-making across a range of areas is made possible by this study, which advances effective algorithms designed for complicated and unpredictable network environments.

Network Analysis of Dyadic Burdens, Psychological Disorders, Psychological Resilience, and Illness- or Caregiving-Related Beliefs in Patients With Chronic Heart Failure and Their Caregivers.

Previous research has examined the dyadic health components consisting of dyadic burdens, psychological disorders, psychological resilience, and illness- or caregiving-related beliefs independently from each other in patients with chronic heart failure (CHF) and their caregivers, but there is a need for further insights into their interconnections. We aimed to explore the interconnections among dyadic health components in patients with CHF and their caregivers. We conducted a cross-sectional study, recruiting in a total of 355 patients with CHF and their 355 respective caregivers, totaling 710 individuals across the dyads. Assessments were conducted on symptom burden, caregiver burden, anxiety, depression, psychological resilience, perceived control, and caregiver self-efficacy. Network analysis was used regarding these constructs as nodes and their associations as edges. The strongest edge weight was observed between patients' anxiety and depression, followed by caregivers' anxiety and depression. Patients' depression exhibited the strongest edge weight with dyadic burdens. Caregiver burden was independently correlated with all nodes. Patients' symptom burden had fewer associations with the nodes within the caregiver community. Patients' anxiety, depression, and psychological resilience demonstrated the strongest and most influential correlations with other nodes. The findings illustrated extensive interconnections among dyadic health components in CHF dyads. These findings underscored the significance of managing and intervening with patients and caregivers as a dyadic whole. Given the strong and frequent associations of patients' anxiety, depression, and psychological resilience with other nodes in the network, interventions targeting these nodes may enhance the overall network health of CHF dyads.

Enhancing quality-driven iot operations for smart cities: A comprehensive approach for routing, security, and optimization

This study introduces a tailored data acquisition and communication framework for IoT smart applications, focusing on enhancing efficiency and system performance. The proposed Quality-Driven IoT Routing (EQR-SC) for smart cities utilizes IoT-enabled wireless sensor networks. Additionally, a noteworthy contribution is the introduction of the Chaotic Firefly Optimization (CFOA) algorithm for IoT sensor cluster formation, potentially optimizing the organization and efficiency of IoT sensor networks in smart cities. Trust-based cluster Head Selection is enhanced by employing the Weighted Clustering Algorithm (WCA), which assigns weights to nodes based on trustworthiness and relevant metrics to select reliable cluster heads. The proposal of a lightweight data encryption technique enhances data security among IoT sensors, ensuring the privacy and integrity of transmitted information. To optimize pathfinding within the IoT platform, the research employs the Bellman-Ford algorithm, ensuring efficient data routing while accommodating negative edge weights when necessary. Finally, a thorough performance analysis, conducted through network simulation (NS2), provides insights into the effectiveness of the proposed OQR-SC technique, allowing for valuable comparisons with existing state-of-the-art methods.

Deciphering cell types by integrating scATAC-seq data with genome sequences.

The single-cell assay for transposase-accessible chromatin using sequencing (scATAC-seq) technology provides insight into gene regulation and epigenetic heterogeneity at single-cell resolution, but cell annotation from scATAC-seq remains challenging due to high dimensionality and extreme sparsity within the data. Existing cell annotation methods mostly focus on the cell peak matrix without fully utilizing the underlying genomic sequence. Here we propose a method, SANGO, for accurate single-cell annotation by integrating genome sequences around the accessibility peaks within scATAC data. The genome sequences of peaks are encoded into low-dimensional embeddings, and then iteratively used to reconstruct the peak statistics of cells through a fully connected network. The learned weights are considered as regulatory modes to represent cells, and utilized to align the query cells and the annotated cells in the reference data through a graph transformer network for cell annotations. SANGO was demonstrated to consistently outperform competing methods on 55 paired scATAC-seq datasets across samples, platforms and tissues. SANGO was also shown to be able to detect unknown tumor cells through attention edge weights learned by the graph transformer. Moreover, from the annotated cells, we found cell-type-specific peaks that provide functional insights/biological signals through expression enrichment analysis, cis-regulatory chromatin interaction analysis and motif enrichment analysis.

Fitting Distances by Tree Metrics Minimizing the Total Error within a Constant Factor

We consider the numerical taxonomy problem of fitting a positive distance function \({\mathcal {D}:{S\choose 2}\rightarrow \mathbb {R}_{\gt 0}}\) by a tree metric. We want a tree T with positive edge weights and including S among the vertices so that their distances in T match those in \(\mathcal {D}\) . A nice application is in evolutionary biology where the tree T aims to approximate thebranching process leading to the observed distances in \(\mathcal {D}\) [Cavalli-Sforza and Edwards 1967]. We consider the total error, that is, the sum of distance errors over all pairs of points. We present a deterministic polynomial time algorithm minimizing the total error within a constant factor. We can do this both for general trees and for the special case of ultrametrics with a root having the same distance to all vertices in S . The problems are APX-hard, so a constant factor is the best we can hope for in polynomial time. The best previous approximation factor was O ((log n )(log log n )) by Ailon and Charikar [2005], who wrote “determining whether an O (1) approximation can be obtained is a fascinating question.”