Graph Matching Problem Research Articles

From Spin Glasses to Negative-Weight Percolation.

Spin glasses are prototypical random systems modelling magnetic alloys. One important way to investigate spin glass models is to study domain walls. For two dimensions, this can be algorithmically understood as the calculation of a shortest path, which allows for negative distances or weights. This led to the creation of the negative weight percolation (NWP) model, which is presented here along with all necessary basics from spin glasses, graph theory and corresponding algorithms. The algorithmic approach involves a mapping to the classical matching problem for graphs. In addition, a summary of results is given, which were obtained during the past decade. This includes the study of percolation transitions in dimension from up to and beyond the upper critical dimension , also for random graphs. It is shown that NWP is in a different universality class than standard percolation. Furthermore, the question of whether NWP exhibits properties of Stochastic–Loewner Evolution is addressed and recent results for directed NWP are presented.

Distributed Online Learning of Fog Computing Under Nonuniform Device Cardinality

Processing data around the point of capture, fog computing can support computationally demanding Internet-of-Things (IoT) services. Distributed online optimization is important given the size of IoT, but challenging due to time variations of random traffic and nonuniform connectivity (or cardinality) of edge servers and IoT devices. This paper presents a distributed online learning approach to asymptotically minimizing the time-average cost of fog computing in the absence of the a-priori knowledge on traffic randomness, for light-weight, and delay-tolerant application scenarios. Stochastic gradient descent is exploited to decouple the optimizations between time slots. A graph matching problem is then formulated for every time slot by decoupling and unifying the nonuniform cardinalities, and solved in a distributed manner by developing a new linear (1/2)-approximation method. We prove that the optimality loss resulting from the distributed approximate graph matching method can be compensated and diminish by increasing the learning time. Corroborated by simulations, the proposed distributed online learning is asymptotically optimal and superior to the state of the art in terms of throughput and energy efficiency.

Hypergraph Matching With an Entropy Barrier Function

Graph matching problem has been widely used in many fields. Due to the discrete property of graph matching problem, the continuous relaxation will result in a bad influence on the solution. Thus, we propose a hypergraph matching model with an entropy barrier function considering the discrete property of the assignment matrix. The model is solved by a nonmonotone active set projected Newton method whose nonmonotone line search technique has a good ability to accept the projected Newton stepsize. We evaluate our method through several experiments, which indicates that our method has better matching results than others.

Energy Disaggregation of Appliances Consumptions Using HAM Approach

Non-intrusive load monitoring (NILM) makes it possible for users to track the energy consumption of a household. In this paper, we present a new hybrid energy disaggregation approach named HAM. This event-based load disaggregation algorithm uses an improved multi-layer Hungarian algorithm to match appliances transient features and a supervised adaptive resonance theory mapping neural network (ARTMAP) to cluster steady features. This approach using a modified dual sliding window-based cumulative sum control chart algorithm (DSWC) to detect the transient event first, and we convert the multi-dimensional electrical features of appliances into a bipartite graph matching problem. This way, an improved Hungarian algorithm is proposed to find the perfect matching when appliances are in different states. For classification and identification, the ARTMAP network is used to learn and classify the steady features of appliances. Furthermore, we introduced a grey correlation evaluation and multiple matching strategies to increase the fitness and accuracy of the approach. Experimental results demonstrate that the proposed HAM outperforms the comparative algorithm in both our resident appliances dataset (RAD) and BLUED public dataset. The proposed approach could also facilitate NILM more applicable for common households.

Machine Learning Guided Atom Mapping of Metabolic Reactions.

Atom mapping of a chemical reaction is a mapping between the atoms in the reactant molecules and the atoms in the product molecules. It encodes the underlying reaction mechanism and, as such, constitutes essential information in computational studies in drug design. Various techniques have been investigated for the automatic computation of the atom mapping of a chemical reaction, approaching the problem as a graph matching problem. The graph abstraction of the chemical problem, though, eliminates crucial chemical information. There have been efforts for enhancing the graph representation by introducing the bond stabilities as edge weights, as they are estimated based on experimental evidence. Here, we present a fully automated optimization-based approach, named AMLGAM (Automated Machine Learning Guided Atom Mapping), that uses machine learning techniques for the estimation of the bond stabilities based on the chemical environment of each bond. The optimization method finds the reaction mechanism which favors the breakage/formation of the less stable bonds. We evaluated our method on a manually curated data set of 382 chemical reactions and ran our method on a much larger and diverse data set of 7400 chemical reactions. We show that the proposed method improves the accuracy over existing techniques based on results published by earlier studies on a common data set and is capable of handling unbalanced reactions.

On the chance-constrained minimum spanning k-core problem

A graph is called a k-core if every vertex has at least k neighbors. If the parameter k is sufficiently large relative to the number of vertices, a k-core is guaranteed to possess 2-hop reachability between all pairs of vertices. Furthermore, it is guaranteed to preserve those pairwise distances under arbitrary single-vertex deletion. Hence, the concept of a k-core can be used to produce 2-hop survivable network designs, specifically to design inter-hub networks. Formally, given an edge-weighted graph, the minimum spanningk-core problem seeks a spanning subgraph of the given graph that is a k-core with minimum total edge weight. For any fixed k, this problem is equivalent to a generalized graph matching problem and can be solved in polynomial time. This article focuses on a chance-constrained version of the minimum spanning k-core problem under probabilistic edge failures. We first show that this probabilistic version is NP-hard, and we conduct a polyhedral study to strengthen the formulation. The quality of bounds produced by the strengthened formulation is demonstrated through a computational study.

Seeded graph matching

Given two graphs, the graph matching problem is to align the two vertex sets so as to minimize the number of adjacency disagreements between the two graphs. The seeded graph matching problem is the graph matching problem when we are first given a partial alignment that we are tasked with completing. In this article, we modify the state-of-the-art approximate graph matching algorithm “FAQ” of Vogelstein et al. (2015) to make it a fast approximate seeded graph matching algorithm, adapt its applicability to include graphs with differently sized vertex sets, and extend the algorithm so as to provide, for each individual vertex, a nomination list of likely matches. We demonstrate the effectiveness of our algorithm via simulation and real data experiments; indeed, knowledge of even a few seeds can be extremely effective when our seeded graph matching algorithm is used to recover a naturally existing alignment that is only partially observed.

Exploiting multi-layer graph factorization for multi-attributed graph matching

Multi-layer graph matching methods effectively solve multi-attributed graph matching problems based on the multi-layer structure adopted to address the ambiguity and uncertainty arisen from the attribute integration. However, despite of its effectiveness for matching multi-attributed graphs, the approach has a long way to apply in the practical environment due to the scalability problem caused by a huge matrix for describing the multi-layer structure. In this paper, we propose a novel multi-layer graph matching algorithm based on the multi-layer graph factorization to address the issue. The main contribution of this research is three-fold. First, we propose a factorization method that decomposes the huge multi-layer matrix into several small matrices for efficiency. Second, we reformulate the original multi-layer matching problem into two relaxed problems by using the factorized matrices. Third, based on the relaxed problems, we propose a multi-layer graph matching algorithm inspired from the convex-concave relaxation procedure. In our extensive experiments on the synthetic and real-image datasets, the proposed method exhibits better performance than state-of-the-art algorithms based on the single-layer structure.

Finding Correlated Patterns via High-Order Matching for Multiple Sourced Biological Data.

The emergence of multidimensional genomic data poses new challenges in data analysis. Finding correlated patterns within multiple-sourced biological data is useful in understanding potential interactions between the multimodal genomic data. Multidimensional genomic data contain multiple genomic data types, and different types of genomic data have different scales and units. These data cannot simply be aggregated for analysis. To address this issue, a correlated pattern discovery model incorporating prior knowledge is proposed. Tensor similarity is used to measure the correlation between common patterns. The model is combined with prior knowledge, the expression of which is transformed into constraints. Efficient numerical solutions are designed and analyzed. The proposed method is shown to perform robustly and effectively with both simulated data and real biological data. We conduct experiments on five real cancer data sets to reveal various cancer subtypes. A survival analysis of these subtypes confirms the effectiveness of the model. We introduce a correlated pattern discovery model incorporating prior knowledge. This model is meaningful for the realization of personalized diagnoses by doctors in the treatment of cancer and other diseases. The problem of finding correlated patterns from multiple-sourced biological data was formulated as a high-order graph matching problem, and the prior knowledge data were seamlessly incorporated into the matching model.

Subgraph learning for graph matching

Graph matching is a powerful tool for computer vision, distance measure and machine learning. However, many factors influences the accuracy of matching. The outliers is a key problem in the process of matching. In this paper, a novel approach is proposed to handle graph matching problem based on Markov Chain Monte Carlo framework. By constructing a target distribution, the proposed can perform a process of sampling to maximize the graph matching objective. In this process, our method can effectively save matching pairwise under one-to-one matching constraints and also avoid the effect of outliers and deformation. The corresponding experiments on synthetic graphs, real images and view-based 3D model retrieval demonstrate the superiority of the proposed method.

Complete Your Mobility: Linking Trajectories Across Heterogeneous Mobility Data Sources

Nowadays, human activities and movements are recorded by a variety of tools, forming different trajectory sets which are usually isolated from one another. Thus, it is very important to link different trajectories of one person in different sets to provide massive information for facilitating trajectory mining tasks. Most prior work took advantages of only one dimensional information to link trajectories and can link trajectories in a one-to-many manner (providing several candidate trajectories to link to one specific trajectory). In this paper, we propose a novel approach called one-to-one constraint trajectory linking with multi-dimensional information (OCTL) that links the corresponding trajectories of one person in different sets in a one-to-one manner. We extract multidimensional features from different trajectory datasets for corresponding relationships prediction, including spatial, temporal and spatio-temporal information, which jointly describe the relationships between trajectories. Using these features, we calculate the corresponding probabilities between trajectories in different datasets. Then, we formulate the link inference problem as a bipartite graph matching problem and employ effective methods to link one trajectory to another. Moreover, the advantages of our approach are empirically verified on two real-world trajectory sets with convincing results.

Information Processing and Wireless Energy Harvesting in Interference-Aware Public Safety Networks

In a public safety environment, user equipments (UEs) located within the coverage area of evolved NodeB, relay network services to out-of-coverage UEs. However, relay UEs in public safety environments are typically energy constrained and cannot operate indefinitely without recharging. Radio frequency energy harvesting has been proposed as a solution for recharging wireless UEs. In this paper, we propose a scheme for extending the lifetime of a public safety network by wirelessly charging relay UEs. In addition, we propose a relay selection method considering the battery status of relay UEs. The proposed relay selection is defined as a bipartite graph matching problem and the optimal relay is obtained through matching games technique. The proposed scheme not only improves the network lifetime but also extend the network coverage. We also conduct system level simulations to evaluate the performance of the proposed scheme. Simulation results show that the overall performance of the system is improved in terms of achievable throughput and network lifetime.

Approach for sequential image interpretation using a priori binary perceptual topological and photometric knowledge and k-means-based segmentation.

The proposed approach exploits a priori known qualitative inclusion and photometric relationships between image regions, represented by oriented graphs. Our work assumes a sequential image segmentation procedure where regions are progressively segmented and recognized by associating them with corresponding nodes in graphs related to the prior knowledge. The main contribution concerns the parameterization of the k-means clustering algorithm, to be used during the segmentation procedure, and the graph-matching-based identification of resulting clusters, corresponding to regions declared in graphs. The parameterization of k-means is based on known relationships as well as on regions that have been segmented and recognized at previous steps. Parameters are the region of interest within which k-means clustering is constrained, the number of clusters, and seeding constraints. Photometric relationships built from resulting clusters are matched with a priori known relationships to identify each cluster, this being formulated as an exact graph-matching problem. The potential of this approach is studied in four use cases involving real gray-scale and color images with dedicated sequential analysis procedures. Processing results are compared with those obtained without the proposed parameterization of k-means, as well as with some other clustering approaches. Results show the relevance of our approach, in particular in terms of segmentation accuracy, computation time, and seeding reliability.

Multi-Attributed Graph Matching With Multi-Layer Graph Structure and Multi-Layer Random Walks.

This paper addresses the multi-attributed graph matching problem, which considers multiple attributes jointly while preserving the characteristics of each attribute for graph matching. Since most of conventional graph matching algorithms integrate multiple attributes to construct a single unified attribute in an oversimplified manner, the information from multiple attributes is often not completely utilized. In order to solve this problem, we propose a novel multi-layer graph structure that can preserve the characteristics of each attribute in separated layers, and also propose a multi-attributed graph matching algorithm based on random walk centrality with the proposed multi-layer graph structure. We compare the proposed algorithm with other state-of-the-art graph matching algorithms based on a single-layer structure using synthetic and real data sets and demonstrate the superior performance of the proposed multi-layer graph structure and the multi-attributed graph matching algorithm.

Learning error-correcting graph matching with a multiclass neural network

Many tasks in computer vision and pattern recognition are formulated as graph matching problems. Despite the NP-hard nature of such problems, fast and accurate approximations have led to significant progress in a wide range of applications. However, learning graph matching from observed data, remains a challenging issue. In practice, the node correspondences ground truth is rarely available. This paper presents an effective scheme for optimizing the graph matching problem in a classification context. For this, we propose a representation that is based on a parametrized model graph, and optimize the associated parameters. The objective of the optimization problem is to increase the classification rate. Experimental results on seven public datasets demonstrate the effectiveness (in terms of accuracy and speed) of our approach compared to four reference graph classifiers.

Maximum Weight Matching Using Odd-Sized Cycles: Max-Product Belief Propagation and Half-Integrality

We study the maximum weight matching (MWM) problem for general graphs through the max-product belief propagation (BP) and related Linear Programming (LP). The BP approach provides distributed heuristics for finding the maximum a posteriori (MAP) assignment in a joint probability distribution represented by a graphical model (GM), and respective LPs can be considered as continuous relaxations of the discrete MAP problem. It was recently shown that a BP algorithm converges to the correct MAP/MWM assignment under a simple GM formulation of MWM, as long as the corresponding LP relaxation is tight. First, under the motivation for forcing the tightness condition, we consider a new GM formulation of MWM, say C-GM, using non-intersecting odd-sized cycles in the graph; the new corresponding LP relaxation, say C-LP, becomes tight for more MWM instances. However, the tightness of C-LP now does not guarantee such convergence and correctness of the new BP on C-GM. To address the issue, we introduce a novel graph transformation applied to C-GM, which results in another GM formulation of MWM, and prove that the respective BP on it converges to the correct MAP/MWM assignment, as long as C-LP is tight. Finally, we also show that C-LP always has half-integral solutions, which leads to an efficient BP-based MWM heuristic consisting of making sequential, “cutting plane”, modifications to the underlying GM. Our experiments show that this BP-based cutting plane heuristic performs, as well as that based on traditional LP solvers.

A local branching heuristic for solving a Graph Edit Distance Problem

The Graph Edit Distance (GED) problem is a very interesting problem that relates to Graph Matching (GM) problems, wherever a graph models a problem-defined pattern. Solving the GED problem leads to compute the matching between two given graphs by minimizing their GED, which is also called a dissimilarity measure between them. It has many application domains such as structural pattern recognition and chemical Structure-Activity Relationships (SAR). GEDEnA (Edges no Attributes) is a sub-problem of the GED problem that deals with a special type of graphs where edges do not carry attributes. They are both NP-hard problems and it is difficult to find the optimal solution in reasonable time. Many heuristics exist in the literature that give suboptimal solutions. Some other works have used mathematical programming tools to model and solve these two problems. The present work takes advantage of a powerful Mixed Integer Linear Program (MILP) formulation and proposes a heuristic called Local Branching to solve the GEDEnA problem. Mainly, a MILP model is iteratively modified by adding additional constraints to define neighborhoods in the solution space, which are explored using a black-box solver. A problem-dependent exploration is performed to find efficient solutions. Lastly, the proposed heuristic is evaluated considering two factors: its computational time and solution quality against the heuristics available in the literature and exact methods. The computational experiments reported in this paper show that the proposed local branching heuristic strongly outperforms the existing heuristics for the GEDEnA problem.

Scene Video Text Tracking With Graph Matching

Video has become one of the dominant data resources with the development of the Internet. As a result, the structured sorting of videos, which can be used for storage and extraction, represents a growing concern in the community. In particular, the text within videos can carry rich semantic information, leading to many novel studies wherein text tracking and recognition are performed. One essential step in text tracking involves template matching. In general, the adjacent matrices are modeled to represent the extracted tracking object features. Then, often, the Hungarian algorithm is applied to find the correspondence pairs between consecutive frames. In many works, text features are extracted based on morphological features such as color histograms and aspect ratios. However, under those features, similar text objects are not sufficiently distinguishable to make a distinction between them. To address this issue, we regard the template matching task as a graph matching problem. The main novelty involves a graph matching approach that utilizes the relationship between two trajectories or two objects, whereby a graph matching solver can be readily used in our tracking system. By utilizing the content information, the mismatch between the same object among different frames is effectively reduced. The experimental results demonstrate that the tracker with the graph matching method tends to increase the valid correspondence of trajectories and candidate objects.

Exact Recovery with Symmetries for the Doubly Stochastic Relaxation

Graph matching, or quadratic assignment, is the problem of labeling the vertices of two graphs so that they are as similar as possible. A common method for approximately solving the NP-hard graph matching problem is relaxing it to a convex optimization problem over the set of doubly stochastic (DS) matrices. Recent analysis has shown that for almost all pairs of isomorphic and asymmetric graphs, the DS relaxation succeeds in correctly retrieving the isomorphism between the graphs. Our goal in this paper is to analyze the case of symmetric isomorphic graphs. This goal is motivated by shape matching applications where the graphs of interest usually have reflective symmetry. For symmetric problems the graph matching problem has multiple isomorphisms, and so convex relaxations admit all convex combinations of these isomorphisms as viable solutions. If the convex relaxation does not admit any additional superfluous solution, we say that it is convex exact. We show that convex exactness depends strongly on the symmetry group of the graphs; for a fixed symmetry group $G$, the DS relaxation either will be convex exact for almost all pairs of isomorphic graphs with symmetry group $G$, or will fail for all such pairs. We show that for reflective groups with at least one full orbit, convex exactness holds almost everywhere. The importance of this result stems from the fact that such groups are the most common symmetry groups in shape matching applications. Finally, we provide some simple examples of nonreflective symmetry groups for which convex exactness always fails. These examples also show that the property of having generic convex exactness is not invariant under group isomorphisms. When convex exactness holds, the isomorphisms of the graphs are the extreme points of the convex solution set and the task of finding an isomorphism in polynomial time is tractable. We discuss several common algorithms capable of retrieving an isomorphism in this case. We also show that interior point algorithms will not return an isomorphism in this case but will return the centroid of the set of isomorphisms, which gives a “fuzzy encoding” of the symmetries of the shape. In fact, for any symmetry group $G$, the centroid solution can be recovered efficiently for almost all pairs of isomorphic graphs with symmetry group $G$.

The intensional semantic conceptual graph matching algorithm based on conceptual sub-graph weight self-adjustment

Semantic computing is an important task in the research on natural language processing. On solving the problem of the inaccurate conceptual graph matching, this paper proposes an algorithm to compute the similarity of conceptual graphs, based on conceptual sub-graph weight self-adjustment. The algorithm works by basing itself on the intensional logic model of Chinese concept connotation, using intensional semantic conceptual graph as knowledge representation method and combining itself with the computation method of E-A-V structures. When computing the similarity of conceptual graphs, the algorithm can give the homologous weight to the sub-graph according to the proportion of how much information the sub-graph contains in the whole conceptual graph. Therefore, it can achieve better similarity results, which has also been proved in the experiments of this paper.