Maximum Common Subgraph Problem Research Articles

A Method for Matching Information of Substation Secondary Screen Cabinet Terminal Block Based on Artificial Intelligence

The matching of schematic diagrams and physical information of terminal blocks in substation secondary screen cabinets plays a crucial role in the operation and maintenance of substations. To enhance the automation level of this task and reduce labor costs, a method for identifying and matching information of terminal blocks in substation secondary screen cabinets based on artificial intelligence is investigated in this paper. Initially, multi-layer object detection networks, tailored to the characteristics of both the schematic diagrams and the physical entities in substation secondary screen cabinets, are designed for the precise extraction of information. Subsequently, network topologies for both the schematic and physical systems are established using the Neo4j database, which allows for the digital storage of information in the substation secondary screen cabinet systems. Finally, the branch-and-bound method, improved by the application of a multi-modular graph convolutional network (MGCN) and deep Q-network (DQN), is employed to solve the maximum common subgraph (MCS) problem, resulting in the rapid and efficient matching of schematic and physical data.

Formulations for the maximum common edge subgraph problem

The Maximum Common Edge Subgraph Problem (MCES), given two graphs G and H, asks about finding a maximum common subgraph between those two graphs with a maximal number of edges. We present new integer programming formulations for MCES and compare the performances of those formulations with earlier formulations from the literature.

Recursive-Parallel Algorithm for Solving the Maximum Common Subgraph Problem

The paper proposes an algorithm for solving the problem of finding the maximum common subgraph. Both the sequential and the parallel version of the algorithm, their software implementation are described, and an experimental study of their effectiveness is carried out. This problem is one of the most famous NP-complete problems. Its solution may be required when solving many practical problems related to the study of complex structures. We solve it in a statement when we need to find all possible isomorphisms of the found common subgraph. Due to the extremely high complexity of the problem, it is natural to want to speed up its solution by parallelizing the algorithm. To organize parallel computing, the author used the RPM_ParLib library. It allows to develop effective applications for parallel computing on a local network under the control of the runtime environment .NET Framework.The library supports a recursive-parallel programming style and provides effective work distribution and dynamic load balancing of computational modules during program execution. It can be used for applications written in any programming language supported by the .NET Framework. The purpose of the numerical experiment was to study the acceleration achieved due to the recursive-parallel organization of calculations. For the experiment, the author developed a special application in the C# language designed to generate various sets of initial data with specified parameters. The paper describes the features of the generated initial pairs of graphs, as well as the results obtained during the experiment.

The Multi-Maximum and Quasi-Maximum Common Subgraph Problem

The Maximum Common Subgraph problem has been long proven NP-hard. Nevertheless, it has countless practical applications, and researchers are still searching for exact solutions and scalable heuristic approaches. Driven by applications in molecular science and cyber-security, we concentrate on the Maximum Common Subgraph among an indefinite number of graphs. We first extend a state-of-the-art branch-and-bound procedure working on two graphs to N graphs. Then, given the high computational cost of this approach, we trade off complexity for accuracy, and we propose a set of heuristics to approximate the exact solution for N graphs. We analyze sequential, parallel multi-core, and parallel-many core (GPU-based) approaches, exploiting several leveraging techniques to decrease the contention among threads, improve the workload balance of the different tasks, reduce the computation time, and increase the final result size. We also present several sorting heuristics to order the vertices of the graphs and the graphs themselves. We compare our algorithms with a state-of-the-art method on publicly available benchmark sets. On graph pairs, we are able to speed up the exact computation by a 2× factor, pruning the search space by more than 60%. On sets of more than two graphs, all exact solutions are extremely time-consuming and of a complex application in many real cases. On the contrary, our heuristics are far less expensive (as they show a lower-bound for the speed up of 10×), have a far better asymptotic complexity (with speed ups up to several orders of magnitude in our experiments), and obtain excellent approximations of the maximal solution with 98.5% of the nodes on average.

Comparison of Atom Maps

The computation of reliable, chemically correct atom maps from educt/product pairs has turned out to be a difficult problem in cheminformatics because the chemically correct solution is not necessarily an optimal solution for combinatorial formulations such as maximum common subgraph problems. As a consequence, competing models have been devised and compared in extensive benchmarking studies. Due to isomorphisms among products and educts it is not immediately obvious, however, when two atom maps for a given educt/product pairs are the same. We formalize here the equivalence of atom maps and show that equivalence of atom maps is in turn equivalent to the isomorphism of labeled auxiliary graphs. In particular, we demonstrate that Fujita's Imaginary Transition State can be used for this purpose. Numerical experiments show that practical feasibility. Generalizations to the equivalence of subgraph matches, double pushout graph transformation rules, and mechanisms of multi-step reactions are discussed briefly.

The Maximum Common Subgraph Problem: A Parallel and Multi-Engine Approach

The maximum common subgraph of two graphs is the largest possible common subgraph, i.e., the common subgraph with as many vertices as possible. Even if this problem is very challenging, as it has been long proven NP-hard, its countless practical applications still motivates searching for exact solutions. This work discusses the possibility to extend an existing, very effective branch-and-bound procedure on parallel multi-core and many-core architectures. We analyze a parallel multi-core implementation that exploits a divide-and-conquer approach based on a thread pool, which does not deteriorate the original algorithmic efficiency and it minimizes data structure repetitions. We also extend the original algorithm to parallel many-core GPU architectures adopting the CUDA programming framework, and we show how to handle the heavily workload-unbalance and the massive data dependency. Then, we suggest new heuristics to reorder the adjacency matrix, to deal with “dead-ends”, and to randomize the search with automatic restarts. These heuristics can achieve significant speed-ups on specific instances, even if they may not be competitive with the original strategy on average. Finally, we propose a portfolio approach, which integrates all the different local search algorithms as component tools; such portfolio, rather than choosing the best tool for a given instance up-front, takes the decision on-line. The proposed approach drastically limits memory bandwidth constraints and avoids other typical portfolio fragility as CPU and GPU versions often show a complementary efficiency and run on separated platforms. Experimental results support the claims and motivate further research to better exploit GPUs in embedded task-intensive and multi-engine parallel applications.

A Learning Based Branch and Bound for Maximum Common Subgraph Related Problems

The performance of a branch-and-bound (BnB) algorithm for maximum common subgraph (MCS) problem and its related problems, like maximum common connected subgraph (MCCS) and induced Subgraph Isomorphism (SI), crucially depends on the branching heuristic. We propose a branching heuristic inspired from reinforcement learning with a goal of reaching a tree leaf as early as possible to greatly reduce the search tree size. Experimental results show that the proposed heuristic consistently and significantly improves the current best BnB algorithm for the MCS, MCCS and SI problems. An analysis is carried out to give insight on why and how reinforcement learning is useful in the new branching heuristic.

Improved Hardness of Maximum Common Subgraph Problems on Labeled Graphs of Bounded Treewidth and Bounded Degree

We consider the maximum common connected edge subgraph problem and the maximum common connected induced subgraph problem for simple graphs with labeled vertices (or labeled edges). The former is to find a connected graph with the maximum number of edges that is isomorphic to a subgraph of each of the two input graphs. The latter is to find a common connected induced subgraph with the maximum number of vertices. We prove that both problems are NP-hard for 3-outerplanar labeled graphs even if the maximum vertex degree is bounded by 4. Since the reductions used in the proofs construct graphs with treewidth at most 4, both problems are NP-hard also for such graphs, which significantly improves the previous hardness results for graphs with treewidth 11. We also present improved exponential-time algorithms for both problems on labeled graphs of bounded treewidth and bounded vertex degree.

Improving graphs of cycles approach to structural similarity of molecules.

This paper focuses on determining the structural similarity of two molecules, i.e., the similarity of the interconnection of all the elementary cycles in the corresponding molecular graphs. In this paper, we propose and analyze an algorithmic approach based on the resolution of the Maximum Common Edge Subgraph (MCES) problem with graphs representing the interaction of cycles molecules. Using the ChEBI database, we compare the effectiveness of this approach in terms of structural similarity and computation time with two calculations of similarity of molecular graphs, one based on the MCES, the other on the use of different fingerprints (Daylight, ECFP4, ECFP6, FCFP4, FCFP6) to measure Tanimoto coefficient. We also analyze the obtained structural similarity results for a selected subset of molecules.

On maximum common subgraph problems in series–parallel graphs

The complexity of the maximum common connected subgraph problem in partial k-trees is still not fully understood. Polynomial-time solutions are known for degree-bounded outerplanar graphs, a subclass of the partial 2-trees. On the other hand, the problem is known to be NP-hard in vertex-labeled partial 11-trees of bounded degree. We consider series–parallel graphs, i.e., partial 2-trees. We show that the problem remains NP-hard in biconnected series–parallel graphs with all but one vertex of degree 3 or less. A positive complexity result is presented for a related problem of high practical relevance which asks for a maximum common connected subgraph that preserves blocks and bridges of the input graphs. We present a polynomial time algorithm for this problem in series–parallel graphs, which utilizes a combination of BC- and SP-tree data structures to decompose both graphs.

CytoMCS: A Multiple Maximum Common Subgraph Detection Tool for Cytoscape.

Comparative analysis of biological networks is a major problem in computational integrative systems biology. By computing the maximum common edge subgraph between a set of networks, one is able to detect conserved substructures between them and quantify their topological similarity. To aid such analyses we have developed CytoMCS, a Cytoscape app for computing inexact solutions to the maximum common edge subgraph problem for two or more graphs. Our algorithm uses an iterative local search heuristic for computing conserved subgraphs, optimizing a squared edge conservation score that is able to detect not only fully conserved edges but also partially conserved edges. It can be applied to any set of directed or undirected, simple graphs loaded as networks into Cytoscape, e.g. protein-protein interaction networks or gene regulatory networks. CytoMCS is available as a Cytoscape app at http://apps.cytoscape.org/apps/cytomcs.

Between Subgraph Isomorphism and Maximum Common Subgraph

When a small pattern graph does not occur inside a larger target graph, we can ask how to find "as much of the pattern as possible" inside the target graph. In general, this is known as the maximum common subgraph problem, which is much more computationally challenging in practice than subgraph isomorphism. We introduce a restricted alternative, where we ask if all but k vertices from the pattern can be found in the target graph. This allows for the development of slightly weakened forms of certain invariants from subgraph isomorphism which are based upon degree and number of paths. We show that when k is small, weakening the invariants still retains much of their effectiveness. We are then able to solve this problem on the standard problem instances used to benchmark subgraph isomorphism algorithms, despite these instances being too large for current maximum common subgraph algorithms to handle. Finally, by iteratively increasing k, we obtain an algorithm which is also competitive for the maximum common subgraph

Recognizing Topological Analogy in Semantic Network

Recognizing topological analogy between the parts of semantic network seems to be very important step in the process of semantic categorization and interpretation of data that are embedded into the semantic network. Considering the semantic network as a set of graphs, recognition of topological analogy between the parts of semantic network can be treated as maximum common subgraph problem which falls in the group of exact graph matching problems. In this paper authors propose a new algorithm for maximum common subgraph detection aimed to a specific semantic network called Active Semantic Model (ASM). This semantic network can be represented as the set of labeled directed multigraphs with unique node labels. The structure of these graphs is specific because associations or edges are labeled with several attributes and some of them are related to nodes connected by edge. That kind of association-oriented structure enables associations or edges to play key role in the process of semantic categorization and interpretation of data. Furthermore, this kind of structure enables modeling semantic contexts in a form of semantically designated graphs (of associations). Proposed algorithm is capable of recognizing simultaneously maximum common subgraph of input graph and each of the graphs representing different contexts in ASM semantic network.

The maximum common edge subgraph problem: A polyhedral investigation

In the Maximum Common Edge Subgraph Problem (MCES), given two graphs G and H with the same number of vertices, one has to find a common subgraph of G and H (not necessarily induced) with the maximum number of edges. This problem arises in parallel programming environments, and was first defined in Bokhari (1981) [2]. This paper presents a new integer programming formulation for the MCES and a polyhedral study of this model. Several classes of valid inequalities are identified, most of which are shown to define facets. These findings were incorporated into a branch&cut algorithm we implemented. Experimental results with this algorithm are reported.

Using local similarity measures to efficiently address approximate graph matching

In this paper, we investigate heuristics for Approximate Graph Matching (AGM), in particular when it can be formulated as a Maximum Common Edge Subgraph (MCES) problem. First, we observe empirically that initializing a local search with a tiny subset of a known optimal solution always results in much better solutions than starting with an empty solution. The main challenge could then be to retrieve such small subsets for any problem instance. For this purpose, we propose several local similarity measures and evaluate their ability to predict node matches which could be used to start a local search. The resulting algorithm (SIM-T) is a classic tabu algorithm that is initialized by a greedy procedure relying mainly, in its earliest steps, on similarity measures.We conducted experiments on a large collection of random graphs of various orders (from 50 to 3000 nodes) and densities. Results obtained are mostly excellent, especially on similar pairs of labeled graphs. Comparisons made with two recent state-of-the-art algorithms–“BP” and “PATH”–indicate a superiority of our approach, in terms of both scores and computation times.

On the use of similarity metrics for approximate graph matching

Abstract In this paper, we investigate heuristics for approximate graph matching, in particular its formulation as a Maximum Common Edge Subgraph problem. Our experiments suggest that a small percentage of accurate node matches is sufficient to get near optimal solutions using a simple hill-climbing. The real challenge could then be to somehow drag the search in this zone. For this purpose, we discuss the use of similarity measures. We present and assess the performance of two similarity measures. Very good results were obtained on labeled graphs.

A branch&cut algorithm for the maximum common edge subgraph problem

Abstract We investigate the Maximum Common Edge Subgraph Problem (MCES) defined as follows. Given two graphs G and H with the same number of vertices, find a common subgraph of G and H (not necessary induced) with the maximum number of edges. This problem arises in parallel programming environments, and was first defined by Bokhari in [S. Bokhari, On the mapping problem, IEEE Trans. Comput., C-30(3), 1981]. We present a new integer programming formulation for the MCES problem and carry out a polyhedral investigation of this model. A number of valid inequalities are identified, most of which are facet-defining. We also report on computational experiments.

On parameterized complexity of the Multi-MCS problem

We introduce the maximum common subgraph problem for multiple graphs ( Multi-MCS) inspired by various biological applications such as multiple alignments of gene sequences, protein structures, metabolic pathways, or protein–protein interaction networks. Multi-MCS is a generalization of the two-graph Maximum Common Subgraph problem (MCS). On the basis of the framework of parameterized complexity theory, we derive the parameterized complexity of Multi-MCS for various parameters for different classes of graphs. For example, for directed graphs with labeled vertices, we prove that the parameterized m -Multi-MCS problem is W [2]-hard, while the parameterized k -Multi-MCS problem is W [ t ] -hard ( ∀ t ≥ 1 ), where m and k are the size of the maximum common subgraph and the number of multiple graphs, respectively. We show similar results for other parameterized versions of the Multi-MCS problem for directed graphs with vertex labels and undirected graphs with vertex and edge labels by giving linear FPT reductions of the problems from parameterized versions of the longest common subsequence problem. Likewise, for unlabeled undirected graphs, we show that a parameterized version of the Multi-MCS problem with a fixed number of input graphs is W [1]-complete by showing a linear FPT reduction to and from a parameterized version of the maximum clique problem.

Structural pattern matching of nonribosomal peptides

BackgroundNonribosomal peptides (NRPs), bioactive secondary metabolites produced by many microorganisms, show a broad range of important biological activities (e.g. antibiotics, immunosuppressants, antitumor agents). NRPs are mainly composed of amino acids but their primary structure is not always linear and can contain cycles or branchings. Furthermore, there are several hundred different monomers that can be incorporated into NRPs. The NORINE database, the first resource entirely dedicated to NRPs, currently stores more than 700 NRPs annotated with their monomeric peptide structure encoded by undirected labeled graphs. This opens a way to a systematic analysis of structural patterns occurring in NRPs. Such studies can investigate the functional role of some monomeric chains, or analyse NRPs that have been computationally predicted from the synthetase protein sequence. A basic operation in such analyses is the search for a given structural pattern in the database.ResultsWe developed an efficient method that allows for a quick search for a structural pattern in the NORINE database. The method identifies all peptides containing a pattern substructure of a given size. This amounts to solving a variant of the maximum common subgraph problem on pattern and peptide graphs, which is done by computing cliques in an appropriate compatibility graph.ConclusionThe method has been incorporated into the NORINE database, available at . Less than one second is needed to search for a pattern in the entire database.

A DNA-based graph encoding scheme with its applications to graph isomorphism problems

Feynman first proposed DNA-based computation in 1961, but his idea was not implemented by experiment for a few decades. By properly manipulating DNA strands as the input instance of the Hamiltonian path problem, Adleman succeeded in solving the problem in a test tube. Since the experimental demonstration of its feasibility, DNA-based computing has been applied to a number of decision or combinatorial optimization problems. In this paper, we propose a DNA-based graph encoding scheme which can be used to solve some intractable graph problems, such as the subgraph isomorphism problem and its generalized problem – the maximum common subgraph problem, which are known to be NP-complete problems, in the Adleman–Lipton model using polynomial number of basic biological operations.