Edges In Subgraph Research Articles

Estimates of the Number of Edges in Subgraphs of Johnson Graphs

Estimates of the Number of Edges in Subgraphs of Johnson Graphs

Estimates of the Number of Edges in Subgraphs of Johnson Graphs

В работе мы рассматриваем специальные дистанционные графы и оцениваем число ребер в их подграфах. Полученные оценки улучшают известные результаты. Библиография: 19 названий.

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.

How do centrality measures help to predict similarity patterns in molecular chemical structural graphs?

The proposed work uses centrality measures based heuristic method to improve the efficiency of the solution for the similarity search problem in molecular chemical graphs by effectively identifying central candidate or representative candidate nodes, which simplify the complex processes involved while detecting a large-sized maximal common connected edge subgraph. After analyzing the structure of the two input molecular chemical graphs, a Tensor Product graph is created. This newly built graph is further analyzed to get the similarity pattern of the input graphs. It is an open problem to decide which centrality measure selects the best central candidate node in Tensor Product graphs to get a large maximal common connected edge graph. Since each centrality measure is analyses, the given graph is uniquely based on its own specific aspects. The proposed work offers directions on using various centrality measures to determine a big-sized maximal common connected subgraph for two molecular chemical input graphs. It also analyses seven centrality measures to select the best candidate node in the Tensor Product graph of two input chemical molecular graphs. Based on the obtained results, the betweenness centrality and degree centrality measures exclusively help to get large-sized similarity patterns.

Evaluating Mineral Lattices as Evolutionary Proxies for Metalloprotein Evolution.

Protein coordinated iron-sulfur clusters drive electron flow within metabolic pathways for organisms throughout the tree of life. It is not known how iron-sulfur clusters were first incorporated into proteins. Structural analogies to iron-sulfide minerals present on early Earth, suggest a connection in the evolution of both proteins and minerals. The availability of large protein and mineral crystallographic structure data sets, provides an opportunity to explore co-evolution of proteins and minerals on a large-scale using informatics approaches. However, quantitative comparisons are confounded by the infinite, repeating nature of the mineral lattice, in contrast to metal clusters in proteins, which are finite in size. We address this problem using the Niggli reduction to transform a mineral lattice to a finite, unique structure that when translated reproduces the crystal lattice. Protein and reduced mineral structures were represented as quotient graphs with the edges and nodes corresponding to bonds and atoms, respectively. We developed a graph theory-based method to calculate the maximum common connected edge subgraph (MCCES) between mineral and protein quotient graphs. MCCES can accommodate differences in structural volumes and easily allows additional chemical criteria to be considered when calculating similarity. To account for graph size differences, we use the Tversky similarity index. Using consistent criteria, we found little similarity between putative ancient iron-sulfur protein clusters and iron-sulfur mineral lattices, suggesting these metal sites are not as evolutionarily connected as once thought. We discuss possible evolutionary implications of these findings in addition to suggesting an alternative proxy, mineral surfaces, for better understanding the coevolution of the geosphere and biosphere.

Lower Bound on the Minimum Number of Edges in Subgraphs of Johnson Graphs

Lower Bound on the Minimum Number of Edges in Subgraphs of Johnson Graphs

Estimate of the Number of Edges in Subgraphs of a Johnson Graph

New estimates for the minimum number of edges in subgraphs of a Johnson graph are obtained.

Complexity indices for the traveling salesman problem continued

We consider the symmetric traveling salesman problem (TSP) with instances represented by complete graphs G with distances between cities as edge weights. A complexity index is an invariant of an instance I by which we predict the execution time of an exact TSP algorithm for I. In the paper [5] we have considered some short edge subgraphs of G and defined several new invariants related to their connected components. Extensive computational experiments with instances on 50 vertices with the uniform distribution of integer edge weights in the interval [1,100] show that there exists correlation between the sequences of selected invariants and the sequence of execution times of the well-known TSP Solver Concorde. In this paper we extend these considerations for instances up to 100 vertices.

Cohort analytics: efficiency and applicability

The abundant availability of health-care data calls for effective analysis methods to help medical experts gain a better understanding of their patients and their health. The focus of existing work has been largely on prediction. In this paper, we introduce Core, a framework for cohort “representation” and “exploration.” Our contributions are twofold: First, we formalize cohort representation as the problem of aggregating the trajectories of its patients. This problem is challenging because cohorts often consist of hundreds of patients who underwent medical actions of various types at different points in time. We prove that producing a representative cohort trajectory is NP-complete with a reduction in the multiple sequence alignment problem. We propose a heuristic that extends the Needleman–Wunsch algorithm for sequence matching to handle temporal sequences. To further improve cohort representation efficiency, we introduce “trajectory families” and “stratified sampling.” Our second contribution is formalizing the problem of cohort exploration as finding a set of cohorts that are similar to a cohort of interest and that maximize entropy. This problem is challenging because the potential number of similar cohorts is huge. We prove NP-completeness with a reduction in the maximum edge subgraph problem. To address complexity, we develop a multi-staged approach based on limiting the search space to “contrast cohorts.” To speed up the computation of cohort similarity, we use “event sets” that are inspired from the double dictionary encoding proposed for keyword search. Moreover, we explore the usefulness and efficiency of Core using an extensive set of qualitative and quantitative experiments on two real health-care datasets. In a user study with medical experts, we show that Core reduces time-to-insight from hours to seconds and helps them find better insights than baseline approaches. Moreover, we show that the obtained cohort representations offer the right trade-off between quality and performance. We study the benefits of trajectory families and stratified sampling for cohort representation and show their applicability on large and heterogeneous cohorts. We also show the benefit of event sets for cohort exploration in providing interactive performance.

An Optimal Algorithm for Enumerating Connected Convex Subgraphs in Acyclic Digraphs

Reconfigurable computing system is emerging as an important computing system for satisfying the present and future computing demands in performance and flexibility. Extensible processor is a representative implementation of reconfigurable computing. In this context, custom instruction enumeration problem is one of the most computationally difficult problems involved in custom instruction synthesis for extensible processors. Custom instruction enumeration problem is essentially enumerating connected convex subgraphs from a given application graph. In this brief, we propose a provable optimal algorithm for enumerating connected convex subgraphs in acyclic digraphs in the sense of time complexity. The running time of the proposed algorithm is theoretically proved to be O(Σ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C∈CC(G)</sub> (|V(C)|+|E(C)|)), where CC(G) denotes the set of connected convex subgraphs in directed acyclic graph G, |V(C)| and |E(C)| denote the number of vertices and the number of edges in subgraph C respectively. Experimental results show that the proposed algorithm is more efficient than the state-of-the-art algorithms in terms of runtime.

Rational roots of all‐terminal reliability

AbstractGiven a connected graph G whose vertices are perfectly reliable and whose edges each fail independently with probability q ∈ [0, 1], the (all‐terminal) reliability of G is the probability that the resulting subgraph of operational edges contains a spanning tree (this probability is always a polynomial in q). The location of the roots of reliability polynomials has been well studied, with particular interest in finding those with the largest moduli. In this paper, we will discuss a related problem—among all reliability polynomials of graphs on n vertices, what can we say about the rational roots? We prove that (for n ≥ 2), the rational roots are −1, − 1/2, − 1/3,…, − 1/(n − 1), 1. Moreover, we show that for n ≥ 3, the root of minimum modulus among all graphs of order n is rational, and determine all roots of smallest moduli and the corresponding graphs. Finally, we provide the first nontrivial mathematical property that distinguishes, via reliability, the class of simple graphs (i.e., those without loops and multiple edges) from that of graphs in general.

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.

Estimate of the Number of Edges in Special Subgraphs of a Distance Graph

The classical problem of estimating the number of edges in a subgraph of a special distance graph is considered. Old results are significantly improved.

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.

Graph comparison via nonlinear quantum search

In this paper we present an efficiently scaling quantum algorithm which finds the size of the maximum common edge subgraph for a pair of arbitrary graphs and thus provides a meaningful measure of graph similarity. The algorithm makes use of a two-part quantum dynamic process: in the first part we obtain information crucial for the comparison of two graphs through linear quantum computation. However, this information is hidden in the quantum system with vanishingly small amplitude that even quantum algorithms such as Grover's search are not fast enough to distill the information efficiently. In order to extract the information we call upon techniques in nonlinear quantum computing to provide the speed-up necessary for an efficient algorithm. The linear quantum circuit requires $\mathcal{O}(n^3 \log^3 (n) \log \log (n))$ elementary quantum gates and the nonlinear evolution under the Gross-Pitaevskii equation has a time scaling of $\mathcal{O}(\frac{1}{g} n^2 \log^3 (n) \log \log (n))$, where $n$ is the number of vertices in each graph and $g$ is the strength of the Gross-Pitaveskii non-linearity. Through this example, we demonstrate the power of nonlinear quantum search techniques to solve a subset of NP-hard problems.

Leveraging multi-modal fusion for graph-based image annotation

Considering each of the visual features as one modality in image annotation task, efficient fusion of different modalities is essential in graph-based learning. Traditional graph-based methods consider one node for each image and combine its visual features into a single descriptor before constructing the graph. In this paper, we propose an approach that constructs a subgraph for each modality in such a way that edges of subgraph are determined using a search-based approach that handles class-imbalance challenge in the annotation datasets. Multiple subgraphs are then connected to each other to have a supergraph. This follows by introducing a learning framework to infer the tags of unannotated images on the supergraph. The proposed approach takes advantages of graph-based semi-supervised learning and multi-modal representation simultaneously. We evaluate the performance of the proposed approach on different datasets. The results reveal that the proposed approach improves the accuracy of annotation systems.

The Densest $k$-Subhypergraph Problem

The densest $k$-subgraph (D$k$S) problem and its corresponding minimization problem smallest $p$-edge subgraph (S$p$ES) have come to play a central role in approximation algorithms. This is due bot...

Complexity indices for the traveling salesman problem based on short edge subgraphs

We present the extension of our previous work on complexity indices for the traveling salesman problem (TSP). Since we study the symmetric traveling salesman problem, the instances are represented by complete graphs G with distances between cities as the edge weights. A complexity index is an invariant of an instance I by which the execution time of an exact TSP algorithm for I can be predicted. We consider some subgraphs of G consisting of short edges and define several new invariants related to their connected components. For computational experiments we have used the well-known TSP Solver Concorde. Experiments with instances on 50 vertices with the uniform distribution of integer edge weights in the interval [1, 100] show that there exists a notable correlation between the sequences of selected invariants and the sequence of execution times of the TSP Solver Concorde. We provide logical explanations of these phenomena.

New insights on GA-H reduced graphs

Let GA be a hereditary family of graphs and H a family of acyclically directed graphs. A graph G(V,E) is a GA-Hreduced graph if it can be obtained from a graph GA(V,D)∈GA by deleting the edges of an edge subgraph H(V,E′)∈H. The present paper complements reference [9] by proving that the following properties are equivalent: Property 1: u→v∈E′ implies NGA[u]⊆NGA[v]. Property 2: u→v∈E′ and v—w∉E∪E′=D implies u—w∉E∪E′=D. This equivalence implies that most of the algorithms in the above reference apply to the GA-H reduced graphs where H is the family of transitive edge subgraphs of the closed neighborhoods containment graphs of the graphs in GA. We also describe a polynomial time algorithm for maximum weight independent set under the weaker condition that H(V,E′) is a locally transitive edge subgraph of the closed neighborhoods containment graph of GA(V,D).

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.