Graph Decomposition Research Articles

Distributed fast boundary element methods for Helmholtz problems

We present an approach for a distributed memory parallelization of the boundary element method. The given mesh is decomposed into submeshes and the respective matrix blocks are distributed among computational nodes (processes). The distribution which takes care of the load balancing during the system matrix assembly and matrix-vector multiplication is based on the cyclic graph decomposition. Moreover, since the individual matrix blocks are approximated using the adaptive cross approximation method, we describe its modification capable of dealing with zero blocks in the double-layer operator matrix since these are usually problematic when using the original adaptive cross approximation algorithm. Convergence and scalability of the method are demonstrated on the half- and full-space sound scattering problems modeled by the Helmholtz equation.

Decomposition of complete graphs into connected unicyclic graphs with eight edges and pentagon

<p>A <span class="math"><em>G</em></span>-decomposition of the complete graph <span class="math"><em>K</em><sub><em>n</em></sub></span> is a family of pairwise edge disjoint subgraphs of <span class="math"><em>K</em><sub><em>n</em></sub></span>, all isomorphic to <span class="math"><em>G</em></span>, such that every edge of <span class="math"><em>K</em><sub><em>n</em></sub></span> belongs to exactly one copy of <span class="math"><em>G</em></span>. Using standard decomposition techniques based on <span class="math"><em>ρ</em></span>-labelings, introduced by Rosa in 1967, and their modifications we show that each of the ten non-isomorphic connected unicyclic graphs with eight edges containing the pentagon decomposes the complete graph <span class="math"><em>K</em><sub><em>n</em></sub></span> whenever the necessary conditions are satisfied.</p>

Coloring decompositions of complete geometric graphs

A decomposition of a non-empty simple graph $G$ is a pair $[G,P]$, such that $P$ is a set of non-empty induced subgraphs of $G$, and every edge of $G$ belongs to exactly one subgraph in $P$. The chromatic index $\chi'([G,P])$ of a decomposition $[G,P]$ is the smallest number $k$ for which there exists a $k$-coloring of the elements of $P$ in such a way that: for every element of $P$ all of its edges have the same color, and if two members of $P$ share at least one vertex, then they have different colors. A long standing conjecture of Erd\H{o}s-Faber-Lov\'asz states that every decomposition $[K_n,P]$ of the complete graph $K_n$ satisfies $\chi'([K_n,P])\leq n$. In this paper we work with geometric graphs, and inspired by this formulation of the conjecture, we introduce the concept of chromatic index of a decomposition of the complete geometric graph. We present bounds for the chromatic index of several types of decompositions when the vertices of the graph are in general position. We also consider the particular case in which the vertices are in convex position and present bounds for the chromatic index of a few types of decompositions.

Hamilton decompositions of one-ended Cayley graphs

We prove that any one-ended, locally finite Cayley graph G(Γ,S), where Γ is an abelian group and S is a finite generating set of non-torsion elements, admits a decomposition into edge-disjoint Hamiltonian (i.e. spanning) double-rays. In particular, the n-dimensional grid Zn admits a decomposition into n edge-disjoint Hamiltonian double-rays for all n∈N.

Deleting Vertices to Graphs of Bounded Genus

We show that a problem of deleting a minimum number of vertices from a graph to obtain a graph embeddable on a surface of a given Euler genus is solvable in time 2^{C_g cdot k^2 log k} n^{mathcal {O}(1)}, where k is the size of the deletion set, C_g is a constant depending on the Euler genus g of the target surface, and n is the size of the input graph. On the way to this result, we develop an algorithm solving the problem in question in time 2^{mathcal {O}((t+g) log (t+g))} n given a tree decomposition of the input graph of width t. The results generalize previous algorithms for the surface being a sphere by Marx and Schlotter (Algorithmica 62(3–4):807–822, 2012. https://doi.org/10.1007/s00453-010-9484-z), Kawarabayashi (in: 50th annual IEEE symposium on foundations of computer science, FOCS 2009, IEEE Computer Society, pp 639–648, 2009. https://doi.org/10.1109/FOCS.2009.45) and Jansen et al. (in: Chekuri (ed) 25th annual ACM-SIAM symposium on discrete algorithms, SODA 2014, SIAM, pp 1802–1811, 2014. https://doi.org/10.1137/1.9781611973402.130).

A memory efficient maximal clique enumeration method for sparse graphs with a parallel implementation

Maximal clique enumeration (MCE) is a widely studied problem that plays a crucial role in structure mining of undirected graphs. The increasing scale of real-world graphs has brought the challenges of high memory cost and high CPU workload to the problem. In this paper, we propose a memory efficient method named CMC-bit for MCE on sparse graphs. It reduces the memory cost via minimizing the candidate cliques and representing them by the data structure bitset. It generates an appropriate order for the vertex set according to two optimized principles to reduce the CPU cost. We further design a partition-based CMC-bit algorithm with a one-side extending strategy to solve the memory-limited problem. We parallelize the CMC-bit algorithm based on MapReduce with a range-based partition strategy to make an optimal trade-off between the shuffling workload of graph decomposition and load balance in the Reduce phase. We conduct extensive experiments on 30 real-world datasets. The results demonstrate that both the CMC-bit algorithm and its parallel implementation significantly outperform the respective state-of-the-art algorithms in speed. We also show that the parallel CMC-bit algorithm achieves good performance on the scalability with respect to both the reducer number and the CPU number.

Contagion of network products in small-world networks

We formulate a model in which agents embedded in an exogenous social network decide whether to adopt a new network product or not. In the theoretical part of the paper, we characterize the stochastically stable equilibria for complete networks and cycles. For an arbitrary network structure, we develop a novel graph decomposition method to characterize the set of recurrent communication states, which is a superset of stochastically stable equilibria of the adoption game presented in our model. In the simulation part, we study the contagion process of a network product in small-world networks that systematically represent social networks. We simulate a generalization of the Morris (Rev Econ Stud 67(1):57–78, 2000) Contagion model that can explain the chasm between early adopters and early majority. Our numerical analysis shows that the failure of a new network product is less likely in a highly cliquish network. In addition, the contagion process reaches to steady state faster in random networks than in highly cliquish networks. It turns out that marketers should work with mixed marketing strategies, which will result in a full contagion of a network product and faster contagion rates with a higher probability.

Modular decomposition of graphs and the distance preserving property

Given a graph G, a subgraph H is isometric if dH(u,v)=dG(u,v) for every pair u,v∈V(H), where d is the distance function. A graph G is distance preserving (dp) if it has an isometric subgraph of every possible order. A graph is sequentially distance preserving (sdp) if its vertices can be ordered such that deleting the first i vertices results in an isometric subgraph, for all i≥1. We introduce a generalisation of the lexicographic product of graphs, which can be used to non-trivially describe graphs. This generalisation is the inverse of the modular decomposition of graphs, which divides the graph into disjoint clusters called modules. Using these operations, we give a necessary and sufficient condition for graphs to be dp. Finally, we show that the Cartesian product of a dp graph and an sdp graph is dp.

Colourings of cubic graphs inducing isomorphic monochromatic subgraphs

AbstractA ‐bisection of a bridgeless cubic graph is a ‐colouring of its vertex set such that the colour classes have the same cardinality and all connected components in the two subgraphs induced by the colour classes ( monochromatic components in what follows) have order at most . Ban and Linial Conjectured that every bridgeless cubic graph admits a ‐bisection except for the Petersen graph. A similar problem for the edge set of cubic graphs has been studied: Wormald conjectured that every cubic graph with has a ‐edge colouring such that the two monochromatic subgraphs are isomorphic linear forests (ie, a forest whose components are paths). Finally, Ando conjectured that every cubic graph admits a bisection such that the two induced monochromatic subgraphs are isomorphic. In this paper, we provide evidence of a strong relation of the conjectures of Ban‐Linial and Wormald with Ando's Conjecture. Furthermore, we also give computational and theoretical evidence in their support. As a result, we pose some open problems stronger than the above‐mentioned conjectures. Moreover, we prove Ban‐Linial's Conjecture for cubic‐cycle permutation graphs. As a by‐product of studying ‐edge colourings of cubic graphs having linear forests as monochromatic components, we also give a negative answer to a problem posed by Jackson and Wormald about certain decompositions of cubic graphs into linear forests.

General equitable decompositions for graphs with symmetries

Using the theory of equitable decompositions it is possible to decompose a matrix M appropriately associated with a given graph. The result is a collection of smaller matrices whose collective eigenvalues are the same as the eigenvalues of the original matrix M. This is done by decomposing the matrix over a graph symmetry. Previously it was shown that a matrix can be equitably decomposed over any uniform, basic, or separable automorphism. Here we extend this theory to show that it is possible to equitably decompose a matrix over any automorphism of a graph, without restriction. Moreover, we give a step-by-step procedure which can be used to generate such a decomposition. We also prove under mild conditions that if a matrix M is equitably decomposed the resulting divisor matrix, which is the divisor matrix of the associated equitable partition, will have the same spectral radius as the original matrix M.

Identifying similar networks using structural hierarchy

Comparing structural similarities among complex networks is an important task in several scientific and social science applications. Existing techniques for quantifying network similarity range from network-centric methods that consider global network topology to node-centric methods that consider local node-level sub-structures.In this paper, we address the research gap between computationally expensive network-centric approaches and myopic node-centric network comparison methods by introducing a novel approach to quantify network similarity based on hierarchical graph decomposition. The approach adequately captures both global and local topology and is motivated by the observation that networks from diverse domains such as physical, chemical, biological and social systems exhibit an inherent structural hierarchy that emerges from local dyadic and triadic interactions. The proposed algorithm, Network Similarity via graph Decomposition (NSD), extracts network signatures from hierarchical decomposition of networks and uses Canberra distance to quantify the similarity between signatures. We use two well-known graph decomposition methods to expose network hierarchy resulting in two variations of NSD. We find that our approach groups similar networks better than competing algorithms. Experimentation using 40 real-world networks, 15 massive networks, and 30 large synthetic networks establishes that the proposed methodology is effective, scalable, sensitive and applicable to wide variety of networks.

Associated primes of powers of edge ideals and ear decompositions of graphs

In this paper, we give a complete description of the associated primes of every power of the edge ideal in terms of generalized ear decompositions of the graph. This result establishes a surprising relationship between two seemingly unrelated notions of commutative algebra and combinatorics. It covers all previous major results in this topic and has several interesting consequences.

Research on multi-focus image fusion algorithm based on total variation and quad-tree decomposition

In order to get the focus region from the two or more source image more effectively and to represent the source image completely and effectively, a decomposition strategy based on the combination of total variation and quad-tree and an improved fusion method of focus region detection are proposed. The theory of total variation and quad-tree is applied to the fusion of multi-focus images. Firstly, two registered experimental images are decomposed into the optimal block decomposition graph by total variation and quad-tree decomposition, respectively. For each block, the initial focus region decision map of the source image is found by using improved Sum-Modified-Laplacian, and the final focus region decision map is obtained by consistency test and morphological processing for the initial focus region decision map. Furthermore, compared with other algorithms, the improved algorithm has more advantages in extracting the focus area, because of improved focus evaluation function and more accurate detection of the focus area. According to the fusion source image of the final focus region decision map, the results of four sets of experiments show that the fusion quality and effect are significantly improved compared with the existing algorithm.

The application of graph decomposition to development of large scale agent-based economic models

In this work we describe the application of the graph decomposition algorithms for the development of a scalable high-performance agent-based model of population of Russia described in terms of demography, migration and transport flows. The simulated system consists of agents representing individuals and sets of links to other agents, which represent the social interactions of individual. Individual agents in the model participate in several independent processes, for which different sets of social links is important such as family and neighbors. To perform a load balancing of agents between cluster computer nodes the METIS graph decomposition algorithm was used. These algorithms allow to split the graph of agents and links into parts of similar size with least possible number of links between them. A number of numerical experiments was carried out for test model to estimate the influence of the parameters of the model on scalability.

Finding small-width connected path decompositions in polynomial time

A connected path decomposition of a simple graph G is a path decomposition (X1,…,Xl) such that the subgraph of G induced by X1∪⋯∪Xi is connected for each i∈{1,…,l}. The connected pathwidth of G is then the minimum width over all connected path decompositions of G. We prove that for each fixed k, the connected pathwidth of any input graph can be computed in polynomial-time. This answers an open question raised by Fedor V. Fomin during the GRASTA 2017 workshop, since connected pathwidth is equivalent to the connected (monotone) node search game.

Connecting Subspace Learning and Extreme Learning Machine in Speech Emotion Recognition

Speech emotion recognition (SER) is a powerful tool for endowing computers with the capacity to process information about the affective states of users in human-machine interactions. Recent research has shown the effectiveness of graph embedding-based subspace learning and extreme learning machine applied to SER, but there are still various drawbacks in these two techniques that limit their application. Regarding subspace learning, the change from linearity to nonlinearity is usually achieved through kernelization, whereas extreme learning machines only take label information into consideration at the output layer. In order to overcome these drawbacks, this paper leverages extreme learning machines for dimensionality reduction and proposes a novel framework to combine spectral regression-based subspace learning and extreme learning machines. The proposed framework contains three stages-data mapping, graph decomposition, and regression. At the data mapping stage, various mapping strategies provide different views of the samples. At the graph decomposition stage, specifically designed embedding graphs provide a possibility to better represent the structure of data through generating virtual coordinates. Finally, at the regression stage, dimension-reduced mappings are achieved by connecting the virtual coordinates and data mapping. Using this framework, we propose several novel dimensionality reduction algorithms, apply them to SER tasks, and compare their performance to relevant state-of-the-art methods. Our results on several paralinguistic corpora show that our proposed techniques lead to significant improvements.

Packing tree degree sequences

A degree sequence $D=(d_1,\dots, d_n)$ is a series of non-ne\-ga\-tive integers. A degree sequence is graphical if there exists a vertex labeled graph $G$ in which the degree of vertex $v_i$ is exactly $d_i$ for $i=1,\dots,n$. The graph $G$ is called a realization of $D$. The \emph{edge-disjoint realization problem} is the following: given a $c \times n$ degree matrix $D=\{(d_{1,1},\dots, d_{1,n} ),$ $(d_{2,1},\dots,d_{2,n} ),\dots, (d_{c,1},\dots, d_{c,n})\}$, in which each row of the matrix is a graphical degree sequence, decide if there exists pairwise edge-disjoint realizations of these degree sequences on the same vertex set $V=\{v_1,v_2,\dots,v_n\}$. Such set of edge-disjoint graphs is called a realization of the degree matrix. A realization can also be presented as an edge colored simple graph, in which the edges with a given color form a realization of the degree sequence in a given row of the degree matrix. It is known that the edge-disjoint realization problem is $\NP$-complete even if the number of rows is three and the degrees on each vertex sum up to $n-1$, that is, when a decomposition of the complete graph is required into three subgraphs with prescribed degrees. Moreover, the problem is also $\NP$-complete when the number of rows is two and the sum of the degrees on some of the vertices is less than $n-1$. However, special cases that are computationally tractable are also of interest. A classical result of Kundu \cite{kundutree} shows that deciding if two \emph{tree degree sequences} have edge-disjoint realizations is in $\P$. Motivated by the aforementioned result, we consider special cases of the problem for two tree degree sequences. We show that if there is no node which have degree one in both degree sequences (that is, the requested trees do not share a common leaf node) then they always have edge-disjoint caterpillar realizations. By using a probabilistic method, we prove that two tree degree sequences always have edge-disjoint realizations if each vertex is a leaf in at least one of the trees. This theorem can be extended to more trees: we show that the edge-disjoint realization problem is in $\P$ for an arbitrary number of tree sequences with the property that each vertex is a non-leaf in at most one of the trees, i.e., in every column of the degree matrix there is at most one entry which is greater than one. We also consider the following variant of the edge-disjoint realization problem: given two graphical degree sequences $D_1$ and $D_2$ such that $D_2$ is a tree degree sequence, decide if there exists edge-disjoint realizations of $D_1$ and $D_2$ where the realization of $D_2$ does not need to be a tree. We show that this problem is already $\NP$-complete. Counting, or just estimating the number of distinct realizations of degree sequences is challenging in general. We show that efficient approximations for the number of solutions as well as an almost uniform sampler exist for two tree degree sequences if each vertex is a leaf in at least one of the trees.

Regular Decomposition of Large Graphs: Foundation of a Sampling Approach to Stochastic Block Model Fitting

We analyze the performance of regular decomposition, a method for compression of large and dense graphs. This method is inspired by Szemerédi’s regularity lemma (SRL), a generic structural result of large and dense graphs. In our method, stochastic block model (SBM) is used as a model in maximum likelihood fitting to find a regular structure similar to the one predicted by SRL. Another ingredient of our method is Rissanen’s minimum description length principle (MDL). We consider scaling of algorithms to extremely large size of graphs by sampling a small subgraph. We continue our previous work on the subject by proving some experimentally found claims. Our theoretical setting does not assume that the graph is generated from a SBM. The task is to find a SBM that is optimal for modeling the given graph in the sense of MDL. This assumption matches with real-life situations when no random generative model is appropriate. Our aim is to show that regular decomposition is a viable and robust method for large graphs emerging, say, in Big Data area.

Decompositions of graphs based on a new graph product

Recently, we have introduced a new graph product, motivated by applications in the context of synchronising periodic real-time processes. This vertex-removing synchronised product (VRSP) is based on modifications of the well-known Cartesian product, and closely related to the synchronised product due to Wöhrle and Thomas. Here, we recall the definition of the VRSP and use it to define two different decompositions of graphs. Although our main results apply to directed labelled acyclic multigraphs, the VRSP can also be used to decompose any undirected graph of order at least 4 into two smaller graphs.

A parallel space–time boundary element method for the heat equation

In this paper we introduce a new parallel solver for the weakly singular space–time boundary integral equation for the heat equation. The space–time boundary mesh is decomposed into a given number of submeshes. Pairs of the submeshes represent dense blocks in the system matrices, which are distributed among computational nodes by an algorithm based on a cyclic decomposition of complete graphs ensuring load balance. In addition, we employ vectorization and threading in shared memory to ensure intra-node efficiency. We present scalability experiments on different CPU architectures to evaluate the performance of the proposed parallelization techniques. All levels of parallelism allow us to tackle large problems and lead to an almost optimal speedup.