ABSTRACT In the classical version of Traveling salesman problem , the targets which have to be visited are stationary but in real life there are large numbers of instances where the targets are in motion . The combination of biogeography-based optimization algorithm and EAX can provide high efficiency in solving optimization problems , including the traveling salesman problem ( TSP ) . This paper proposed a combination of those approaches to solve traveling salesman problem . The new hybrid approach was examined with standard datasets for TSP in TSPLIB . On the other hand , the ( min , + ) matrix product which is at the heart of many fundamental graph problems such as All-Pairs Shortest Paths , has received only minor no(1 ) improvements over its brute-force cubic running time and is widely conjectured to require n3−o(1 ) time . There is a plethora of matrix products and graph problems whose complexity seems to lie in the middle of these two problems . For instance , the Min-Max matrix product , the Minimum Witness matrix product , All-Pairs Shortest Paths in directed unweighted graphs and determining whether an edge-colored graph contains a monochromatic triangle , can all be solved in Õ(n(3+ω)/2 ) time . In the model of local computation algorithms ( LCAs ) , we aim to compute the queried part of the output by examining only a small ( sublinear ) portion of the input . Many recently developed LCAs on graph problems achieve time and space complexities with very low dependence on n , the number of vertices . Nonetheless , these complexities are generally at least exponential in d , the upper bound on the degree of the input graph . Secondly , the basic differences between the three classical solutions to the MST problem are discussed . Finally , algorithms solving several other graph problems , based of Jarnik 's approach to the solution of the MST problem , are discussed in conjunction with the properties of the Breadth-First-Search Tree and Depth-First-Search Tree . Important features of DBFO are that the bacteria agents do not depend on the local heuristic information but estimates new exploration schemes depending upon the previous experience and covered path analysis . This makes the algorithm better in combination generation for graph-based problems and combination generation for NP hard problems . In order to achieve these goals , the problem is defined as an integer problem , in which , feasibility of solutions is tested using a profit/cost relationship . This permits to solve PSP as a maximization of a single objective through an Ant Colony System Algorithm ( ACS ) , an efficient solver for graph- based problems . Experimental results reveal that ACS reaches 74 % of efficiency on solving instances randomly generated ; obtaining purchasing plans as a result . Our algorithmic contribution is a new \emph{deterministic } method of restructuring the input matrices in a sparsity-aware manner , which assigns each node with element-wise multiplication tasks that are not necessarily consecutive but are balanced , yielding communication-efficient multiplication . Moreover , this new deterministic method for restructuring matrices may be used to restructure the adjacency matrix of input graphs , enabling faster solutions for graph related problems . As an example , we present a new deterministic algorithm which solves the triangle listing problem in $ O(m/n^{5/3 } + 1)$ rounds , a complexity that was previously obtained by a \emph{randomized } algorithm [ Pandurangan et al. , SPAA 2018 ] and matches the lower bound of $ \tilde{\Omega}(n^{1/3})$ when $ m = n^2 $ of [ Izumi and Le Gall , PODC 2017 , Pandurangan et al. , SPAA 2018 ] . Our algorithmic contribution is a new deterministic method of restructuring the input matrices in a sparsity-aware manner , which assigns each node with element-wise multiplication tasks that are not necessarily consecutive but guarantee a balanced element distribution , providing for communication-efficient multiplication . Moreover , this new deterministic method for restructuring matrices may be used to restructure the adjacency matrix of input graphs , enabling faster deterministic solutions for graph related problems . As an example , we present a new sparsity aware , deterministic algorithm which solves the triangle listing problem in O ( m / n 5 / 3 + 1 ) rounds , a complexity that was previously obtained by a randomized algorithm [ Pandurangan et al. ( 2018 ) [ 26 ] ] , and that matches the known lower bound of Ω ˜ ( n 1 / 3 ) when m = n 2 of [ Izumi and Le Gall ( 2017 ) [ 19 ] , Pandurangan et al. ( 2018 ) [ 26 ] ] . Each processor in host network simulates the operations of at least one processor in guest network . Among all parallel machines , the mesh and torus are two major architecture which have been used for parallel computation , image processing , computer vision , numerical analysis , and graph algorithm problems . We examine here techniques for simulation , or embedding , of these two machines . That is why we also decided to generate graphs that were ' random ' under certain constraints that would better test our algorithms . Lastly , we will discuss how the generator could be used to help others simulate their graph algorithm problems . In this paper we address the problem of determining a CPE-based and a network-based layout of VPN tunnels while taking into account two factors : the cost of the links over which the VPN tunnels are established and the cost of the core routers that serve as end points for the VPN . We define related graph algorithm problems , analyze their complexity , and present heuristics for solving these problems efficiently . GPregel is a BSP model in graph processing such as Pregel from Google . It harnesses a lightweight compiler to hide the underlying complexity of the parallel processing details and simplifies programming , so that it greatly reduces the difficulty in utilizing the GPU to solve graph computing problems . Moreover , GPregel develops several optimizations for enhancing the performance , including ( 1 ) a special storage model for BSP model running on GPU , which overcomes the execution divergence and irregular memory access by coarse-grained designs ; ( 2 ) a warp-level optimal strategy Parallelized-Messages-Sending and a thread-level optimal strategy Threads-Merge-Executing to accelerate the computations of high degree vertexes and low degree vertexes respectively ; ( 3 ) messages copy mechanism optimization that utilizes a shared array and a rolling array to speed up the messages copy . In this thesis we consider some combinatorial optimisation problems on graphs and hypergraphs , and we study the existence ( or non-existence ) of polynomial kernels for these problems . In particular , we describe a generic kernelization for a theoretical class of graph problems , which can be used to derive the existence of a polynomial kernel for many graph problems of interest . There is a gap between research results of the enthusiastic innovators in the field of information technology and the current weak integration of the use of computers into mathematics teaching . This paper examines what exactly the real potentials of using some mathematics computer software are to support mathematics teaching and learning in graph-oriented problems , more specifically we try to estimate the value added impact of computer use in the mathematics learning process . While electronic computation has been used by mathematicians for five decades , it has been in the hands of teachers and learners for at most three decades but the real breakthrough of decentralised and personalised micro-computer-based computing has been widely available for less than two decades . We give a column generation based branch and bound algorithm for coalition structure generation over graphs problem using valuation functions for which this problem is proven to be NP-complete . For a given graph G = ( V;E ) and a valuation function w : 2^V - > R , the problem is to find the most valuable coalition structure ( or partition ) of V. We consider two cases : first when the value of a coalition is the sum of the weights of its edges which can be positive or negative , second when the value of a coalition takes account of both inter- and intra-coalitional disagreements and agreements , respectively . To that end , we describe a general framework for supervised learning on graphs called Message Passing Neural Networks ( MPNNs ) that simply abstracts the commonalities between several of the most promising existing neural models for graph structured data , in order to make it easier to understand the relationships between them and come up with novel variations . Given how many researchers have published models that fit into the MPNN framework , we believe that the community should push this general approach as far as possible on practically important graph problems and only suggest new variations that are well motivated by applications , such as the application we consider here : predicting the quantum mechanical properties of small organic molecules ( see task schematic in figure [ reference ] ) . In general , the search for practically effective machine learning ( ML ) models in a given domain proceeds through a sequence of increasingly realistic and interesting benchmarks . Here , ( 1 ) is mostly achieved by previous work on Graph Neural Networks scarselli2009graph ; we make several minor adaptations of this framework , including changing it to use modern practices around Recurrent Neural Networks . ( 2 ) is important because we desire outputs from graph - structured problems that are not solely individual classifications . In these cases , the challenge is how to learn features on the graph that encode the partial output sequence that has already been produced ( e.g. , the path so far if outputting a path ) and that still needs to be produced ( e.g. , the remaining path ) . A feedback neural network for solving graph coloring problem is presented . The circuit has an associated transcendental energy function that ensures fast convergence to the exact solution . The final main component of this work is experimental in nature . A group of undergraduates was instructed to solve the graph coloring problem . Each one used our software system to control the color of one node in a network , and their objective was to arrive at a valid coloring . We present an algorithmic framework ( including a single data structure ) that is extended into linear-time algorithms to solve several NP-complete graph problems ( i.e. , INDEPENDENT SET , M AXIMUM CUT , GRAPH COLORING , HAMILTONIAN CYCLE , and DISJOINT PATHS ) . The linearity is achieved assuming the provision of a branch decomposition of the instance graph . This separation enables us to efficiently process each vertex interval by just loading the corresponding log . Over the current state of the art out-of-core graph processing framework , our evaluation results show that the PartitionedVC framework improves performance by up to $ 16.40\times$ , $ 1.13\times$ , $ 1.64\times$ , $ 1.38\times$ , and $ 2.76\times$ for the widely used breadth-first search , pagerank , community detection , graph coloring , and the maximal independent set applications , respectively . We also discuss how our results are related to the famous Hajos ' conjecture and Hadwiger 's conjecture . We point out that it is Unique-Game hard to obtain an O(k/log2 k)-approximation algorithm for graph-coloring of graphs with maximum degree at most k − 2 [ 6 ] , and hence it is also Unique-Game hard to obtain an O(k/log2 k)-approximation algorithm for graph-coloring of graphs without a Kk-subdivision or without a Kk-immersion . Therefore it really makes sense to consider an additive approximation algorithm for graph coloring of these family of graphs ( which is in contrast to a 2-approximation algorithm for graph-coloring of H-minor-free graphs [ 13 ] ) . Using our structure theorem , we give an approximation algorithm for coloring of a graph G without a fixed graph H as a totally odd subdivision , using 2χ(G ) + 6(|H| − 1 ) colors , where χ(G ) is chromatic number of G. The same conclusion is true if we replace " totally odd " by " parity " . We point out that it is Unique-Game hard to obtain an O(k/log2 k)-approximation algorithm for graph-coloring of graphs with maximum degree at most k − 2 [ 2 ] , and hence it is also Unique-Game hard to obtain an O(k/log2 k)-approximation algorithm for graph-coloring of graphs even without a Kk-subdivision ( i.e , without the parity constraint ) . Thus our additive error Θ(k ) is most likely best possible up to constant . This relaxation has an integrality gap of $ \frac{6}{7}\simeq 0.8571 $ , showing that our guarantee is almost tight . For maximum graph homomorphism , we show that a $ \bigl({\ensuremath{\frac{1}{2}}}+{\ensuremath{\varepsilon}}_0)$-approximation algorithm , for any constant e0>0 , implies an algorithm for distinguishing between certain average-case instances of the subgraph isomorphism problem that appear to be hard . Complementing this , we give a $ \bigl({\ensuremath{\frac{1}{2}}}+\Omega(\frac{1}{|H|\log{|H|}})\bigr)$-approximation algorithm . Subgraph isomorphism , or finding matching patterns in a graph , is a classic graph problem that has many practical use cases . There are even commercialized solutions for this problem such as RDF databases with their support for SPARQL queries . Here we consider two generalizations of classical tree pattern matching , which consists of determining if one tree is isomorphic to a subgraph of another . For the embedding problems of subgraph isomorphism and topological embedding , we present algorithms for determining a largest tree embeddable in two trees T and T ' ( or a largest subtree ) and a smallest tree in which each of T and T ' can be embedded ( or a smallest supertree ) . Both subtrees and supertrees can be used in a variety of different applications . A number of matching algorithms for graphs with unique node labels are developed . It is shown that problems such as graph isomorphism , subgraph isomorphism , maximum common subgraph ( MCS ) and graph edit distance ( GED ) have a computational complexity that is only quadratic in the number of nodes . Moreover , computing the median of a set of graphs is only linear in the cardinality of the set . SP-graphs are simple and layered , making the main phases of workflow easier to distinguish . Additionally , from a more formal point of view , polynomial-time algorithms for performing complex graph-based operations ( e.g. , comparing workflows , which is directly related to the problem of subgraph homomorphism ) can be designed when workflows have SP-structures while such operations are related to an NP-hard problem for DAG structures without any restriction on their structures . The SPFlow rewriting and provenance-preserving algorithm and its associated tool are thus introduced . Finally a SPARQL query is represented as a multigraph , and the SPARQL querying problem is reduced to the subgraph homomorphism problem . AMbER exploits structural properties of the query multigraph as well as the proposed indexes , in order to tackle the problem of subgraph homomorphism . The performance of AMbER , in comparison with state-of-the-art systems , has been extensively evaluated over several RDF benchmarks . Abstract The mechanism of interpretation ( subgraph homomorphism , homomorphic embedding , general coloring ) gives rise to a “ geography ” of graphs . The position of symmetric graphs in this geography is investigated . However , the existing approaches to mine frequent subgraphs have significant drawbacks in terms of efficiency . In this paper , we focus on real-time frequent subgraph mining and propose an efficient customized data structure and technique to reduce subgraph isomorphism checking as well as a supergraph based optimized descendant generation algorithm . Extensive performance analyses prove the efficiency of our algorithm over the existing methods . In this research , we develop a novel algorithm that improves AGM in two aspects . First , the exact subgraph-isomorphism checking is replaced by comparing the shape descriptors composed from the point coordinates corresponding to the nodes of the subgraphs in question . Second , a new approach for generating frequent subgraph candidates is adopted , which allows large frequent subgraphs to be found in fewer iterations . By exploring some properties of graph generators , two effective pruning techniques , backward edge pruning and forward edge pruning , are proposed to prune the branches of the well-known DFS code enumeration tree that do not contain graph generators . To further improve the efficiency , an effective index structure , ADI++ , is also devised to facilitate the subgraph isomorphism checking . We experimentally evaluate various aspects of FOGGER using both real and synthetic datasets . Querying large graphs to retrieve information in permissible time is an emerging research problem , and it has roots in various application domains , mainly to analyse large networks . For a given query graph , the aim of subgraph isomorphism finding in a data graph is to identify all its subgraphs that are isomorphic to the query graph , and it has become the central problem for querying large graphs . Though different research groups have proposed many techniques using graph compression , postponing cartesian products , and candidate region exploration in the recent past , most of them show exponential behaviour for some query graphs on a large data graph . In this paper , we propose a subgraph isomorphism finding method , SubISO , which uses an objective function based on the eccentricity and some isomorphic invariants of the vertices of the query graph to minimize the number and size of the candidate regions in the data graph . Since subgraph isomorphism finding has a large number of solutions , especially in a large graph with repeated node labels , we propose to limit the maximum number of recursive calls of the generic subgraph search function to complete the execution of SubISO and return at most k-relevant matches . The SubISO finds at most k-relevant solutions in reasonable elapsed time over the queries for which existing state-of-the-art methods show exponential behaviour . On the remaining set of graphs , a subgraph isomorphism algorithm is applied to verify the graphs that contain the query graph . A second category , so called no-filter , verify ( NFV ) , invested in optimizing the subgraph isomorphism process , by employing a lightweight index primarily to locate candidate vertices on the graphs in the dataset . The current trend is to totally dismiss the FTV methods and employ NFV methods instead . In this paper , we present Taxogram , a taxonomy-superimposed graph mining algorithm that can efficiently discover frequent graph structures in a database of taxonomy-superimposed graphs . Taxogram has two advantages : ( i ) It performs a subgraph isomorphism test once per class of patterns which are structurally isomorphic , but have different labels , and ( ii ) it reconciles standard graph mining methods with taxonomy-based graph mining and takes advantage of well-studied methods in the literature . Taxogram has three stages : ( a ) relabeling nodes in the input database , ( b ) mining pattern classes/families and constructing associated occurrence indices , and ( c ) computing patterns and eliminating useless ( i.e. , over-generalized ) patterns by post-processing occurrence indices . In this paper , we propose a new algorithm called Mutual-Match to solve the problem of uncertain subgraph query under the situation where both the graph dataset and the query are uncertain . Considering that the subgraph isomorphism verification is an NP-hard problem , and it will be more complex on uncertain graph data , the Mutual-Match algorithm uses MapReduce processes to find out all the subgraph matches and can adapt to dynamic graphs well . Experimental results on both real-world and synthetic datasets show the correctness and effectiveness of our proposed methods . However , directly publishing the findings obtained from triangle counts may bring potential privacy concern , which raises great challenges and opportunities for privacy-preserving triangle counting . In this paper , we choose to use differential privacy to protect triangle counting for large scale graphs . To reduce the large sensitivity caused in large graphs , we propose a novel graph projection method that can be used to obtain an upper bound for sensitivity in different distributions . Triangle counting is one of the most basic graph applications to solve many real-world problems in a wide variety of domains . Exploring the massive parallelism of the Graphics Processing Unit ( GPU ) to accelerate the triangle counting is prevail . Briefly , a triangle encompasses three vertices from a graph , where every vertex possesses at least one incidental edge to the other two vertices from the triangle . Consequently , list intersection , which identifies the incidental edges , becomes the core algorithm for triangle counting . At the meantime , attracted by the enormous parallel computing potential of Graphics Processing Units ( GPUs ) , numerous efforts have been devoted to deploy triangle counting algorithms on GPUs . Our algorithm is the most scalable and parallel among all existing GPU implementations and also outperforms all existing CPU distributed implementations . This work specifically focuses on leveraging our implementation on the triangle counting problem for the Subgraph Isomorphism Graph Challenge 2019 , demonstrating a geometric mean speedup over the 2018 champion of $ 3.84 \times $ . Triangle counting is a representative graph problem that shows the challenges of improving graph algorithm performance using algorithmic techniques and adopting graph algorithms to new architectures . In this paper , we describe an update to the linear-algebraic formulation of the triangle counting problem . Our new approach relies on fine-grained tasking based on a tile layout . This article studies I/O-efficient algorithms for the triangle listing problem and the triangle counting problem , whose solutions are basic operators in dealing with many other graph problems . In the former problem , given an undirected graph G , the objective is to find all the cliques involving 3 vertices in G. In the latter problem , the objective is to report just the number of such cliques without having to enumerate them . Evidently , the task of triple sampling has not been explored in depth , although sampling is a more fundamental task than counting , and the former is useful for solving various other tasks , including counting . In recent years , some works on triple sampling have been proposed that are based on direct sampling , solely for the purpose of triangle count approximation . They sample only from a uniform distribution , and are not effective for sampling triples from an arbitrary user-defined distribution . We revisit the well-studied problem of triangle count estimation in graph streams . Given a graph represented as a stream of m edges , our aim is to compute a (1 + -ε)-approximation to the triangle count T , using a small space algorithm . The range searching problems which allow partition trees where every query enters only a sublinear number of nodes are characterized as those with finite Vapnik - Chervonenk is dimension . The concrete combinatorial bounds obtained imply — among others — that every set of n points in the plane < list><item > can be stored in an & Ogr;(n ) space data structure which allows triangle counting queries in & Ogr;(√n · < italic > log</italic><supscrpt>3</supscrpt > n ) time , and < /item><item > can be stored in an & Ogr;(n · < italic > log</italic > n ) space data structure which allows disk counting queries in & Ogr;(√n · < italic > log</italic><supscrpt>3</supscrpt > n ) time ; < /item></list > the preprocessing time for the data structures is polynomial . Recent results by Chazelle entail that these bounds for space and query time are optimal up to polylog — factors . They introduced constraints that restrict the shapes of objects ( e.g. , requiring the pixels of a vertex to form a 2D box ) and how the representations of different objects can intersect . In this way , one can solve a variety of NP-hard graph problems , including pathwidth , bandwidth , optimum st-orientation , area-minimal ( bar-k ) visibility representation , boxicity-k graphs and others . For example , in a grid-based drawing of a visibility representation , each vertex is represented by a horizontal box of height 1 and each edge is represented by a vertical box of width 1 . A large number of NP-hard graph problems become polynomial-time solvable on graph classes where the mim-width is bounded and quickly computable . Hence , when solving such problems on special graph classes , it is helpful to know whether the graph class under consideration has bounded mim-width . Our newly developing theory of bidimensional graph problems provides general techniques for designing efficient fixed-parameter algorithms and approximation algorithms for NP-hard graph problems in broad classes of graphs . This theory applies to graph problems that are bidimensional in the sense that ( 1 ) the solution value for the k × k grid graph ( and similar graphs ) grows with k , typically as Ω(k2 ) , and ( 2 ) the solution value goes down when contracting edges and optionally when deleting edges . We present an algorithmic framework ( including a single data structure ) that is extended into linear-time algorithms to solve several NP-complete graph problems ( i.e. , INDEPENDENT SET , M AXIMUM CUT , GRAPH COLORING , HAMILTONIAN CYCLE , and DISJOINT PATHS ) . The linearity is achieved assuming the provision of a branch decomposition of the instance graph . The algorithm uses a linear number of processors and O(log n ) time . Many NP-complete graph problems are known to be solvable in polynomial time , when restricted to graphs with treewidth ≤ k , k constant . From the results in this paper , it follows that most of these problems are also in NC , when restricted to graphs with treewidth bounded by a constant . In recent years many efforts have been done to solve NP-complete and time consuming problems with the help of DNA computing . In this paper , we use sticker model ( one of the most well-known models of DNA computing ) to present three DNA algorithms for solving three different NP-complete graph-based problems for the first time : domatic partition , kernel and induced path . Also we have simulated these algorithms to show their correctness . This paper originates from the observation that many classical NP graph problems , including some NP-complete problems , are actually of very low nondeterministic time complexity . In order to formalize this observation , we define the complexity class vertex NLIN , which collects the graph problems computable on a nondeterministic RAM in time O(n ) . While many parallel applications exhibit good spatial locality , other important codes in areas like graph problem-solving or CAD do not . Often , these irregular codes contain small records accessed via pointers . The following graph addressing problem was studied by Graham and Pollak in devising a routing scheme for Pierce ’s loop switching network . Let G be a graph with n vertices .
ABSTRACT In the classical version of Traveling salesman problem , the targets which have to be visited are stationary but in real life there are large numbers of instances where the targets are in motion .
The combination of biogeography-based optimization algorithm and EAX can provide high efficiency in solving optimization problems , including the traveling salesman problem ( TSP ) . This paper proposed a combination of those approaches to solve traveling salesman problem . The new hybrid approach was examined with standard datasets for TSP in TSPLIB .
On the other hand , the ( min , + ) matrix product which is at the heart of many fundamental graph problems such as All-Pairs Shortest Paths , has received only minor no(1 ) improvements over its brute-force cubic running time and is widely conjectured to require n3−o(1 ) time . There is a plethora of matrix products and graph problems whose complexity seems to lie in the middle of these two problems . For instance , the Min-Max matrix product , the Minimum Witness matrix product , All-Pairs Shortest Paths in directed unweighted graphs and determining whether an edge-colored graph contains a monochromatic triangle , can all be solved in Õ(n(3+ω)/2 ) time .
In the model of local computation algorithms ( LCAs ) , we aim to compute the queried part of the output by examining only a small ( sublinear ) portion of the input . Many recently developed LCAs on graph problems achieve time and space complexities with very low dependence on n , the number of vertices . Nonetheless , these complexities are generally at least exponential in d , the upper bound on the degree of the input graph .
Secondly , the basic differences between the three classical solutions to the MST problem are discussed . Finally , algorithms solving several other graph problems , based of Jarnik 's approach to the solution of the MST problem , are discussed in conjunction with the properties of the Breadth-First-Search Tree and Depth-First-Search Tree .
Important features of DBFO are that the bacteria agents do not depend on the local heuristic information but estimates new exploration schemes depending upon the previous experience and covered path analysis . This makes the algorithm better in combination generation for graph-based problems and combination generation for NP hard problems .
In order to achieve these goals , the problem is defined as an integer problem , in which , feasibility of solutions is tested using a profit/cost relationship . This permits to solve PSP as a maximization of a single objective through an Ant Colony System Algorithm ( ACS ) , an efficient solver for graph- based problems . Experimental results reveal that ACS reaches 74 % of efficiency on solving instances randomly generated ; obtaining purchasing plans as a result .
Our algorithmic contribution is a new \emph{deterministic } method of restructuring the input matrices in a sparsity-aware manner , which assigns each node with element-wise multiplication tasks that are not necessarily consecutive but are balanced , yielding communication-efficient multiplication . Moreover , this new deterministic method for restructuring matrices may be used to restructure the adjacency matrix of input graphs , enabling faster solutions for graph related problems . As an example , we present a new deterministic algorithm which solves the triangle listing problem in $ O(m/n^{5/3 } + 1)$ rounds , a complexity that was previously obtained by a \emph{randomized } algorithm [ Pandurangan et al. , SPAA 2018 ] and matches the lower bound of $ \tilde{\Omega}(n^{1/3})$ when $ m = n^2 $ of [ Izumi and Le Gall , PODC 2017 , Pandurangan et al. , SPAA 2018 ] .
Our algorithmic contribution is a new deterministic method of restructuring the input matrices in a sparsity-aware manner , which assigns each node with element-wise multiplication tasks that are not necessarily consecutive but guarantee a balanced element distribution , providing for communication-efficient multiplication . Moreover , this new deterministic method for restructuring matrices may be used to restructure the adjacency matrix of input graphs , enabling faster deterministic solutions for graph related problems . As an example , we present a new sparsity aware , deterministic algorithm which solves the triangle listing problem in O ( m / n 5 / 3 + 1 ) rounds , a complexity that was previously obtained by a randomized algorithm [ Pandurangan et al. ( 2018 ) [ 26 ] ] , and that matches the known lower bound of Ω ˜ ( n 1 / 3 ) when m = n 2 of [ Izumi and Le Gall ( 2017 ) [ 19 ] , Pandurangan et al. ( 2018 ) [ 26 ] ] .
Each processor in host network simulates the operations of at least one processor in guest network . Among all parallel machines , the mesh and torus are two major architecture which have been used for parallel computation , image processing , computer vision , numerical analysis , and graph algorithm problems . We examine here techniques for simulation , or embedding , of these two machines .
That is why we also decided to generate graphs that were ' random ' under certain constraints that would better test our algorithms . Lastly , we will discuss how the generator could be used to help others simulate their graph algorithm problems .
In this paper we address the problem of determining a CPE-based and a network-based layout of VPN tunnels while taking into account two factors : the cost of the links over which the VPN tunnels are established and the cost of the core routers that serve as end points for the VPN . We define related graph algorithm problems , analyze their complexity , and present heuristics for solving these problems efficiently .
GPregel is a BSP model in graph processing such as Pregel from Google . It harnesses a lightweight compiler to hide the underlying complexity of the parallel processing details and simplifies programming , so that it greatly reduces the difficulty in utilizing the GPU to solve graph computing problems . Moreover , GPregel develops several optimizations for enhancing the performance , including ( 1 ) a special storage model for BSP model running on GPU , which overcomes the execution divergence and irregular memory access by coarse-grained designs ; ( 2 ) a warp-level optimal strategy Parallelized-Messages-Sending and a thread-level optimal strategy Threads-Merge-Executing to accelerate the computations of high degree vertexes and low degree vertexes respectively ; ( 3 ) messages copy mechanism optimization that utilizes a shared array and a rolling array to speed up the messages copy .
In this thesis we consider some combinatorial optimisation problems on graphs and hypergraphs , and we study the existence ( or non-existence ) of polynomial kernels for these problems . In particular , we describe a generic kernelization for a theoretical class of graph problems , which can be used to derive the existence of a polynomial kernel for many graph problems of interest .
There is a gap between research results of the enthusiastic innovators in the field of information technology and the current weak integration of the use of computers into mathematics teaching . This paper examines what exactly the real potentials of using some mathematics computer software are to support mathematics teaching and learning in graph-oriented problems , more specifically we try to estimate the value added impact of computer use in the mathematics learning process . While electronic computation has been used by mathematicians for five decades , it has been in the hands of teachers and learners for at most three decades but the real breakthrough of decentralised and personalised micro-computer-based computing has been widely available for less than two decades .
We give a column generation based branch and bound algorithm for coalition structure generation over graphs problem using valuation functions for which this problem is proven to be NP-complete . For a given graph G = ( V;E ) and a valuation function w : 2^V - > R , the problem is to find the most valuable coalition structure ( or partition ) of V. We consider two cases : first when the value of a coalition is the sum of the weights of its edges which can be positive or negative , second when the value of a coalition takes account of both inter- and intra-coalitional disagreements and agreements , respectively .
To that end , we describe a general framework for supervised learning on graphs called Message Passing Neural Networks ( MPNNs ) that simply abstracts the commonalities between several of the most promising existing neural models for graph structured data , in order to make it easier to understand the relationships between them and come up with novel variations . Given how many researchers have published models that fit into the MPNN framework , we believe that the community should push this general approach as far as possible on practically important graph problems and only suggest new variations that are well motivated by applications , such as the application we consider here : predicting the quantum mechanical properties of small organic molecules ( see task schematic in figure [ reference ] ) . In general , the search for practically effective machine learning ( ML ) models in a given domain proceeds through a sequence of increasingly realistic and interesting benchmarks .
Here , ( 1 ) is mostly achieved by previous work on Graph Neural Networks scarselli2009graph ; we make several minor adaptations of this framework , including changing it to use modern practices around Recurrent Neural Networks . ( 2 ) is important because we desire outputs from graph - structured problems that are not solely individual classifications . In these cases , the challenge is how to learn features on the graph that encode the partial output sequence that has already been produced ( e.g. , the path so far if outputting a path ) and that still needs to be produced ( e.g. , the remaining path ) .
A feedback neural network for solving graph coloring problem is presented . The circuit has an associated transcendental energy function that ensures fast convergence to the exact solution .
The final main component of this work is experimental in nature . A group of undergraduates was instructed to solve the graph coloring problem . Each one used our software system to control the color of one node in a network , and their objective was to arrive at a valid coloring .
We present an algorithmic framework ( including a single data structure ) that is extended into linear-time algorithms to solve several NP-complete graph problems ( i.e. , INDEPENDENT SET , M AXIMUM CUT , GRAPH COLORING , HAMILTONIAN CYCLE , and DISJOINT PATHS ) . The linearity is achieved assuming the provision of a branch decomposition of the instance graph .
This separation enables us to efficiently process each vertex interval by just loading the corresponding log . Over the current state of the art out-of-core graph processing framework , our evaluation results show that the PartitionedVC framework improves performance by up to $ 16.40\times$ , $ 1.13\times$ , $ 1.64\times$ , $ 1.38\times$ , and $ 2.76\times$ for the widely used breadth-first search , pagerank , community detection , graph coloring , and the maximal independent set applications , respectively .
We also discuss how our results are related to the famous Hajos ' conjecture and Hadwiger 's conjecture . We point out that it is Unique-Game hard to obtain an O(k/log2 k)-approximation algorithm for graph-coloring of graphs with maximum degree at most k − 2 [ 6 ] , and hence it is also Unique-Game hard to obtain an O(k/log2 k)-approximation algorithm for graph-coloring of graphs without a Kk-subdivision or without a Kk-immersion . Therefore it really makes sense to consider an additive approximation algorithm for graph coloring of these family of graphs ( which is in contrast to a 2-approximation algorithm for graph-coloring of H-minor-free graphs [ 13 ] ) .
Using our structure theorem , we give an approximation algorithm for coloring of a graph G without a fixed graph H as a totally odd subdivision , using 2χ(G ) + 6(|H| − 1 ) colors , where χ(G ) is chromatic number of G. The same conclusion is true if we replace " totally odd " by " parity " . We point out that it is Unique-Game hard to obtain an O(k/log2 k)-approximation algorithm for graph-coloring of graphs with maximum degree at most k − 2 [ 2 ] , and hence it is also Unique-Game hard to obtain an O(k/log2 k)-approximation algorithm for graph-coloring of graphs even without a Kk-subdivision ( i.e , without the parity constraint ) . Thus our additive error Θ(k ) is most likely best possible up to constant .
This relaxation has an integrality gap of $ \frac{6}{7}\simeq 0.8571 $ , showing that our guarantee is almost tight . For maximum graph homomorphism , we show that a $ \bigl({\ensuremath{\frac{1}{2}}}+{\ensuremath{\varepsilon}}_0)$-approximation algorithm , for any constant e0>0 , implies an algorithm for distinguishing between certain average-case instances of the subgraph isomorphism problem that appear to be hard . Complementing this , we give a $ \bigl({\ensuremath{\frac{1}{2}}}+\Omega(\frac{1}{|H|\log{|H|}})\bigr)$-approximation algorithm .
Subgraph isomorphism , or finding matching patterns in a graph , is a classic graph problem that has many practical use cases . There are even commercialized solutions for this problem such as RDF databases with their support for SPARQL queries .
Here we consider two generalizations of classical tree pattern matching , which consists of determining if one tree is isomorphic to a subgraph of another . For the embedding problems of subgraph isomorphism and topological embedding , we present algorithms for determining a largest tree embeddable in two trees T and T ' ( or a largest subtree ) and a smallest tree in which each of T and T ' can be embedded ( or a smallest supertree ) . Both subtrees and supertrees can be used in a variety of different applications .
A number of matching algorithms for graphs with unique node labels are developed . It is shown that problems such as graph isomorphism , subgraph isomorphism , maximum common subgraph ( MCS ) and graph edit distance ( GED ) have a computational complexity that is only quadratic in the number of nodes . Moreover , computing the median of a set of graphs is only linear in the cardinality of the set .
SP-graphs are simple and layered , making the main phases of workflow easier to distinguish . Additionally , from a more formal point of view , polynomial-time algorithms for performing complex graph-based operations ( e.g. , comparing workflows , which is directly related to the problem of subgraph homomorphism ) can be designed when workflows have SP-structures while such operations are related to an NP-hard problem for DAG structures without any restriction on their structures . The SPFlow rewriting and provenance-preserving algorithm and its associated tool are thus introduced .
Finally a SPARQL query is represented as a multigraph , and the SPARQL querying problem is reduced to the subgraph homomorphism problem . AMbER exploits structural properties of the query multigraph as well as the proposed indexes , in order to tackle the problem of subgraph homomorphism . The performance of AMbER , in comparison with state-of-the-art systems , has been extensively evaluated over several RDF benchmarks .
Abstract The mechanism of interpretation ( subgraph homomorphism , homomorphic embedding , general coloring ) gives rise to a “ geography ” of graphs . The position of symmetric graphs in this geography is investigated .
However , the existing approaches to mine frequent subgraphs have significant drawbacks in terms of efficiency . In this paper , we focus on real-time frequent subgraph mining and propose an efficient customized data structure and technique to reduce subgraph isomorphism checking as well as a supergraph based optimized descendant generation algorithm . Extensive performance analyses prove the efficiency of our algorithm over the existing methods .
In this research , we develop a novel algorithm that improves AGM in two aspects . First , the exact subgraph-isomorphism checking is replaced by comparing the shape descriptors composed from the point coordinates corresponding to the nodes of the subgraphs in question . Second , a new approach for generating frequent subgraph candidates is adopted , which allows large frequent subgraphs to be found in fewer iterations .
By exploring some properties of graph generators , two effective pruning techniques , backward edge pruning and forward edge pruning , are proposed to prune the branches of the well-known DFS code enumeration tree that do not contain graph generators . To further improve the efficiency , an effective index structure , ADI++ , is also devised to facilitate the subgraph isomorphism checking . We experimentally evaluate various aspects of FOGGER using both real and synthetic datasets .
Querying large graphs to retrieve information in permissible time is an emerging research problem , and it has roots in various application domains , mainly to analyse large networks . For a given query graph , the aim of subgraph isomorphism finding in a data graph is to identify all its subgraphs that are isomorphic to the query graph , and it has become the central problem for querying large graphs . Though different research groups have proposed many techniques using graph compression , postponing cartesian products , and candidate region exploration in the recent past , most of them show exponential behaviour for some query graphs on a large data graph . In this paper , we propose a subgraph isomorphism finding method , SubISO , which uses an objective function based on the eccentricity and some isomorphic invariants of the vertices of the query graph to minimize the number and size of the candidate regions in the data graph . Since subgraph isomorphism finding has a large number of solutions , especially in a large graph with repeated node labels , we propose to limit the maximum number of recursive calls of the generic subgraph search function to complete the execution of SubISO and return at most k-relevant matches . The SubISO finds at most k-relevant solutions in reasonable elapsed time over the queries for which existing state-of-the-art methods show exponential behaviour .
On the remaining set of graphs , a subgraph isomorphism algorithm is applied to verify the graphs that contain the query graph . A second category , so called no-filter , verify ( NFV ) , invested in optimizing the subgraph isomorphism process , by employing a lightweight index primarily to locate candidate vertices on the graphs in the dataset . The current trend is to totally dismiss the FTV methods and employ NFV methods instead .
In this paper , we present Taxogram , a taxonomy-superimposed graph mining algorithm that can efficiently discover frequent graph structures in a database of taxonomy-superimposed graphs . Taxogram has two advantages : ( i ) It performs a subgraph isomorphism test once per class of patterns which are structurally isomorphic , but have different labels , and ( ii ) it reconciles standard graph mining methods with taxonomy-based graph mining and takes advantage of well-studied methods in the literature . Taxogram has three stages : ( a ) relabeling nodes in the input database , ( b ) mining pattern classes/families and constructing associated occurrence indices , and ( c ) computing patterns and eliminating useless ( i.e. , over-generalized ) patterns by post-processing occurrence indices .
In this paper , we propose a new algorithm called Mutual-Match to solve the problem of uncertain subgraph query under the situation where both the graph dataset and the query are uncertain . Considering that the subgraph isomorphism verification is an NP-hard problem , and it will be more complex on uncertain graph data , the Mutual-Match algorithm uses MapReduce processes to find out all the subgraph matches and can adapt to dynamic graphs well . Experimental results on both real-world and synthetic datasets show the correctness and effectiveness of our proposed methods .
However , directly publishing the findings obtained from triangle counts may bring potential privacy concern , which raises great challenges and opportunities for privacy-preserving triangle counting . In this paper , we choose to use differential privacy to protect triangle counting for large scale graphs . To reduce the large sensitivity caused in large graphs , we propose a novel graph projection method that can be used to obtain an upper bound for sensitivity in different distributions .
Triangle counting is one of the most basic graph applications to solve many real-world problems in a wide variety of domains . Exploring the massive parallelism of the Graphics Processing Unit ( GPU ) to accelerate the triangle counting is prevail .
Briefly , a triangle encompasses three vertices from a graph , where every vertex possesses at least one incidental edge to the other two vertices from the triangle . Consequently , list intersection , which identifies the incidental edges , becomes the core algorithm for triangle counting . At the meantime , attracted by the enormous parallel computing potential of Graphics Processing Units ( GPUs ) , numerous efforts have been devoted to deploy triangle counting algorithms on GPUs .
Our algorithm is the most scalable and parallel among all existing GPU implementations and also outperforms all existing CPU distributed implementations . This work specifically focuses on leveraging our implementation on the triangle counting problem for the Subgraph Isomorphism Graph Challenge 2019 , demonstrating a geometric mean speedup over the 2018 champion of $ 3.84 \times $ .
Triangle counting is a representative graph problem that shows the challenges of improving graph algorithm performance using algorithmic techniques and adopting graph algorithms to new architectures . In this paper , we describe an update to the linear-algebraic formulation of the triangle counting problem . Our new approach relies on fine-grained tasking based on a tile layout .
This article studies I/O-efficient algorithms for the triangle listing problem and the triangle counting problem , whose solutions are basic operators in dealing with many other graph problems . In the former problem , given an undirected graph G , the objective is to find all the cliques involving 3 vertices in G. In the latter problem , the objective is to report just the number of such cliques without having to enumerate them .
Evidently , the task of triple sampling has not been explored in depth , although sampling is a more fundamental task than counting , and the former is useful for solving various other tasks , including counting . In recent years , some works on triple sampling have been proposed that are based on direct sampling , solely for the purpose of triangle count approximation . They sample only from a uniform distribution , and are not effective for sampling triples from an arbitrary user-defined distribution .
We revisit the well-studied problem of triangle count estimation in graph streams . Given a graph represented as a stream of m edges , our aim is to compute a (1 + -ε)-approximation to the triangle count T , using a small space algorithm .
The range searching problems which allow partition trees where every query enters only a sublinear number of nodes are characterized as those with finite Vapnik - Chervonenk is dimension . The concrete combinatorial bounds obtained imply — among others — that every set of n points in the plane < list><item > can be stored in an & Ogr;(n ) space data structure which allows triangle counting queries in & Ogr;(√n · < italic > log</italic><supscrpt>3</supscrpt > n ) time , and < /item><item > can be stored in an & Ogr;(n · < italic > log</italic > n ) space data structure which allows disk counting queries in & Ogr;(√n · < italic > log</italic><supscrpt>3</supscrpt > n ) time ; < /item></list > the preprocessing time for the data structures is polynomial . Recent results by Chazelle entail that these bounds for space and query time are optimal up to polylog — factors .
They introduced constraints that restrict the shapes of objects ( e.g. , requiring the pixels of a vertex to form a 2D box ) and how the representations of different objects can intersect . In this way , one can solve a variety of NP-hard graph problems , including pathwidth , bandwidth , optimum st-orientation , area-minimal ( bar-k ) visibility representation , boxicity-k graphs and others . For example , in a grid-based drawing of a visibility representation , each vertex is represented by a horizontal box of height 1 and each edge is represented by a vertical box of width 1 .
A large number of NP-hard graph problems become polynomial-time solvable on graph classes where the mim-width is bounded and quickly computable . Hence , when solving such problems on special graph classes , it is helpful to know whether the graph class under consideration has bounded mim-width .
Our newly developing theory of bidimensional graph problems provides general techniques for designing efficient fixed-parameter algorithms and approximation algorithms for NP-hard graph problems in broad classes of graphs . This theory applies to graph problems that are bidimensional in the sense that ( 1 ) the solution value for the k × k grid graph ( and similar graphs ) grows with k , typically as Ω(k2 ) , and ( 2 ) the solution value goes down when contracting edges and optionally when deleting edges .
We present an algorithmic framework ( including a single data structure ) that is extended into linear-time algorithms to solve several NP-complete graph problems ( i.e. , INDEPENDENT SET , M AXIMUM CUT , GRAPH COLORING , HAMILTONIAN CYCLE , and DISJOINT PATHS ) . The linearity is achieved assuming the provision of a branch decomposition of the instance graph .
The algorithm uses a linear number of processors and O(log n ) time . Many NP-complete graph problems are known to be solvable in polynomial time , when restricted to graphs with treewidth ≤ k , k constant . From the results in this paper , it follows that most of these problems are also in NC , when restricted to graphs with treewidth bounded by a constant .
In recent years many efforts have been done to solve NP-complete and time consuming problems with the help of DNA computing . In this paper , we use sticker model ( one of the most well-known models of DNA computing ) to present three DNA algorithms for solving three different NP-complete graph-based problems for the first time : domatic partition , kernel and induced path . Also we have simulated these algorithms to show their correctness .
This paper originates from the observation that many classical NP graph problems , including some NP-complete problems , are actually of very low nondeterministic time complexity . In order to formalize this observation , we define the complexity class vertex NLIN , which collects the graph problems computable on a nondeterministic RAM in time O(n ) .
While many parallel applications exhibit good spatial locality , other important codes in areas like graph problem-solving or CAD do not . Often , these irregular codes contain small records accessed via pointers .
The following graph addressing problem was studied by Graham and Pollak in devising a routing scheme for Pierce ’s loop switching network . Let G be a graph with n vertices .
Dynamic version of Traveling Salesman Problem Solving Traveling Salesman Problem based on Biogeography-based Optimization and Edge Assembly Cross-over Monochromatic Triangles, Intermediate Matrix Products, and Convolutions Local Computation Algorithms for Graphs of Non-constant Degrees The Minimum Spanning Tree Problem: Jarník's solution in historical and present context Discrete bacteria foraging optimization algorithm for graph based problems – a transition from continuous to discrete An Ant Colony System Metaheuristic Algorithm for Solving a Bi-Objective Purchasing Scheduling Problem Sparse Matrix Multiplication and Triangle Listing in the Congested Clique Model Sparse matrix multiplication and triangle listing in the Congested Clique model Simulation problems in parallel networks A Random Graph Generator On the cost of virtual private networks A BSP model graph processing system on many cores Polynomial kernels for graph and hypergraph optimisation problems Using the computer to visualise graph-oriented problems A Branch and Bound Algorithm for Coalition Structure Generation over Graphs Neural Message Passing for Quantum Chemistry Gated Graph Sequence Neural Networks A Non–Linear Feedback Neural Network for Graph Coloring The effects of network topology on strategic behavior Linear-time algorithms for graphs with bounded branchwidth PartitionedVC: Partitioned External Memory Graph Analytics Framework for SSDs List-coloring graphs without subdivisions and without immersions Totally odd subdivisions and parity subdivisions: Structures and Coloring Approximation Algorithms for Graph Homomorphism Problems PGX.ISO: Parallel and Efficient In-Memory Engine for Subgraph Isomorphism Finding Largest Subtrees and Smallest Supertrees Matching graphs with unique node labels Designing scientific workflow following a structure and provenance-aware strategy. (Conception de workflows scientifiques fondée sur la structure et la provenance) Querying RDF Data Using A Multigraph-based Approach Symmetric graphs and interpretations An Efficient Approach for Mining Frequent Subgraphs Automatic discovery of common design structures in CAD models FOGGER: an algorithm for graph generator discovery SubISO: A Scalable and Novel Approach for Subgraph Isomorphism Search in Large Graph Towards Hybrid Methods for Graph Pattern Queries Taxonomy-superimposed graph mining Uncertain Subgraph Query Processing over Uncertain Graphs Privacy-Preserving Triangle Counting in Large Graphs Fast Triangle Counting on GPU H-INDEX: Hash-Indexing for Parallel Triangle Counting on GPUs Fast BFS-Based Triangle Counting on GPUs Linear Algebra-Based Triangle Counting via Fine-Grained Tasking on Heterogeneous Environments : (Update on Static Graph Challenge) I/O-Efficient Algorithms on Triangle Listing and Counting Sampling Triples from Restricted Networks using MCMC Strategy How the Degeneracy Helps for Triangle Counting in Graph Streams Partition trees for triangle counting and other range searching problems PiGra- A Tool for Pixelated Graph Representations Bounding the Mim-Width of Hereditary Graph Classes The Bidimensionality Theory and Its Algorithmic Applications Linear-time algorithms for graphs with bounded branchwidth NC-Algorithms for Graphs with Small Treewidth Using sticker model of DNA computing to solve domatic partition, kernel and induced path problems Graph properties checkable in linear time in the number of vertices Speeding up irregular applications in shared-memory multiprocessors: memory binding and group prefetching On the Loop Switching Addressing Problem
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