ReFlO: an interactive tool for pipe-and-filter domain specification and program generation

ReFlO is a framework and interactive tool to record and systematize domain knowledge used by experts to derive complex pipe-and-filter (PnF) applications. Domain knowledge is encoded as transformations that alter PnF graphs by refinement (adding more details), flattening (removing modular boundaries), and optimization (substituting inefficient PnF graphs with more efficient ones). All three kinds of transformations arise in reverse-engineering legacy PnF applications. We present the conceptual foundation and tool capabilities of ReFlO, illustrate how parallel PnF applications are designed and generated, and how domain-specific libraries of transformations are developed.     

A linear-time algorithm for finding Hamiltonian (<Emphasis Type="Italic">s</Emphasis>, <Emphasis Type="Italic">t</Emphasis>)-paths in odd-sized rectangular grid graphs with a rectangular hole

A grid graph \(G_{\mathrm{g}}\) is a finite vertex-induced subgraph of the two-dimensional integer grid \(G^\infty \). A rectangular grid graph R(m, n) is a grid graph with horizontal size m and vertical size n. A rectangular grid graph with a rectangular hole is a rectangular grid graph R(m, n) such that a rectangular grid subgraph R(k, l) is removed from it. The Hamiltonian path problem for general grid graphs is NP-complete. In this paper, we give necessary conditions for the existence of a Hamiltonian path between two given vertices in an odd-sized rectangular grid graph with a rectangular hole. In addition, we show that how such paths can be computed in linear time.     

Mathematical models and routing algorithms for CAD technological preparation of cutting processes

Resource-conscious technologies for cutting sheet material include the ICP and ECP technologies that allow for aligning fragments of the contours of cutouts. In this work, we show the mathematical model for the problem of cutting out parts with these technologies and algorithms for finding cutting tool routes that satisfy technological constraints. We give a solution for the problem of representing a cutting plan as a plane graph G = (V,F,E), which is a homeomorphic image of the cutting plan. This has let us formalize technological constraints on the trajectory of cutting the parts according to the cutting plan and propose a series of algorithms for constructing a route in the graph G = (V,F,E), which is an image of an admissible trajectory. Using known coordinates of the preimages of vertices of graph G = (V,F,E) and the locations of fragments of the cutting plan that are preimages of edges of graph G = (V,F,E), the resulting route in the graph G = (V,E) can be interpreted as the cutting tool’s trajectory.The proposed algorithms for finding routes in a connected graph G have polynomial computational complexity. To find the optimal route in an unconnected graph G, we need to solve, for every dividing face f of graph G, a travelling salesman problem on the set of faces incident to f.     

MDS codes in Doob graphs

The Doob graph D(m, n), where m > 0, is a Cartesian product of m copies of the Shrikhande graph and n copies of the complete graph K ₄ on four vertices. The Doob graph D(m, n) is a distance-regular graph with the same parameters as the Hamming graph H(2m + n, 4). We give a characterization of MDS codes in Doob graphs D(m, n) with code distance at least 3. Up to equivalence, there are m ³/36+7m ²/24+11m/12+1?(m mod 2)/8?(m mod 3)/9 MDS codes with code distance 2m + n in D(m, n), two codes with distance 3 in each of D(2, 0) and D(2, 1) and with distance 4 in D(2, 1), and one code with distance 3 in each of D(1, 2) and D(1, 3) and with distance 4 in each of D(1, 3) and D(2, 2).     

Graph summarization with quality guarantees

We study the problem of graph summarization. Given a large graph we aim at producing a concise lossy representation (a summary) that can be stored in main memory and used to approximately answer queries about the original graph much faster than by using the exact representation. In this work we study a very natural type of summary: the original set of vertices is partitioned into a small number of supernodes connected by superedges to form a complete weighted graph. The superedge weights are the edge densities between vertices in the corresponding supernodes. To quantify the dissimilarity between the original graph and a summary, we adopt the reconstruction error and the cut-norm error. By exposing a connection between graph summarization and geometric clustering problems (i.e., k-means and k-median), we develop the first polynomial-time approximation algorithms to compute the best possible summary of a certain size under both measures. We discuss how to use our summaries to store a (lossy or lossless) compressed graph representation and to approximately answer a large class of queries about the original graph, including adjacency, degree, eigenvector centrality, and triangle and subgraph counting. Using the summary to answer queries is very efficient as the running time to compute the answer depends on the number of supernodes in the summary, rather than the number of nodes in the original graph.     

Answering subgraph queries over massive disk resident graphs

Due to its wide applications, subgraph query has attracted lots of attentions in database community. In this paper, we focus on subgraph query over a single large graph G, i.e., finding all embeddings of query Q in G. Different from existing feature-based approaches, we map all edges into a two-dimensional space R ² and propose a bitmap structure to index R ². At run time, we find a set of adjacent edge pairs (AEP) or star-style patterns (SSP) to cover Q. We develop edge join (EJ) algorithms to address both AEP and SSP subqueries. Based on the bitmap index, our method can optimize I/O and CPU cost. More importantly, our index has the linear space complexity instead of exponential complexity in feature-based approaches, which indicates that our index can scale well with respect to large data size. Furthermore, our index has light maintenance overhead, which has not been considered in most of existing work. Extensive experiments show that our method significantly outperforms existing ones in both online and offline processing with respect to query response time, index building time, index size and index maintenance overhead.     

Making Doubling Metrics Geodesic

The starting point of our research is the following problem: given a doubling metric ?=(V,d), can one (efficiently) find an unweighted graph G′=(V′,E′) with V?V′ whose shortest-path metric d′ is still doubling, and which agrees with d on V×V? While it is simple to show that the answer to the above question is negative if distances must be preserved exactly. However, allowing a (1+ε) distortion between d and d′ enables us bypass this hurdle, and obtain an unweighted graph G′ with doubling dimension at most a factor O(log?ε ^?1) times the doubling dimension of G.More generally, this paper gives algorithms that construct graphs G′ whose convex (or geodesic) closure has doubling dimension close to that of ?, and the shortest-path distances in G′ closely approximate those of ? when restricted to V×V. Similar results are shown when the metric ? is an additive (tree) metric and the graph G′ is restricted to be a tree.     

Embedding cycles and paths on solid grid graphs

Despite many algorithms for embedding graphs on unbounded grids, only a few results on embedding graphs on restricted grids have been published. In this paper, we study the problem of embedding paths and cycles on solid grid graphs. We show that a cycle of length k is unit-length embeddable on a solid grid graph G if k is an even integer between four and the length of the longest cycle of G. In addition, our result shows that a path of length k is unit-length embeddable on G, between its two given vertices s and t, if \(k\le L\) and \(k\equiv L (\mathrm{mod}\ 2)\), in which L is the length of the longest path between s and t. Our presented two algorithms show that such embeddings can be found in linear time for cycles and quadratic time for paths, with respect to the size of graph G. In the case of rectangular grid graphs, the running time of the algorithms can be improved to O(k) and O\((k^2)\), respectively. In addition, we extend our results to \(m\times n\times o\) 3D grids. A application of our result is in the interconnection network mapping in parallel processing.     

HD-GDD: high dimensional graph dominance drawing approach for reachability query

Efficiently answering reachability queries, which checks whether one vertex can reach another in a directed graph, has been studied extensively during recent years. However, the size of the graph that people are facing and generating nowadays is growing so rapidly that simple algorithms, such as BFS and DFS, are no longer feasible. Although Refined Online Search algorithms can scale to large graphs, they all suffer from the false positive problem. In this paper, we analyze the cause of false positive and propose an efficient High Dimensional coordinate generating method based on Graph Dominance Drawing (HD-GDD) to answer reachability queries in linear or even constant time. We conduct experiments on different graph structures and different graph sizes to fully evaluate the performance and behavior of our proposal. Empirical results demonstrate that our method outperforms state-of-the-art algorithms and can handle extensive graphs.     

(<Emphasis Type="Italic">a</Emphasis>, <Emphasis Type="Italic">d</Emphasis>)-Distance Antimagic Labeling of Some Types of Graphs

We analyze the necessary existence conditions for (a, d)-distance antimagic labeling of a graph G = (V, E) of order n. We obtain theorems that expand the family of not (a, d) -distance antimagic graphs. In particular, we prove that the crown P _n ○ P ₁ does not admit an (a, 1)-distance antimagic labeling for n ≥ 2 if a ≥ 2. We determine the values of a at which path P _n can be an (a, 1)-distance antimagic graph. Among regular graphs, we investigate the case of a circulant graph.     

Processing SPARQL queries over distributed RDF graphs

We propose techniques for processing SPARQL queries over a large RDF graph in a distributed environment. We adopt a “partial evaluation and assembly” framework. Answering a SPARQL query Q is equivalent to finding subgraph matches of the query graph Q over RDF graph G. Based on properties of subgraph matching over a distributed graph, we introduce local partial match as partial answers in each fragment of RDF graph G. For assembly, we propose two methods: centralized and distributed assembly. We analyze our algorithms from both theoretically and experimentally. Extensive experiments over both real and benchmark RDF repositories of billions of triples confirm that our method is superior to the state-of-the-art methods in both the system’s performance and scalability.     

Bimagic Vertex Labelings

The notion of the equivalence of vertex labelings on a given graph is introduced. The equivalence of three bimagic labelings for regular graphs is proved. A particular solution is obtained for the problem of the existence of a 1-vertex bimagic vertex labeling of multipartite graphs, namely, for graphs isomorphic with K_{n, n, m}. It is proved that the sequence of bi-regular graphs K_n(ij)?=?((K_n???1???M)?+?K₁)???(u_nu_i)???(u_nu_j) admits 1-vertex bimagic vertex labeling, where u_i, u_j is any pair of non-adjacent vertices in the graph K_n???1???M, u_n is a vertex of K₁, M is perfect matching of the complete graph K_n???1. It is established that if an r-regular graph G of order n is distance magic, then graph G + G has a 1-vertex bimagic vertex labeling with magic constants (n?+?1)(n?+?r)/2?+?n² and (n?+?1)(n?+?r)/2?+?nr. Two new types of graphs that do not admit 1-vertex bimagic vertex labelings are defined.     

The Complexity Ecology of Parameters: An Illustration Using Bounded Max Leaf Number

In the framework of parameterized complexity, exploring how one parameter affects the complexity of a different parameterized (or unparameterized problem) is of general interest. A well-developed example is the investigation of how the parameter treewidth influences the complexity of (other) graph problems. The reason why such investigations are of general interest is that real-world input distributions for computational problems often inherit structure from the natural computational processes that produce the problem instances (not necessarily in obvious, or well-understood ways). The max leaf number ml(G) of a connected graph G is the maximum number of leaves in a spanning tree for G. Exploring questions analogous to the well-studied case of treewidth, we can ask: how hard is it to solve 3-Coloring, Hamilton Path, Minimum Dominating Set, Minimum Bandwidth or many other problems, for graphs of bounded max leaf number? What optimization problems are W[1]-hard under this parameterization? We do two things:

1. (1)
   We describe much improved FPT algorithms for a large number of graph problems, for input graphs G for which ml(G)≤k, based on the polynomial-time extremal structure theory canonically associated to this parameter. We consider improved algorithms both from the point of view of kernelization bounds, and in terms of improved fixed-parameter tractable (FPT) runtimes O ^*(f(k)).
    
2. (2)
   The way that we obtain these concrete algorithmic results is general and systematic. We describe the approach, and raise programmatic questions.
    

     

Constant-Degree Graph Expansions that Preserve Treewidth

Many hard algorithmic problems dealing with graphs, circuits, formulas and constraints admit polynomial-time upper bounds if the underlying graph has small treewidth. The same problems often encourage reducing the maximal degree of vertices to simplify theoretical arguments or address practical concerns. Such degree reduction can be performed through a sequence of splittings of vertices, resulting in an expansion of the original graph. We observe that the treewidth of a graph may increase dramatically if the splittings are not performed carefully. In this context we address the following natural question: is it possible to reduce the maximum degree to a constant without substantially increasing the treewidth?We answer the above question affirmatively. We prove that any simple undirected graph G=(V,E) admits an expansion G′=(V′,E′) with the maximum degree ≤3 and tw(G′)≤tw(G)+1, where tw(?) is the treewidth of a graph. Furthermore, such an expansion will have no more than 2|E|+|V| vertices and 3|E| edges; it can be computed efficiently from a tree-decomposition of G. We also construct a family of examples for which the increase by 1 in treewidth cannot be avoided.     

Top-<Emphasis Type="Italic">k</Emphasis> overlapping densest subgraphs

Finding dense subgraphs is an important problem in graph mining and has many practical applications. At the same time, while large real-world networks are known to have many communities that are not well-separated, the majority of the existing work focuses on the problem of finding a single densest subgraph. Hence, it is natural to consider the question of finding the top-k densest subgraphs. One major challenge in addressing this question is how to handle overlaps: eliminating overlaps completely is one option, but this may lead to extracting subgraphs not as dense as it would be possible by allowing a limited amount of overlap. Furthermore, overlaps are desirable as in most real-world graphs there are vertices that belong to more than one community, and thus, to more than one densest subgraph. In this paper we study the problem of finding top-k overlapping densest subgraphs, and we present a new approach that improves over the existing techniques, both in theory and practice. First, we reformulate the problem definition in a way that we are able to obtain an algorithm with constant-factor approximation guarantee. Our approach relies on using techniques for solving the max-sum diversification problem, which however, we need to extend in order to make them applicable to our setting. Second, we evaluate our algorithm on a collection of benchmark datasets and show that it convincingly outperforms the previous methods, both in terms of quality and efficiency.     

Approximating the Sparsest k-Subgraph in Chordal Graphs

Given a simple undirected graph G = (V, E) and an integer k < |V|, the Sparsest k-Subgraph problem asks for a set of k vertices which induces the minimum number of edges. As a generalization of the classical independent set problem, Sparsest k-Subgraph is ????-hard and even not approximable unless ?????? in general graphs. Thus, we investigate Sparsest k-Subgraph in graph classes where independent set is polynomial-time solvable, such as subclasses of perfect graphs. Our two main results are the ????-hardness of Sparsest k-Subgraph on chordal graphs, and a greedy 2-approximation algorithm. Finally, we also show how to derive a P T A S for Sparsest k-Subgraph on proper interval graphs.     

Testing list <Emphasis Type="Italic">H</Emphasis>-homomorphisms

In the List H- Homomorphism Problem, for a graph H that is a parameter of the problem, an instance consists of an undirected graph G with a list constraint \({L(v) \subseteq V(H)}\) for each variable \({v \in V(G)}\), and the objective is to determine whether there is a list H-homomorphism \({f:V(G) \to V(H)}\), that is, \({f(v) \in L(v)}\) for every \({v \in V(G)}\) and \({(f(u),f(v)) \in E(H)}\) whenever \({(u,v) \in E(G)}\).We consider the problem of testing list H-homomorphisms in the following weighted setting: An instance consists of an undirected graph G, list constraints L, weights imposed on the vertices of G, and a map \({f:V(G) \to V(H)}\) given as an oracle access. The objective is to determine whether f is a list H-homomorphism or far from any list H-homomorphism. The farness is measured by the total weight of vertices \({v \in V(G)}\) for which f(v) must be changed so as to make f a list H-homomorphism. In this paper, we classify graphs H with respect to the number of queries to f required to test the list H-homomorphisms. Specifically, we show that (i) list H-homomorphisms are testable with a constant number of queries if and only if H is a reflexive complete graph or an irreflexive complete bipartite graph and (ii) list H-homomorphisms are testable with a sublinear number of queries if and only if H is a bi-arc graph.     

On (<Emphasis Type="Italic">a</Emphasis>, <Emphasis Type="Italic">d</Emphasis> )-Distance Anti-Magic and 1-Vertex Bimagic Vertex Labelings of Certain Types of Graphs

The results for the corona P _n ?°?P₁ are generalized, which make it possible to state that P _n ?°?P₁ is not an ( a, d)-distance antimagic graph for arbitrary values of a and d. A condition for the existence of an ( a, d)-distance antimagic labeling of a hypercube Q _n is obtained. Functional dependencies are found that generate this type of labeling for Q _n . It is proved by the method of mathematical induction that Q _n is a (2 ⁿ ?+?n???1,?n???2) -distance antimagic graph. Three types of graphs are defined that do not allow a 1-vertex bimagic vertex labeling. A relation between a distance magic labeling of a regular graph G and a 1-vertex bimagic vertex labeling of G?∪?G is established.     

Organization of self-diagnosis of the discrete multicomponent systems a structure like bipartite quasicomplete graphs

Organization of an efficient self-diagnosis of the multicomponent computer and communication systems of diverse structures always attracted attention of the researchers and engineers. A method to solve these problems is presented in the paper by way of the example of a system whose structure is modeled by a uniform ordinary bipartite graph of diameter d = 3, any degree s > 1, and any number n of vertices, where n = s(s ? 1) + 1. The method requires checking of (s ? 1)³ graph loops of length eight each, which is smaller than the number s ²(s ? 1) + s of checks of single graph edges.     

Degree-Constrained Graph Orientation: Maximum Satisfaction and Minimum Violation

A degree-constrained graph orientation of an undirected graph G is an assignment of a direction to each edge in G such that the outdegree of every vertex in the resulting directed graph satisfies a specified lower and/or upper bound. Such graph orientations have been studied for a long time and various characterizations of their existence are known. In this paper, we consider four related optimization problems introduced in reference (Asahiro et al. LNCS 7422, 332–343 (2012)): For any fixed non-negative integer W, the problems MAX W-LIGHT, MIN W-LIGHT, MAX W-HEAVY, and MIN W-HEAVY take as input an undirected graph G and ask for an orientation of G that maximizes or minimizes the number of vertices with outdegree at most W or at least W. As shown in Asahiro et al. LNCS 7422, 332–343 (2012)).