Showing papers on "Split graph published in 2014"

Parallel subgraph listing in a large-scale graph

Abstract: Subgraph listing is a fundamental operation to many graph and network analyses. The problem itself is computationally expensive and is well-studied in centralized processing algorithms. However, the centralized solutions cannot scale well to large graphs. Recently, several parallel approaches are introduced to handle the large graphs. Unfortunately, these parallel approaches still rely on the expensive join operations, thus cannot achieve high performance. In this paper, we design a novel parallel subgraph listing framework, named PSgL. The PSgL iteratively enumerates subgraph instances and solves the subgraph listing in a divide-and-conquer fashion. The framework completely relies on the graph traversal, and avoids the explicit join operation. Moreover, in order to improve its performance, we propose several solutions to balance the workload and reduce the size of intermediate results. Specially, we prove the problem of partial subgraph instance distribution for workload balance is NP-hard, and carefully design a set of heuristic strategies. To further reduce the enormous intermediate results, we introduce three independent mechanisms, which are automorphism breaking of the pattern graph, initial pattern vertex selection based on a cost model, and a pruning method based on a light-weight index. We have implemented the prototype of PSgL, and run comprehensive experiments of various graph listing operations on diverse large graphs. The experiments clearly demonstrate that PSgL is robust and can achieve performance gain over the state-of-the-art solutions up to 90%.

Reconfiguration graphs for vertex colourings of chordal and chordal bipartite graphs

Abstract: A k-colouring of a graph G=(V,E) is a mapping c:V?{1,2,?,k} such that c(u)?c(v) whenever uv is an edge. The reconfiguration graph of the k-colourings of G contains as its vertex set the k-colourings of G, and two colourings are joined by an edge if they differ in colour on just one vertex of G. We introduce a class of k-colourable graphs, which we call k-colour-dense graphs. We show that for each k-colour-dense graph G, the reconfiguration graph of the l-colourings of G is connected and has diameter O(|V|2), for all l?k+1. We show that this graph class contains the k-colourable chordal graphs and that it contains all chordal bipartite graphs when k=2. Moreover, we prove that for each k?2 there is a k-colourable chordal graph G whose reconfiguration graph of the (k+1)-colourings has diameter ?(|V|2).

Fast maximum clique algorithms for large graphs

Abstract: We propose a fast, parallel maximum clique algorithm for large sparse graphs that is designed to exploit characteristics of social and information networks. Despite clique's status as an NP-hard problem with poor approximation guarantees, our method exhibits nearly linear runtime scaling over real-world networks ranging from 1000 to 100 million nodes. In a test on a social network with 1.8 billion edges, the algorithm finds the largest clique in about 20 minutes. Key to the efficiency of our algorithm are an initial heuristic procedure that finds a large clique quickly and a parallelized branch and bound strategy with aggressive pruning and ordering techniques. We use the algorithm to compute the largest temporal strong components of temporal contact networks.

Minimum vertex blocker clique problem

Abstract: We study the minimum vertex blocker clique problem VBCP,1 which is to remove a subset of vertices of minimum cardinality in a weighted undirected graph, such that the maximum weight of a clique in the remaining graph is bounded above by a given integer ri¾?1. Cliques are among the most popular concepts used to model cohesive clusters in different graph-based applications, such as social, biological, and communication networks. The general case of VBCP on weighted graphs is known to be NP-hard, and we show that the special case on unweighted graphs is also NP-hard for any fixed integer ri¾?1. We present an analytical lower bound on the cardinality of an optimal solution to VBCP, as well as formulate VBCP as a linear 0-1 program with an exponential number of constraints. Facet-inducing inequalities for the convex hull of feasible solutions to VBCP are also identified. Furthermore, we develop the first exact algorithm for solving VBCP, which solves the proposed formulation by using a row generation approach. Computational results obtained by utilizing this algorithm on a test-bed of randomly generated instances are also provided. © 2014 Wiley Periodicals, Inc. NETWORKS, Vol. 641, 48-64 2014

Multi-graph-view Learning for Graph Classification

Abstract: Graph classification has traditionally focused on graphs generated from a single feature view. In many applications, it is common to have useful information from different channels/views to describe objects, which naturally results in a new representation with multiple graphs generated from different feature views being used to describe one object. In this paper, we formulate a new Multi-Graph-View learning task for graph classification, where each object to be classified contains graphs from multiple graph-views. This problem setting is essentially different from traditional single-graph-view graph classification, where graphs are from one single feature view. To solve the problem, we propose a Cross Graph-View Sub graph Feature based Learning (gCGVFL) algorithm that explores an optimal set of sub graphs, across multiple graph-views, as features to represent graphs. Specifically, we derive an evaluation criterion to estimate the discriminative power and the redundancy of sub graph features across all views, and assign proper weight values to each view to indicate its importance for graph classification. The iterative cross graph-view sub graph scoring and graph-view weight updating form a closed loop to find optimal sub graphs to represent graphs for multi-graph-view learning. Experiments and comparisons on real-world tasks demonstrate the algorithm's performance.

Interval scheduling and colorful independent sets

Abstract: Numerous applications in scheduling, such as resource allocation or steel manufacturing, can be modeled using the NP-hard Independent Set problem (given an undirected graph and an integer k, find a set of at least k pairwise non-adjacent vertices). Here, one encounters special graph classes like 2-union graphs (edge-wise unions of two interval graphs) and strip graphs (edge-wise unions of an interval graph and a cluster graph), on which Independent Set remains NP-hard but admits constant-ratio approximations in polynomial time. We study the parameterized complexity of Independent Set on 2-union graphs and on subclasses like strip graphs. Our investigations significantly benefit from a new structural "compactness" parameter of interval graphs and novel problem formulations using vertex-colored interval graphs. Our main contributions are: 1. We show a complexity dichotomy: restricted to graph classes closed under induced subgraphs and disjoint unions, Independent Set is polynomial-time solvable if both input interval graphs are cluster graphs, and is NP-hard otherwise. 2. We chart the possibilities and limits of effective polynomial-time preprocessing (also known as kernelization). 3. We extend Halldorsson and Karlsson (2006)'s fixed-parameter algorithm for Independent Set on strip graphs parameterized by the structural parameter "maximum number of live jobs" to show that the problem (also known as Job Interval Selection) is fixed-parameter tractable with respect to the parameter k and generalize their algorithm from strip graphs to 2-union graphs. Preliminary experiments with random data indicate that Job Interval Selection with up to fifteen jobs and 5*10^5 intervals can be solved optimally in less than five minutes.

Approximation Algorithms for Intersection Graphs

Abstract: We study three complexity parameters that, for each vertex v, are an upper bound for the number of cliques that are sufficient to cover a subset S(v) of its neighbors. We call a graph k-perfectly groupable if S(v) consists of all neighbors, k-simplicial if S(v) consists of the neighbors with a higher number after assigning distinct numbers to all vertices, and k-perfectly orientable if S(v) consists of the endpoints of all outgoing edges from v for an orientation of all edges. These parameters measure in some sense how chordal-like a graph is—the last parameter was not previously considered in literature. The similarity to chordal graphs is used to construct simple polynomial-time approximation algorithms with constant approximation ratio for many NP-hard problems, when restricted to graphs for which at least one of the three complexity parameters is bounded by a constant. As applications we present approximation algorithms with constant approximation ratio for maximum weighted independent set, minimum (independent) dominating set, minimum vertex coloring, maximum weighted clique, and minimum clique partition for large classes of intersection graphs.

Graphs with Extremal Matching Energies and Prescribed Parameters

Abstract: Gutman and Wagner [I. Gutman, S. Wagner, The matching energy of a graph, Discrete Appl. Math. 160 (2012) 2177-2187] defined the matching energy of a graph and gave some properties of the matching energy, especially in characterizing the extremal graphs among some classes of graphs. Further, the graphs with maximum matching energy and given connectivity (resp. chromatic number) were characterized by Li and Yan. In this paper, the unicyclic graphs with fixed girth and the graphs with given clique number are characterized in terms of maximum and minimum matching energy.

Metric dimension and zero forcing number of two families of line graphs

Abstract: Zero forcing number has recently become an interesting graph parameter studied in its own right since its introduction by the “AIM Minimum Rank–Special Graphs Work Group”, whereas metric dimension is a well-known graph parameter. We investigate the metric dimension and the zero forcing number of some line graphs by first determining the metric dimension and the zero forcing number of the line graphs of wheel graphs and the bouquet of circles. We prove that $Z(G) \le 2Z(L(G))$ for a simple and connected graph $G$. Further, we show that $Z(G) \le Z(L(G))$ when $G$ is a tree or when $G$ contains a Hamiltonian path and has a certain number of edges. We compare the metric dimension with the zero forcing number of a line graph by demonstrating a couple of inequalities between the two parameters. We end by stating some open problems.

On toughness and hamiltonicity of 2K2-free graphs

Abstract: The toughness of a (noncomplete) graph G is the minimum value of t for which there is a vertex cut A whose removal yields |A|/t components. Determining toughness is an NP-hard problem for general input graphs. The toughness conjecture of Chvatal, which states that there exists a constant t such that every graph on at least three vertices with toughness at least t is hamiltonian, is still open for general graphs. We extend some known toughness results for split graphs to the more general class of 2K2-free graphs, that is, graphs that do not contain two vertex-disjoint edges as an induced subgraph. We prove that the problem of determining toughness is polynomially solvable and that Chvatal's toughness conjecture is true for 2K2-free graphs.

Robust Hamiltonicity of Dirac graphs

Abstract: A graph is Hamiltonian if it contains a cycle which passes through every vertex of the graph exactly once. A classical theorem of Dirac from 1952 asserts that every graph on n vertices with minimum degree at least n/2 is Hamiltonian. We refer to such graphs as Dirac graphs. In this paper we extend Dirac’s theorem in two directions and show that Dirac graphs are robustly Hamiltonian in a very strong sense. First, we consider a random subgraph of a Dirac graph obtained by taking each edge independently with probability p, and prove that there exists a constant C such that if p ≥ C log n/n, then a.a.s. the resulting random subgraph is still Hamiltonian. Second, we prove that if a (1 : b) Maker-Breaker game is played on a Dirac graph, then Maker can construct a Hamiltonian subgraph as long as the bias b is at most cn/ log n for some absolute constant c > 0. Both of these results are tight up to a constant factor, and are proved under one general framework.

Pushing the Envelope in Graph Compression

Abstract: We improve the state-of-the-art method for the compression of web and other similar graphs by introducing an elegant technique which further exploits the clustering properties observed in these graphs. The analysis and experimental evaluation of our method shows that it outperforms the currently best method of Boldi et al. by achieving a better compression ratio and retrieval time. Our method exhibits vast improvements on certain families of graphs, such as social networks, by taking advantage of their compressibility characteristics, and ensures that the compression ratio will not worsen for any graph, since it easily falls back to the state-of-the-art method.

Contracting chordal graphs and bipartite graphs to paths and trees

Abstract: We study the following two graph modification problems: given a graph G and an integer k, decide whether G can be transformed into a tree or into a path, respectively, using at most k edge contractions. These problems, which we call Tree Contraction and Path Contraction, respectively, are known to be NP-complete in general. We show that on chordal graphs these problems can be solved in O(n+m) and O(nm) time, respectively. As a contrast, both problems remain NP-complete when restricted to bipartite input graphs.

Exploring Subexponential Parameterized Complexity of Completion Problems

Abstract: Let F be a family of graphs. In the F-Completion problem, we are given an n-vertex graph G and an integer k as input, and asked whether at most k edges can be added to G so that the resulting graph does not contain a graph from F as an induced subgraph. It appeared recently that special cases of F-Completion, the problem of completing into a chordal graph known as "Minimum Fill-in", corresponding to the case of F={C_4,C_5,C_6,...}, and the problem of completing into a split graph, i.e., the case of F={C_4,2K_2,C_5}, are solvable in parameterized subexponential time. The exploration of this phenomenon is the main motivation for our research on F-Completion. In this paper we prove that completions into several well studied classes of graphs without long induced cycles also admit parameterized subexponential time algorithms by showing that: - The problem Trivially Perfect Completion is solvable in parameterized subexponential time, that is F-Completion for F={C_4,P_4}, a cycle and a path on four vertices. - The problems known in the literature as Pseudosplit Completion, the case where F={2K_2,C_4}, and Threshold Completion, where F={2K_2,P_4,C_4}, are also solvable in subexponential time. We complement our algorithms for $F$-Completion with the following lower bounds: - For F={2K_2}, F={C_4}, F={P_4}, and F={2K_2,P_4}, F-Completion cannot be solved in time 2^o(k).n^O(1) unless the Exponential Time Hypothesis (ETH) fails. Our upper and lower bounds provide a complete picture of the subexponential parameterized complexity of F-Completion problems for F contained inside {2K_2,C_4,P_4}.

Counting Independent Sets of a Fixed Size in Graphs with a Given Minimum Degree

Abstract: Galvin showed that for all fixed δ and sufficiently large n, the n-vertex graph with minimum degree δ that admits the most independent sets is the complete bipartite graph . He conjectured that except perhaps for some small values of t, the same graph yields the maximum count of independent sets of size t for each possible t. Evidence for this conjecture was recently provided by Alexander, Cutler, and Mink, who showed that for all triples with , no n-vertex bipartite graph with minimum degree δ admits more independent sets of size t than . Here, we make further progress. We show that for all triples with and , no n-vertex graph with minimum degree δ admits more independent sets of size t than , and we obtain the same conclusion for and . Our proofs lead us naturally to the study of an interesting family of critical graphs, namely those of minimum degree δ whose minimum degree drops on deletion of an edge or a vertex.

Frequent subgraph discovery in large attributed streaming graphs

Abstract: The problem of finding frequent subgraphs in large dynamic graphs has so far only considered a dynamic graph as being represented by a series of static snapshots taken at various points in time. This representation of a dynamic graph does not lend itself well to real time processing of real world graphs like social networks or internet traffic which consist of a stream of nodes and edges. In this paper we propose an algorithm that discovers the frequent subgraphs present in a graph represented by a stream of labeled nodes and edges. Our algorithm is efficient and is easily tuned by the user to produce interesting patterns from various kinds of graph data. In our model, updates to the graph arrive in the form of batches which contain new nodes and edges. Our algorithm continuously reports the frequent subgraphs that are estimated to be found in the entire graph as each batch arrives. We evaluate our system using five large dynamic graph datasets: the Hetrec 2011 challenge data, Twitter, DBLP and two synthetic. We evaluate our approach against two popular large graph miners, i.e., SUBDUE and GERM. Our experimental results show that we can find the same frequent subgraphs as a non-incremental approach applied to snapshot graphs, and in less time.

Randomized Experimental Design for Causal Graph Discovery

Abstract: We examine the number of controlled experiments required to discover a causal graph. Hauser and Buhlmann [1] showed that the number of experiments required is logarithmic in the cardinality of maximum undirected clique in the essential graph. Their lower bounds, however, assume that the experiment designer cannot use randomization in selecting the experiments. We show that significant improvements are possible with the aid of randomization - in an adversarial (worst-case) setting, the designer can then recover the causal graph using at most O(log log n) experiments in expectation. This bound cannot be improved; we show it is tight for some causal graphs. We then show that in a non-adversarial (average-case) setting, even larger improvements are possible: if the causal graph is chosen uniformly at random under a Erdos-Renyi model then the expected number of experiments to discover the causal graph is constant. Finally, we present computer simulations to complement our theoretic results. Our work exploits a structural characterization of essential graphs by Andersson et al. [2]. Their characterization is based upon a set of orientation forcing operations. Our results show a distinction between which forcing operations are most important in worst-case and average-case settings.

A contribution to the theory of graph homomorphisms and colorings

Abstract: An oriented graph is a directed graph with no cycle of length at most two. A homomorphism of an oriented graph to another oriented graph is an arc preserving vertex mapping. To push a vertex is to switch the direction of the arcs incident to it. An orientable graph is an equivalence class of oriented graph with respect to the push operation. An orientable graph [−→G] admits a homomorphism to an orientable graph [−→H] if an element of [−→G] admits a homomorphism to an element of [−→H]. A signified graph (G, Σ) is a graph whose edges are assigned either a positive sign or a negative sign, while Σ denotes the set of edges with negative signs assigned to them. A homomorphism of a signified graph to another signified graph is a vertex mapping such that the image of a positive edge is a positive edge and the image of a negative edge is a negative edge. A signed graph [G, Σ] admits a homomorphism to a signed graph [H, Λ] if an element of [G, Σ] admits a homomorphism to an element of [H, Λ]. The oriented chromatic number of an oriented graph −→G is the minimum order of an oriented graph −→H such that −→G admits a homomorphism to −→H. A set R of vertices of an oriented graph −→G is an oriented relative clique if no two vertices of R can have the same image under any homomorphism. The oriented relative clique number of an oriented graph −→G is the maximum order of an oriented relative clique of −→G. An oriented clique or an oclique is an oriented graph whose oriented chromatic number is equal to its order. The oriented absolute clique number of an oriented graph −→G is the maximum order of an oclique contained in −→G as a subgraph. The chromatic number, the relative chromatic number and the absolute chromatic number for orientable graphs, signified graphs and signed graphs are defined similarly. In this thesis we study the chromatic number, the relative clique number and the absolute clique number of the above mentioned four types of graphs. We specifically study these three parameters for the family of outerplanar graphs, of outerplanar graphs with given girth, of planar graphs and of planar graphs with given girth. We also try to investigate the relation between the four types of graphs and prove some results regarding that. In this thesis, we provide tight bounds for the absolute clique number of these families in all these four settings. We provide improved bounds for relative clique numbers for the same. For some of the cases we manage to provide improved bounds for the chromatic number as well. One of the most difficult results that we prove here is that the oriented absolute clique number of the family of planar graphs is at most 15. This result settles a conjecture made by Klostermeyer and MacGillivray in 2003. Using the same technique we manage to prove similar results for orientable planar graphs and signified planar graphs. We also prove that the signed chromatic number of triangle-free planar graphs is at most 25 using the discharging method. This also implies that the signified chromatic number of trianglefree planar graphs is at most 50 improving the previous upper bound. We also study the 2-dipath and oriented L(p, q)-labeling (labeling with a condition for distance one and two) for several families of planar graphs. It was not known if the categorical product of orientable graphs and of signed graphs exists. We prove both the existence and also provide formulas to construct them. Finally, we propose some conjectures and mention some future directions of works to conclude the thesis.

Clique-transversal sets and clique-coloring in planar graphs

Abstract: Let G=(V,E) be a graph. A clique-transversal setD is a subset of vertices of G such that D meets all cliques of G, where a clique is defined as a complete subgraph maximal under inclusion and having at least two vertices. The clique-transversal number, denoted by @t"C(G), of G is the cardinality of a minimum clique-transversal set in G. A k-clique-coloring of G is a k-coloring of its vertices such that no clique is monochromatic. All planar graphs have been proved to be 3-clique-colorable by Mohar and Skrekovski [B. Mohar, R. Skrekovski, The Grotzsch theorem for the hypergraph of maximal cliques, Electron. J. Combin. 6 (1999) #R26]. Erdos et al. [P. Erdos, T. Gallai, Zs. Tuza, Covering the cliques of a graph with vertices, Discrete Math. 108 (1992) 279-289] proposed to find sharp estimates on @t"C(G) for planar graphs. In this paper we first show that every outerplanar graph G of order n(>=2) has @t"C(G)@?3n/5 and the bound is tight. Secondly, we prove that every claw-free planar graph different from an odd cycle is 2-clique-colorable and we present a polynomial-time algorithm to find the 2-clique-coloring. As a by-product of the result, we obtain a tight upper bound on the clique-transversal number for claw-free planar graphs.

Local Algorithms for Sparse Spanning Graphs

Abstract: Constructing a spanning tree of a graph is one of the most basic tasks in graph theory. We consider a relaxed version of this problem in the setting of local algorithms. The relaxation is that the constructed subgraph is a sparse spanning subgraph containing at most $(1+\epsilon)n$ edges (where $n$ is the number of vertices and $\epsilon$ is a given approximation/sparsity parameter). In the local setting, the goal is to quickly determine whether a given edge $e$ belongs to such a subgraph, without constructing the whole subgraph, but rather by inspecting (querying) the local neighborhood of $e$. The challenge is to maintain consistency. That is, to provide answers concerning different edges according to the same spanning subgraph. We first show that for general bounded-degree graphs, the query complexity of any such algorithm must be $\Omega(\sqrt{n})$. This lower bound holds for constant-degree graphs that have high expansion. Next we design an algorithm for (bounded-degree) graphs with high expansion, obtaining a result that roughly matches the lower bound. We then turn to study graphs that exclude a fixed minor (and are hence non-expanding). We design an algorithm for such graphs, which may have an unbounded maximum degree. The query complexity of this algorithm is $poly(1/\epsilon, h)$ (independent of $n$ and the maximum degree), where $h$ is the number of vertices in the excluded minor. Though our two algorithms are designed for very different types of graphs (and have very different complexities), on a high-level there are several similarities, and we highlight both the similarities and the differences.