Stochastic graph as a model for social networks

Social networks are usually modeled and represented as deterministic graphs with a set of nodes as users and edges as connection between users of networks. Due to the uncertain and dynamic nature of user behavior and human activities in social networks, their structural and behavioral parameters are time varying parameters and for this reason using deterministic graphs for modeling and analysis of behavior of users may not be appropriate. In this paper, we propose that stochastic graphs, in which weights associated with edges are random variables, may be a better candidate as a graph model for social network analysis. Thus, we first propose generalization of some network measures for stochastic graphs and then propose six learning automata based algorithms for calculating these measures under the situation that the probability distribution functions of the edge weights of the graph are unknown. Simulations on different synthetic stochastic graphs for calculating the network measures using the proposed algorithms show that in order to obtain good estimates for the network measures, the required number of samples taken from edges of the graph is significantly lower than that of standard sampling method aims to analysis of human behavior in online social networks.     

Approximation Algorithms for Intersection Graphs

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.     

Efficient algorithms for (3, 1) graphs

In this paper, we define a class of graphs which are referred to as (3, 1) graphs. A graph is a member of this class if it has the property that within each set of three vertices, there is at least one edge. We derive a lower bound for the size of a maximum clique in a (3, 1) graph as well as an upper bound for the size of a minimum clique covering. In addition, we show that there exists a linear algorithm for constructing a Hamiltonian circuit in a connected (3, 1) graph and an n⁴-algorithm for finding a minimum coloring in a (3, 1) graph.     

Using an interval graph approach to efficiently solve the housing benefit data retrieval problem

Rappos and Thompson use a set covering formulation and a commercial software package to solve the problem of trying to minimize the number of data sets that have to be read in retrieving all new housing benefit (HB) data entries for a fixed period of time. In this paper, we show that determining the minimum number of data sets that have to be read in retrieving all new HB data entries for a fixed period of time can be solved by finding a minimum size clique cover for an interval graph. Since it is well‐known that a greedy algorithm finds a guaranteed minimum size clique cover for an interval graph, this approach will be more efficient than a set covering approach. Finally, it is obvious that this interval graph formulation and greedy algorithm solution approach is applicable to other data retrieval problems.     

Approximating the maximum weight clique using replicator dynamics

Given an undirected graph with weights on the vertices, the maximum weight clique problem (MWCP) is to find a subset of mutually adjacent vertices (a clique) having the largest total weight. This is a generalization of the problem of finding the maximum cardinality clique of an unweighted graph, which is the special case of the MWCP when all vertex weights are equal. The problem is NP-hard for arbitrary graphs, and so is the problem of approximating it within a constant factor. We present a parallel, distributed heuristic for approximating the MWCP based on dynamics principles. It centers around a continuous characterization of the MWCP (a purely combinatorial problem), and lets it be formulated in terms of continuous quadratic programming. One drawback is the presence of spurious solutions, and we present their characterizations. To avoid them we introduce a regularized continuous formulation of the MWCP and show how it completely solves the problem. The formulation naturally maps onto a parallel, distributed computational network whose dynamical behavior is governed by the replicator equations. These are dynamical systems introduced in evolutionary game theory and population genetics to model evolutionary processes on a macroscopic scale. We present theoretical results which guarantee that the solutions provided by our clique finding replicator network are actually those sought. Experimental results confirm the effectiveness of the proposed approach.     

Incremental <Emphasis Type="Italic">k</Emphasis>-core decomposition: algorithms and evaluation

A k-core of a graph is a maximal connected subgraph in which every vertex is connected to at least k vertices in the subgraph. k-core decomposition is often used in large-scale network analysis, such as community detection, protein function prediction, visualization, and solving NP-hard problems on real networks efficiently, like maximal clique finding. In many real-world applications, networks change over time. As a result, it is essential to develop efficient incremental algorithms for dynamic graph data. In this paper, we propose a suite of incremental k-core decomposition algorithms for dynamic graph data. These algorithms locate a small subgraph that is guaranteed to contain the list of vertices whose maximum k-core values have changed and efficiently process this subgraph to update the k-core decomposition. We present incremental algorithms for both insertion and deletion operations, and propose auxiliary vertex state maintenance techniques that can further accelerate these operations. Our results show a significant reduction in runtime compared to non-incremental alternatives. We illustrate the efficiency of our algorithms on different types of real and synthetic graphs, at varying scales. For a graph of 16 million vertices, we observe relative throughputs reaching a million times, relative to the non-incremental algorithms.     

一种增量式约简方法求解最小顶点覆盖问题

最小顶点覆盖问题是一个应用很广泛的NP难题,针对该问题给出一种增量式属性约简方法。首先将最小顶点覆盖问题转化为一个决策表的最小属性约简问题;利用增量式属性约简思想,随着图中边数的增多,提出一种更新最小顶点覆盖的增量式属性约简算法;该算法时间复杂度低于计算整个图的最小顶点覆盖的时间复杂度,同时针对大规模图问题,可随着边的增加动态更新最小顶点覆盖,因此降低了属性约简的方法求解最小顶点覆盖问题的运行时间;实验结果表明该算法的可行性和有效性。     

Experimental Analysis of Heuristic Algorithms for the Dominating Set Problem

We say a vertex v in a graph G covers a vertex w if v=w or if v and w are adjacent. A subset of vertices of G is a dominating set if it collectively covers all vertices in the graph. The dominating set problem, which is NP-hard, consists of finding a smallest possible dominating set for a graph. The straightforward greedy strategy for finding a small dominating set in a graph consists of successively choosing vertices which cover the largest possible number of previously uncovered vertices. Several variations on this greedy heuristic are described and the results of extensive testing of these variations is presented. A more sophisticated procedure for choosing vertices, which takes into account the number of ways in which an uncovered vertex may be covered, appears to be the most successful of the algorithms which are analyzed. For our experimental testing, we used both random graphs and graphs constructed by test case generators which produce graphs with a given density and a specified size for the smallest dominating set. We found that these generators were able to produce challenging graphs for the algorithms, thus helping to discriminate among them, and allowing a greater variety of graphs to be used in the experiments. Received October 27, 1998; revised March 25, 2001.     

Approximating maximum clique with a Hopfield network

In a graph, a clique is a set of vertices such that every pair is connected by an edge. MAX-CLIQUE is the optimization problem of finding the largest clique in a given graph and is NP-hard, even to approximate well. Several real-world and theory problems can be modeled as MAX-CLIQUE. In this paper, we efficiently approximate MAX-CLIQUE in a special case of the Hopfield network whose stable states are maximal cliques. We present several energy-descent optimizing dynamics; both discrete (deterministic and stochastic) and continuous. One of these emulates, as special cases, two well-known greedy algorithms for approximating MAX-CLIQUE. We report on detailed empirical comparisons on random graphs and on harder ones. Mean-field annealing, an efficient approximation to simulated annealing, and a stochastic dynamics are the narrow but clear winners. All dynamics approximate much better than one which emulates a "naive" greedy heuristic.     

Algorithms for maximum weight induced paths

A family of graphs is a k-bounded-hole family if every graph in the family has no holes with more than k vertices. The problem of finding in a graph a maximum weight induced path has applications in large communication and neural networks when worst case communication time needs to be evaluated; unfortunately this problem is NP-hard even when restricted to bipartite graphs. We show that this problem has polynomial time algorithms for k-bounded-hole families of graphs, for interval-filament graphs and for graphs decomposable by clique cut-sets or by splits into prime subgraphs for which such algorithms exist.     

On Making a Distinguished Vertex of Minimum Degree by Vertex Deletion

For directed and undirected graphs, we study how to make a distinguished vertex the unique minimum-(in)degree vertex through deletion of a minimum number of vertices. The corresponding NP-hard optimization problems are motivated by applications concerning control in elections and social network analysis. Continuing previous work for the directed case, we show that the problem is W[2]-hard when parameterized by the graph’s feedback arc set number, whereas it becomes fixed-parameter tractable when combining the parameters “feedback vertex set number” and “number of vertices to delete”. For the so far unstudied undirected case, we show that the problem is NP-hard and W[1]-hard when parameterized by the “number of vertices to delete”. On the positive side, we show fixed-parameter tractability for several parameterizations measuring tree-likeness. In particular, we provide a dynamic programming algorithm for graphs of bounded treewidth and a vertex-linear problem kernel with respect to the parameter “feedback edge set number”. On the contrary, we show a non-existence result concerning polynomial-size problem kernels for the combined parameter “vertex cover number and number of vertices to delete”, implying corresponding non-existence results when replacing vertex cover number by treewidth or feedback vertex set number.     

On Independent Vertex Sets in Subclasses of Apple-Free Graphs

In a finite undirected graph, an apple consists of a chordless cycle of length at least 4, and an additional vertex which is not in the cycle and sees exactly one of the cycle vertices. A graph is apple-free if it contains no induced subgraph isomorphic to an apple. Apple-free graphs are a common generalization of chordal graphs, claw-free graphs and cographs and occur in various papers. The Maximum Weight Independent Set (MWS) problem is efficiently solvable on chordal graphs, on cographs as well as on claw-free graphs. In this paper, we obtain partial results on some subclasses of apple-free graphs where our results show that the MWS problem is solvable in polynomial time. The main tool is a combination of clique separators with modular decomposition. Our algorithms are robust in the sense that there is no need to recognize whether the input graph is in the given graph class; the algorithm either solves the MWS problem correctly or detects that the input graph is not in the given class.     

On mapping processes to processors in distributed systems

This paper is concerned with the implementation of parallel programs on networks of processors. In particular, we study the use of the network augmenting approach as an implementation tool. According to this approach, the capabilities of a given network of processors can be increased by adding some auxiliary links among the processors. We prove that the minimum set of edges needed to augment a line-like network so that it can accommodate a parallel program is determined by an optimal path cover of the graph representation of the program. Anoptimal path cover of a simple graphG is a set of vertex-disjoint paths that cover all the vertices ofG and has the maximum possible number of edges. We present a linear time optimal path covering algorithm for a class of sparse graphs. This algorithm is of special interest since the optimal path covering problem is NP-complete for general graphs. Our results suggest that a cover and augment scheme can be used for optimal implementation of parallel programs in line-like networks.A preliminary version of this paper was presented at the 6th IEEE Conference on Computer Communications (INFOCOM '87).This reseach is supported in part by National Semiconductor (Israel), Ltd.     

基于局部有限搜索的无向图近似最大团快速求解算法

无向图最大团求解是一个著名的NP-完全问题,解决该问题的经典算法基本上都采用完全精确搜索策略。鉴于NP-完全问题本身所固有的复杂性,这些算法或许仅适用于某些特殊的小规模图,对于具有大规模顶点和边的复杂图还是显得无力,难以适用。针对完全精确搜索策略下的无向图最大团求解算法的大部分时间都用于对图进行额外而无效的查找的问题,采用分划递归技术将图划分为邻接子图和悬挂子图,然后对邻接子图进行递归求解,而对悬挂子图则通过设置搜索范围控制函数进行局部有限搜索。在DIMACS数据集上将所提算法与当前主要的最大团求解算法进行对比实验,结果表明,文中提出的局部有限搜索求解策略能在75%的基准数据上获得最大团,剩下不能得到最大团的数据实际上也可以获得接近于最大团的近似最大团,但算法的平均求解时间仅为目前最大团精确求解算法的20%左右。因此,在很多最大团非精确要求的场景中,所提算法具有极高的应用价值。     

Vertex covering by paths on trees with its applications in machine translation

Given a tree and a set of paths in the tree, the problem of finding a minimum number of paths from the given path set to cover all the vertices in the tree is investigated in the paper. To distinguish from the classical path cover problem, such an optimization problem is referred to as vertex covering by paths. The problem and its edge variant, edge covering by paths, find applications in machine translation. Complexity and algorithmic results are presented for the problem and its edge variant.     

A population-based iterated greedy algorithm for the minimum weight vertex cover problem

Given an undirected, vertex-weighted graph, the goal of the minimum weight vertex cover problem is to find a subset of the vertices of the graph such that the subset is a vertex cover and the sum of the weights of its vertices is minimal. This problem is known to be NP-hard and no efficient algorithm is known to solve it to optimality. Therefore, most existing techniques are based on heuristics for providing approximate solutions in a reasonable computation time.Population-based search approaches have shown to be effective for solving a multitude of combinatorial optimization problems. Their advantage can be identified as their ability to find areas of the space containing high quality solutions. This paper proposes a simple and efficient population-based iterated greedy algorithm for tackling the minimum weight vertex cover problem. At each iteration, a population of solutions is established and refined using a fast randomized iterated greedy heuristic based on successive phases of destruction and reconstruction. An extensive experimental evaluation on a commonly used set of benchmark instances shows that our algorithm outperforms current state-of-the-art approaches.     

A Generalization of AT-Free Graphs and a Generic Algorithm for Solving Triangulation Problems

A subset A of the vertices of a graph G is an asteroidal set if for each vertex a ∈ A a connected component of G-N[a] exists containing A\backslash{a} . An asteroidal set of cardinality three is called asteriodal triple and graphs without an asteriodal triple are called AT-free . The maximum cardinality of an asteroidal set of G , denoted by \an(G) , is said to be the asteriodal number of G . We present a scheme for designing algorithms for triangulation problems on graphs. As a consequence, we obtain algorithms to compute graph parameters such as treewidth, minimum fill-in and vertex ranking number. The running time of these algorithms is a polynomial (of degree asteriodal number plus a small constant) in the number of vertices and the number of minimal separators of the input graph.     

A hybrid algorithmic model for the minimum weight dominating set problem

Iterated greedy algorithms belong to the class of stochastic local search methods. They are based on the simple and effective principle of generating a sequence of solutions by iterating over a constructive greedy heuristic using destruction and construction phases. This paper, first, presents an efficient randomized iterated greedy approach for the minimum weight dominating set problem, where—given a vertex-weighted graph—the goal is to identify a subset of the graphs’ vertices with minimum total weight such that each vertex of the graph is either in the subset or has a neighbor in the subset. Our proposed approach works on a population of solutions rather than on a single one. Moreover, it is based on a fast randomized construction procedure making use of two different greedy heuristics. Secondly, we present a hybrid algorithmic model in which the proposed iterated greedy algorithm is combined with the mathematical programming solver CPLEX. In particular, we improve the best solution provided by the iterated greedy algorithm with the solution polishing feature of CPLEX. The simulation results obtained on a widely used set of benchmark instances shows that our proposed algorithms outperform current state-of-the-art approaches.     

Polyhedral Characteristics of Balanced and Unbalanced Bipartite Subgraph Problems

We study the polyhedral properties of three problems of constructing an optimal complete bipartite subgraph (a biclique) in a bipartite graph. In the first problem, we consider a balanced biclique with the same number of vertices in both parts and arbitrary edge weights. In the other two problems we are dealing with unbalanced subgraphs of maximum and minimum weight with non-negative edges. All three problems are established to be NP-hard. We study the polytopes and the cone decompositions of these problems and their 1-skeletons. We describe the adjacency criterion in the 1-skeleton of the polytope of the balanced complete bipartite subgraph problem. The clique number of the 1-skeleton is estimated from below by a superpolynomial function. For both unbalanced biclique problems we establish the superpolynomial lower bounds on the clique numbers of the graphs of nonnegative cone decompositions. These values characterize the time complexity in a broad class of algorithms based on linear comparisons.     

On the k-path cover problem for cacti

In this paper we investigate the k-path cover problem for graphs, which is to find the minimum number of vertex disjoint k-paths that cover all the vertices of a graph. The k-path cover problem for general graphs is NP-complete. Though notable applications of this problem to database design, network, VLSI design, ring protocols, and code optimization, efficient algorithms are known for only few special classes of graphs. In order to solve this problem for cacti, i.e., graphs where no edge lies on more than one cycle, we introduce the so-called Steiner version of the k-path cover problem, and develop an efficient algorithm for the Steiner k-path cover problem for cacti, which finds an optimal k-path cover for a given cactus in polynomial time.