Approximating Spanning Trees with Inner Nodes Cost

Given a graph G=(V,E) with node weight w:V→R+ and a subset S⊆V, find a minimum total weight tree interconnecting all nodes in S. This is the node-weighted Steiner tree problem which will be studied in this paper. In general, this problem is NP-hard and cannot be approximated by a polynomial time algorithm with performance ratio aln n for any 0<a<1 unless NP⊆DTIME(nO(log n)), where n is the number of nodes in s. In this paper, we are the first to show that even though for unit disk graphs, the problem is still NP-hard and it has a polynomial time constant approximation. We present a 2.5ρ-approximation where ρ is the best known performance ratio for polynomial time approximation of classical Steiner minimum tree problem in graphs. As a corollary, we obtain that there is a polynomial time (9.875+e)-approximation algorithm for minimum weight connected dominating set in unit disk graphs, and also there is a polynomial time (4.875+e)-approximation algorithm for minimum weight connected vertex cover in unit disk graphs.

With the advent of mobile and internet technology, social media has experienced a tremendous growth over the past few years. Social networking site such as Twitter provides the platform for content generation, information dissemination and communications. Summarizing the textual contents or extracting the influential words is a challenging task. Therefore an unsupervised graph based approach is proposed for automatic Keyword Extraction from the collection of Tweets using Node and Edge Weight (KETNEW). The node weight and edge weight depends on various parameters such as tweet frequency, position of node, clustering coefficient, co-occurrence frequency and shared neighbors. The important aspect of our approach is that it is a hybrid approach i.e. a graph based approach (structural approach) which depends upon statistical as well as linguistic features. Apart from this, it also depends on the position of the word in the text. These numerous features made it perform better than existing techniques.

Software fault localization technology aims to identify suspicious statements that cause software failures, which is crucial for ensuring software quality. Spectrum-based software fault location (SBFL) technology calculates the suspiciousness of each statement by analyzing the correlation between statement coverage information and execution results in test cases. SBFL has attracted increasing attention from scholars due to its high efficiency and scalability. However, existing SBFL studies have shown that a large number of statements share the same suspiciousness, which hinders software debuggers from quickly identifying the location of faulty statements. To address this challenge, we propose an SBFL model based on weighted association rule mining and complex networks: FL-WARMCN. The algorithm first uses Jaccard to measure the distance between passing and failing test cases, and applies it as the weight of passing test cases. Next, FL-WARMCN calculates the initial suspiciousness of each statement based on the program spectrum data. Then, the FL-WARMCN model utilizes a weighted association rule mining algorithm to obtain the correlation relationships between statements and models the network based on this. In the network, the suspiciousness of statements is used as node weights, and the correlation between statements is used as edge weights. We chose the eigenvector centrality that takes into account the degree centrality of statements and the importance of neighboring statements to calculate the importance of each statement, and used it as a weight to incorporate into the weighted suspiciousness calculation of the statement. Finally, we applied the FL-WARMCN model for experimental validation on the Defects4J dataset. The results showed that the model was significantly superior to other baselines. In addition, we analyzed the impact of different node and edge weights on model performance.

An approximation algorithm is suggested for the problem of finding a d-regular spanning connected subgraph of maximum weight in a complete undirected weighted n-vertex graph. Probabilistic analysis of the algorithm is carried out for the problem with random input data (some weights of edges) in the case of a uniform distribution of the weights of edges and in the case of a minorized type distribution. It is shown that the algorithm finds an asymptotically optimal solution with time complexity O(n 2) when d = o(n). For the minimization version of the problem, an additional restriction on the dispersion of weights of the graph edges is added to the condition of the asymptotical optimality of the modified algorithm.

In many domains it is necessary to generate surrogate networks, e.g., for hypothesis testing of different properties of a network. Generating surrogate networks typically requires that different properties of the network are preserved, e.g., edges may not be added or deleted and edge weights may be restricted to certain intervals. In this paper we present an efficient property-preserving Markov chain Monte Carlo method termed CycleSampler for generating surrogate networks in which (1) edge weights are constrained to intervals and vertex strengths are preserved exactly, and (2) edge and vertex strengths are both constrained to intervals. These two types of constraints cover a wide variety of practical use cases. The method is applicable to both undirected and directed graphs. We empirically demonstrate the efficiency of the CycleSampler method on real-world data sets. We provide an implementation of CycleSampler in R, with parts implemented in C.

In the context of generating efficient, contention free schedules for inter-node communication through a switch fabric in cluster computing or data center type environments, all-to-all scheduling with equal sized data transfer requests has been studied in the literature [1, 3, 4]. In this paper, we propose a communication scheduling module (CSM) towards generating contention free communication schedules for many-to-many communication with arbitrary sized data. Towards this end, we propose three approximation algorithms - PST, LDT and SDT. From time to time, the CSM first generates a bipartite graph from the set of received requests, then determines which of these three algorithms gives the best approximation factor on this graph and finally executes that algorithm to generate a contention free schedule. Algorithm PST has a worst case run time of O(max (Δ|E|, |E| log (|E|))) and guarantees an approximation factor of 2H2Δ-1, where |E| is the number of edges in the bipartite graph, Δ is the maximum node degree of the bipartite graph and H2Δ-1 is the (2Δ - 1)- th harmonic number. LDT runs in O(|E|2) and has an approximation factor of 2(1 + τ), where τ is a constant defined as a guard band or pause time to eliminate the possibility of contention (in an apparently contention free schedule) caused by system jitter and synchronization inaccuracies between the nodes. SDT gives an approximation factor of 4 log (wmax) and has a worst case run time of O(Δ|E| log (wmax)), where wmax represents the longest communication time in a set of received requests.

This paper focuses on Dynamic Implementation using Array and Linked list. In the data structure concept, the list plays a major role in the allocation of data where it is a general overview of creating a node in a linked list using an array with the initialization of size. The node contains data and is interconnected to the next node in the sequence via their links which perform the search operation where the space and time complexity are not wasted in these analyses. Linked lists are a great way to store a theoretically infinite amount of data with a small and versatile amount of code. The amount of data stored increases linearly with the number of nodes in the list (space-o(n)) where linked lists are more efficient than arrays. It is observed that considering the time complexity when there is a variation in data set linked array is almost reaches close to array whenever dataset is high also linked array has better time complexity than linked list when the searching element is in middle and last. Considering the space complexity linked array is almost 2 times and 4 times less space is needed than array and linked list respectively.

The dominating set problem in graphs asks for a minimum size subset of vertices with the following property: each vertex is required to be either in the dominating set, or adjacent to some vertex in the dominating set. We focus on the related question of finding a connected dominating set of minimum size, where the graph induced by vertices in the dominating set is required to be connected as well. This problem arises in network testing, as well as in wireless communication. Two polynomial time algorithms that achieve approximation factors of 2H(Δ)+2 and H(Δ)+2 are presented, where Δ is the maximum degree and H is the harmonic function. This question also arises in relation to the traveling tourist problem, where one is looking for the shortest tour such that each vertex is either visited or has at least one of its neighbors visited. We also consider a generalization of the problem to the weighted case, and give an algorithm with an approximation factor of (cn+1) \ln n where cn ln k is the approximation factor for the node weighted Steiner tree problem (currently cn = 1.6103 ). We also consider the more general problem of finding a connected dominating set of a specified subset of vertices and provide a polynomial time algorithm with a (c+1) H(Δ) +c-1 approximation factor, where c is the Steiner approximation ratio for graphs (currently c = 1.644 ).

Approximability of constrained LCS

We consider the problem of finding a local optimum for the Max-Cut problem with FLIP-neighborhood, in which exactly one node changes the partition. Schaffer and Yannakakis (SICOMP, 1991) showed PLS-completeness of this problem on graphs with unbounded degree. On the other side, Poljak (SICOMP, 1995) showed that in cubic graphs every FLIP local search takes O(n2) steps, where n is the number of nodes. Due to the huge gap between degree three and unbounded degree, Ackermann, Roglin, and Vocking (JACM, 2008) asked for the smallest d such that on graphs with maximum degree d the local MAX-CUT problem with FLIP-neighborhood is PLS-complete. In this paper, we prove that the computation of a local optimum on graphs with maximum degree five is PLS-complete. Thus, we solve the problem posed by Ackermann et al. almost completely by showing that d is either four or five (unless PLS ⊆ P). On the other side, we also prove that on graphs with degree O(log n) every FLIP local search has probably polynomial smoothed complexity. Roughly speaking, for any instance, in which the edge weights are perturbated by a (Gaussian) random noise with variance &sigma2, every FLIP local search terminates in time polynomial in n and σ-1, with probability 1-n-Ω(1). Putting both results together, we may conclude that although local MAX-CUT is likely to be hard on graphs with bounded degree, it can be solved in polynomial time for slightly perturbated instances with high probability.

Limit laws for the number of triangles in the generalized random graphs with random node weights

Gathering a group of mobile agents is a fundamental task in the field of distributed and mobile systems. This can be made drastically more difficult to achieve when some agents are subject to faults, especially the Byzantine ones that are known as being the worst faults to handle. In this paper we study, from a deterministic point of view, the task of Byzantine gathering in a network modeled as a graph. In other words, despite the presence of Byzantine agents, all the other (good) agents, starting from {possibly} different nodes and applying the same deterministic algorithm, have to meet at the same node in finite time and stop moving. An adversary chooses the initial nodes of the agents (the number of agents may be larger than the number of nodes) and assigns a different positive integer (called label) to each of them. Initially, each agent knows its label. The agents move in synchronous rounds and can communicate with each other only when located at the same node. Within the team, f of the agents are Byzantine. A Byzantine agent acts in an unpredictable and arbitrary way. For example, it can choose an arbitrary port when it moves, can convey arbitrary information to other agents and can change its label in every round, in particular by forging the label of another agent or by creating a completely new one. Besides its label, which corresponds to a local knowledge, an agent is assigned some global knowledge denoted by GK that is common to all agents. In literature, the Byzantine gathering problem has been analyzed in arbitrary n-node graphs by considering the scenario when GK=(n,f) and the scenario when GK=f. In the first (resp. second) scenario, it has been shown that the minimum number of good agents guaranteeing deterministic gathering of all of them is f+1 (resp. f+2). However, for both these scenarios, all the existing deterministic algorithms, whether or not they are optimal in terms of required number of good agents, have the major disadvantage of having a time complexity that is exponential in n and L, where L is the value of the largest label belonging to a good agent. In this paper, we seek to design a deterministic solution for Byzantine gathering that makes a concession on the proportion of Byzantine agents within the team, but that offers a significantly lower complexity. We also seek to use a global knowledge whose the length of the binary representation (that we also call size) is small. In this respect, assuming that the agents are in a strong team i.e., a team in which the number of good agents is at least some prescribed value that is quadratic in f, we give positive and negative results. On the positive side, we show an algorithm that solves Byzantine gathering with all strong teams in all graphs of size at most n, for any integers n and f, in a time polynomial in n and the length |l_{min}| of the binary representation of the smallest label of a good agent. The algorithm works using a global knowledge of size O(log log log n), which is of optimal order of magnitude in our context to reach a time complexity that is polynomial in n and |l_{min}|. Indeed, on the negative side, we show that there is no deterministic algorithm solving Byzantine gathering with all strong teams, in all graphs of size at most n, in a time polynomial in n and |l_{min}| and using a global knowledge of size o(log log log n).

Routing with multiple quality-of-services constraints: An approximation perspective

Motivated by the increasing need for fast distributed processing of large-scale graphs such as the Web graph and various social networks, we study a number of fundamental graph problems in the message-passing model, where we have k machines that jointly perform computation on an arbitrary n-node (typically, n > k) input graph. The graph is assumed to be randomly partitioned among the k ≥ 2 machines (a common implementation in many real world systems). The communication is point-to-point, and the goal is to minimize the time complexity i.e., the number of communication rounds, of solving various fundamental graph problems.We present lower bounds that quantify the fundamental time limitations of distributively solving graph problems. We first show a lower bound of Ω(n/k) rounds for computing a spanning tree (ST) of the input graph. This result also implies the same bound for other fundamental problems such as computing a minimum spanning tree (MST), breadth-first tree (BFS), and shortest paths tree (SPT). We also show an Ω(n/k2) lower bound for connectivity ST verification and other related problems. Our lower bounds develop and use new bounds in random-partition communication complexity.To complement our lower bounds, we also give algorithms for various fundamental graph problems, e.g., PageRank, MST, connectivity, ST verification, shortest paths, cuts, spanners, covering problems, densest subgraph, subgraph isomorphism, finding triangles, etc. We show that problems such as PageRank, MST, connectivity, and graph covering can be solved in O(n/k) time (the notation O hides polylog(n) factors and an additive polylog(n) term); this shows that one can achieve almost linear (in k) speedup, whereas for shortest paths, we present algorithms that run in O(n/Ok) time (for (1 + e)-factor approximation) and in O(n/k) time (for O(log n)-factor approximation) respectively.Our results are a step towards understanding the complexity of distributively solving large-scale graph problems.

SimRank has been proposed to rank web documents based on a graph model on hyperlinks. The existing techniques for conducting SimRank computation adopt an iteration computation paradigm. The most efficient technique has the time complexity O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ) with the space requirement O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) in the worst case for each iteration where n is the number of nodes (web documents). In this paper, we propose novel optimization techniques such that each iteration takes the time O(min{n · m,n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sup> }) and requires space O (n + m) where m is the number of edges in a web-graph model and r ≤ log <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> 7. We also show that our algorithm accelerates the convergence rate of the existing techniques. Moreover, our algorithm not only reduces the time and space complexity of the existing techniques but is also I/O efficient. We conduct extensive experiments on both synthetic and real data sets to demonstrate the efficiency and effectiveness of our iteration techniques.