Vertex Cover Problem—Revised Approximation Algorithm

Abstract: We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Abstract: Social networks, though started as a software tool enabling people to connect with each other, have emerged in recent times as platforms for businesses, individuals and government agencies to conduct a number of activities ranging from marketing to emergency situation management. As a result, a large number of social network analytics tools have been developed for a variety of applications. A snapshot of social networks at any particular time, called a social graph, represents the connectivity of nodes and potentially the flow of information amongst the nodes (or vertices) in the graph. Understanding the flow of information in a social graph plays an important role in social network applications. Two specific problems related to information flow have implications in many social network applications: (a) finding a minimum set of nodes one has to know to recover the whole graph (also known as the vertex cover problem) and (b) determining the minimum set of nodes one required to reach all nodes in the graph within a specific number of hops (we refer this as the vertex reach problem). Finding an optimal solution to these problems is NP-Hard. In this paper, we propose approximation based approaches and show that our approaches outperform existing approaches using both a theoretical analysis and experimental results.

Abstract: he problem of finding Minimum Vertex Cover for graph belongs to the class of NP Complete and plays a key role in Computer Science Theory. The problems which belong to NP Complete set are not solvable in polynomial time in any known way. Since finding Minimum Vertex Cover (MVC) for a graph belongs to NP Complete class; so we are dubious to solve it in any polynomial time algorithm. Such problems are solved by algorithms which promise to give near optimum solution. In this paper we have analyzed and scrutinized such algorithms like greedy algorithm, approximation algorithm, simple genetic algorithm (GA), primal-dual based algorithm (PDB), Alom's algorithm etc. on random directed and undirected graphs and found that all the algorithms give near optimum solution with a negligible performance difference. It was also observed that out of all the above said algorithms Alom's Algorithm is more effective in finding MVC for undirected graphs and for weighted graphs, superior performance is attained by primal-dual based approach.

Abstract: The minimum vertex cover problem (MVCP) is a well-known NP complete combinatorial optimization problem. The aim of this paper is to present an approximation algorithm for minimum vertex cover problem (MVCP). The algorithm construction is based on articulation points/cut vertices and leaf vertices/pendant vertices. The proposed algorithm assures the near optimal or optimal solution for a given graph and can be solved in polynomial time. A numerical example is illustrated to describe the proposed algorithm. Comparative results show that the proposed algorithm is very competitive compared with other existing algorithms.

Abstract: Routing in delay tolerant networks (DTNs) is a challenging problem in networking research. Existing DTN routing solutions have used many approaches to increase the success rate of message delivery, such as meeting probabilities between nodes, packet replication and flooding. One important feature of these protocols is using local connection information to find the “best” path with high likelihood to deliver a packet. In this paper, we propose a new routing protocol called ANBR (Articulation Node Based Routing). From a global view, a general disconnected network can have many small instantaneously clustered mobile nodes. Mobility allows nodes carrying messages to deliver them to other clusters. Selecting appropriate nodes to carry and deliver messages becomes important in order to reduce message delay and overhead. The proposed ANBR tackles this issue by utilizing articulation nodes among a local sub-graph formed by including all neighbors of two “meeting” nodes. Articulation nodes are the articulation points or cut vertices of this local sub-graph, and by definition are the nodes, whose removal will disconnect the graph. Thus, these articulation nodes are more likely to be able to deliver messages outside the local cluster. Packets will be buffered in these nodes and forwarded to other articulation nodes when they meet. The process repeats until messages reach their destinations. We evaluate our algorithm by using real world data from the MIT reality mining project. The simulation results show that ANBR algorithm performs better than related protocols in terms of delivery rate and efficiency.

Abstract: We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Abstract: We consider a binary integer programming formulation (VP) for the weighted vertex packing problem in a simple graph. A sufficient “local” optimality condition for (VP) is given and this result is used to derive relations between (VP) and the linear program (VLP) obtained by deleting the integrality restrictions in (VP). Our most striking result is that those variables which assume binary values in an optimum (VLP) solution retain the same values in an optimum (VP) solution. This result is of interest because variables are (0, 1/2, 1). valued in basic feasible solutions to (VLP) and (VLP) can be solved by a “good” algorithm. This relationship and other optimality conditions are incorporated into an implicit enumeration algorithm for solving (VP). Some computational experience is reported.

Abstract: We propose a heuristic that delivers in $O(n^3 )$ steps a solution for the set covering problem the value of which does not exceed the maximum number of sets covering an element times the optimal value.

Abstract: A local-ratio theorem for approximating the weighted vertex cover problem is presented. It consists of reducing the weights of vertices in certain subgraphs and has the effect of local-approximation. Putting together the Nemhauser-Trotter local optimization algorithm and the local-ratio theorem yields several new approximation techniques which improve known results from time complexity, simplicity and performance-ratio point of view. The main approximation algorithm guarantees a ratio of where K is the smallest integer s.t. † This is an improvement over the currently known ratios, especially for a “practical” number of vertices (e.g. for graphs which have less than 2400, 60000, 10 12 vertices the ratio is bounded by 1.75, 1.8, 1.9 respectively).