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.

Random graphs

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Computers And Intractability A Guide To The Theory Of Np Completeness

Previous research on dimensionality reduction has shown that global and local information is both important for capturing the crucial features of data sets. In this paper, we develop a new approach that can effectively retain both global and local features of a dataset for supervised dimensionality reduction. A new quadratic measure is developed to accurately describe the local features of a dataset and a composite objective is used to balance the global and local features in the dimensionality reduction of labeled datasets. The directions along which the composite objective can be maximized are computed to reduce the dimensionality of the dataset. Testing results on benchmark data sets show that this approach is able to efficiently capture features crucial for classification and generate more accurate results than a few other approaches for dimensionality reduction.

A new algorithm for preserving global and local structures in supervised dimensionality reduction

In this paper, we study the {\sc Dominating Set} problem in random graphs. In a random graph, each pair of vertices are joined by an edge with a probability of $p$, where $p$ is a positive constant less than $1$. We show that, given a random graph in $n$ vertices, a minimum dominating set in the graph can be computed in expected $2^{O(\log_{2}^{2}{n})}$ time. For the parameterized dominating set problem, we show that it cannot be solved in expected $O(f(k)n^{c})$ time unless the minimum dominating set problem can be approximated within a ratio of $o(\log_{2}n)$ in expected polynomial time, where $f(k)$ is a function of the parameter $k$ and $c$ is a constant independent of $n$ and $k$. In addition, we show that the parameterized dominating set problem can be solved in expected $O(f(k)n^{c})$ time when the probability $p$ depends on $n$ and equals to $\frac{1}{g(n)}$, where $g(n)< n$ is a monotonously increasing function of $n$ and its value approaches infinity when $n$ approaches infinity.

pdf/on-the-dominating-set-problem-in-random-graphs-4m3dfvnnip.pdf

On the Dominating Set Problem in Random Graphs

Dimensionality reduction is a problem of fundamental importance in both machine learning and data mining. In this paper, we develop a new approach that can process labeled datasets and accurately reduce their dimensionalities. The approach is based on a new objective that contains information from both the global and local structures of a data set. An iterative approach is used to compute the directions along which the objective can be optimized. Experiments on benchmark data sets show that both the global and local information in a dataset can be effectively captured by this approach and it is thus able to provide more accurate results for dimensionality reduction than some existing approaches.

http://ieeexplore.ieee.org/iel7/7580914/7589673/07589750.pdf

Supervised dimensionality reduction that preserves both global and local information

In this paper, we develop efficient exact and approximate algorithms for computing a maximum independent set in random graphs. In a random graph G, each pair of vertices are joined by an edge with a probability p, where p is a constant between 0 and 1. We show that a maximum independent set in a random graph that contains n vertices can be computed in expected computation time . In addition, we show that, with high probability, the parameterized independent set problem is fixed parameter tractable in random graphs and the maximum independent set in a random graph in n vertices can be approximated within a ratio of in expected polynomial time.

/pdf/on-the-independent-set-problem-in-random-graphs-2nqdwnxyz8.pdf

On the independent set problem in random graphs

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On the independent set problem in random graphs