On the Independent Set and Common Subgraph Problems in Random Graphs

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 $2^{O(\log_{2}^{2}{n})}$. Using techniques based on enumeration, we develop an algorithm that can find a largest common subgraph in two random graphs in $n$ and $m$ vertices ($m \leq n$) in expected computation time $2^{O(n^{\frac{1}{2}}\log_{2}^{\frac{5}{3}}{n})}$. 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 $\frac{2n}{2^{\sqrt{\log_{2}{n}}}}$ in expected polynomial time.