On the Independent Set and Common Subgraph Problems in Random Graphs

TL;DR

This paper introduces efficient algorithms for maximum independent set and common subgraph problems in random graphs, demonstrating their expected computational complexity and approximation ratios, with implications for fixed parameter tractability.

Contribution

It presents novel algorithms with proven expected runtimes for maximum independent set and common subgraph problems in random graphs, including approximation and fixed parameter tractability results.

Findings

Maximum independent set in random graphs can be computed in expected time 2^{O(log^2 n)}.

Largest common subgraph in two random graphs can be found in expected time 2^{O(n^{1/2} log^{5/3} n)}.

Maximum independent set can be approximated within a ratio of 2n / 2^{√log n} in polynomial time.

Abstract

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.