Approximation algorithms for maximum independent set of a unit disk graph

We propose a 2-approximation algorithm for the maximum independent set problem for a unit disk graph. The time and space complexities are O (n³) and O (n²), respectively. For a penny graph, our proposed 2-approximation algorithm works in O (n log n) time using O (n) space. We also propose a polynomial-time approximation scheme (PTAS) for the maximum independent set problem for a unit disk graph. Given an integer k > 1, it produces a solution of size 1/(1+1/k)²|OPT| in O (k⁴n^{σk log k} + n log n) time and O (n + k log k) space, where OPT is the subset of disks in an optimal solution and σ_k ≤ 7k/3 + 2. For a penny graph, the proposed PTAS produces a solution of size 1/(1+1/k)|OPT | in O (2^2σk nk + n log n) time using O (2^σk + n) space.