Approximation Algorithms for Maximum Independent Set Problems and Fractional Coloring Problems on Unit Disk Graphs

Abstract

Unit disk graphs are the intersection graphs of equal sized circles in the plane.

In this paper, we consider the maximum independent set problems on unit disk graphs. When the given unit disk graph is defined on a slab whose width is k, we propose an algorithm for finding a maximum independent set in $\mathrm{O}(n^{4\lceil 2k/ \sqrt{3}\rceil})$ time where n denotes the number of vertices. We also propose a (1 – 1/r)-approximation algorithm for the maximum independent set problems on a (general) unit disk graph whose time complexity is bounded by $\mathrm{O}(rn^{4\lceil 2(r-1)/ \sqrt{3}\rceil})$.

We also propose an algorithm for fractional coloring problems on unit disk graphs. The fractional coloring problem is a continuous version of the ordinary (vertex) coloring problem. Our approach for the independent set problem implies a strongly polynomial time algorithm for the fractional coloring problem on unit disk graphs defined on a fixed width slab. We also propose a strongly polynomial time 2-approximation algorithm for fractional coloring problem on a (general) unit disk graph.