NC algorithms for partitioning planar graphs into induced forests and approximating NP-hard problems

Abstract

It is well known that the vertex set of every planar graph can be partitioned into three subsets each of which induces a forest. Previously, there has been no NC algorithm for computing such a partition. In this paper, we design an optimal NC algorithm for computing such a partition for a given planar graph. It runs in O(log n log* n) time using O(n/(log n log* n)) processors on an EREW PRAM. This algorithm implies optimal NC approximation algorithms for many NP-hard maximum induced subgraph problems on planar graphs with a performance ratio of 3. We also present optimal NC algorithms for partitioning the vertex set of a given K4-free or K2,3-free graph into two subsets each of which induces a forest. As consequences, we obtain optimal NC algorithms for 4-coloring K4-free or K2,3-free graphs which are previously unknown to our knowledge.