Approximation Algorithms for Graph Partition into Bounded Independent Sets

The partition problem of a given graph into three independent sets of minimizing the maximum one is studied in this paper. This problem is NP-hard, even restricted to bipartite graphs. First, a simple $\frac{3}{2}$-approximation algorithm for any 2-colorable graph is presented. An improved $\frac{7}{5}$-approximation algorithm is then designed for a tree. The theoretical proof of the improved algorithm performance ratio is constructive, thus providing an explicit partition approach for each case according to the cardinality of two color classes.