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 $\frac{\mathrm{3}}{\mathrm{2}}$-approximation algorithm for any $\mathrm{2}$-colorable graph is presented. An improved $\frac{\mathrm{7}}{\mathrm{5}}$-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.