Approximate Triclique Coloring for Register Allocation

A graph is said to be a triclique if its vertex set can be partitioned into three cliques i.e if its complement is tripartite. We prove that the coloring problem is NP-complete even when restricted to tricliques whose complements are planar tripartite line graphs. We then proceed to give a factor 1.5-approximation algorithm for the coloring problem on tricliques. In fact, our algorithm works for all graphs G which satisfy (G) 3 and actually guarantees an approximation ratio that is at most 3 4 + n 4(G) . We then indicate how this problem may be useful in the context of the register allocation problem that arises in compiler design. Keywords : Triclique, chromatic number, register allocation graph(RAG), control-flow graph, approximate algorithm. 1 Preliminaries For a graph G = (V; E), a set X V is an independent set if no two vertices in X are adjacent in G and it is a clique if every pair of vertices in X are adjacent in G. A set X V is a vertex-cover if every edge in E ha..